Mary Pyo

Proving Triangles Congruent

Slide Duration:

Section 1: Tools of Geometry
Coordinate Plane

16m 41s

Intro
0:00
The Coordinate System
0:12
Coordinate Plane: X-axis and Y-axis
0:15
1:02
Origin
2:00
Ordered Pair
2:17
Coordinate Plane
2:59
Example: Writing Coordinates
3:01
Coordinate Plane, cont.
4:15
Example: Graphing & Coordinate Plane
4:17
Collinear
5:58
Extra Example 1: Writing Coordinates & Quadrants
7:34
Extra Example 2: Quadrants
8:52
Extra Example 3: Graphing & Coordinate Plane
10:58
Extra Example 4: Collinear
12:50
Points, Lines and Planes

17m 17s

Intro
0:00
Points
0:07
Definition and Example of Points
0:09
Lines
0:50
Definition and Example of Lines
0:51
Planes
2:59
Definition and Example of Planes
3:00
Drawing and Labeling
4:40
Example 1: Drawing and Labeling
4:41
Example 2: Drawing and Labeling
5:54
Example 3: Drawing and Labeling
6:41
Example 4: Drawing and Labeling
8:23
Extra Example 1: Points, Lines and Planes
10:19
Extra Example 2: Naming Figures
11:16
Extra Example 3: Points, Lines and Planes
12:35
Extra Example 4: Draw and Label
14:44
Measuring Segments

31m 31s

Intro
0:00
Segments
0:06
Examples of Segments
0:08
Ruler Postulate
1:30
Ruler Postulate
1:31
5:02
Example and Definition of Segment Addition Postulate
5:03
8:01
Example 1: Segment Addition Postulate
8:04
Example 2: Segment Addition Postulate
11:15
Pythagorean Theorem
12:36
Definition of Pythagorean Theorem
12:37
Pythagorean Theorem, cont.
15:49
Example: Pythagorean Theorem
15:50
Distance Formula
16:48
Example and Definition of Distance Formula
16:49
Extra Example 1: Find Each Measure
20:32
Extra Example 2: Find the Missing Measure
22:11
Extra Example 3: Find the Distance Between the Two Points
25:36
Extra Example 4: Pythagorean Theorem
29:33
Midpoints and Segment Congruence

42m 26s

Intro
0:00
Definition of Midpoint
0:07
Midpoint
0:10
Midpoint Formulas
1:30
Midpoint Formula: On a Number Line
1:45
Midpoint Formula: In a Coordinate Plane
2:50
Midpoint
4:40
Example: Midpoint on a Number Line
4:43
Midpoint
6:05
Example: Midpoint in a Coordinate Plane
6:06
Midpoint
8:28
Example 1
8:30
Example 2
13:01
Segment Bisector
15:14
Definition and Example of Segment Bisector
15:15
Proofs
17:27
Theorem
17:53
Proof
18:21
Midpoint Theorem
19:37
Example: Proof & Midpoint Theorem
19:38
Extra Example 1: Midpoint on a Number Line
23:44
Extra Example 2: Drawing Diagrams
26:25
Extra Example 3: Midpoint
29:14
Extra Example 4: Segment Bisector
33:21
Angles

42m 34s

Intro
0:00
Angles
0:05
Angle
0:07
Ray
0:23
Opposite Rays
2:09
Angles
3:22
Example: Naming Angle
3:23
Angles
6:39
Interior, Exterior, Angle
6:40
Measure and Degrees
7:38
Protractor Postulate
8:37
Example: Protractor Postulate
8:38
11:41
Example: Angle addition Postulate
11:42
Classifying Angles
14:10
Acute Angle
14:16
Right Angles
14:30
Obtuse Angle
14:41
Angle Bisector
15:02
Example: Angle Bisector
15:04
Angle Relationships
16:43
16:47
Vertical Angles
17:49
Linear Pair
19:40
Angle Relationships
20:31
Right Angles
20:32
Supplementary Angles
21:15
Complementary Angles
21:33
Extra Example 1: Angles
24:08
Extra Example 2: Angles
29:06
Extra Example 3: Angles
32:05
Extra Example 4 Angles
35:44
Section 2: Reasoning & Proof
Inductive Reasoning

19m

Intro
0:00
Inductive Reasoning
0:05
Conjecture
0:06
Inductive Reasoning
0:15
Examples
0:55
Example: Sequence
0:56
More Example: Sequence
2:00
Using Inductive Reasoning
2:50
Example: Conjecture
2:51
More Example: Conjecture
3:48
Counterexamples
4:56
Counterexample
4:58
Extra Example 1: Conjecture
6:59
Extra Example 2: Sequence and Pattern
10:20
Extra Example 3: Inductive Reasoning
12:46
Extra Example 4: Conjecture and Counterexample
15:17
Conditional Statements

42m 47s

Intro
0:00
If Then Statements
0:05
If Then Statements
0:06
Other Forms
2:29
Example: Without Then
2:40
Example: Using When
3:03
Example: Hypothesis
3:24
Identify the Hypothesis and Conclusion
3:52
Example 1: Hypothesis and Conclusion
3:58
Example 2: Hypothesis and Conclusion
4:31
Example 3: Hypothesis and Conclusion
5:38
Write in If Then Form
6:16
Example 1: Write in If Then Form
6:23
Example 2: Write in If Then Form
6:57
Example 3: Write in If Then Form
7:39
Other Statements
8:40
Other Statements
8:41
Converse Statements
9:18
Converse Statements
9:20
Converses and Counterexamples
11:04
Converses and Counterexamples
11:05
Example 1: Converses and Counterexamples
12:02
Example 2: Converses and Counterexamples
15:10
Example 3: Converses and Counterexamples
17:08
Inverse Statement
19:58
Definition and Example
19:59
Inverse Statement
21:46
Example 1: Inverse and Counterexample
21:47
Example 2: Inverse and Counterexample
23:34
Contrapositive Statement
25:20
Definition and Example
25:21
Contrapositive Statement
26:58
Example: Contrapositive Statement
27:00
Summary
29:03
Summary of Lesson
29:04
Extra Example 1: Hypothesis and Conclusion
32:20
Extra Example 2: If-Then Form
33:23
Extra Example 3: Converse, Inverse, and Contrapositive
34:54
Extra Example 4: Converse, Inverse, and Contrapositive
37:56
Point, Line, and Plane Postulates

17m 24s

Intro
0:00
What are Postulates?
0:09
Definition of Postulates
0:10
Postulates
1:22
Postulate 1: Two Points
1:23
Postulate 2: Three Points
2:02
Postulate 3: Line
2:45
Postulates, cont..
3:08
Postulate 4: Plane
3:09
Postulate 5: Two Points in a Plane
3:53
Postulates, cont..
4:46
Postulate 6: Two Lines Intersect
4:47
Postulate 7: Two Plane Intersect
5:28
Using the Postulates
6:34
Examples: True or False
6:35
Using the Postulates
10:18
Examples: True or False
10:19
Extra Example 1: Always, Sometimes, or Never
12:22
Extra Example 2: Always, Sometimes, or Never
13:15
Extra Example 3: Always, Sometimes, or Never
14:16
Extra Example 4: Always, Sometimes, or Never
15:03
Deductive Reasoning

36m 3s

Intro
0:00
Deductive Reasoning
0:06
Definition of Deductive Reasoning
0:07
Inductive vs. Deductive
2:51
Inductive Reasoning
2:52
Deductive reasoning
3:19
Law of Detachment
3:47
Law of Detachment
3:48
Examples of Law of Detachment
4:31
Law of Syllogism
7:32
Law of Syllogism
7:33
Example 1: Making a Conclusion
9:02
Example 2: Making a Conclusion
12:54
Using Laws of Logic
14:12
Example 1: Determine the Logic
14:42
Example 2: Determine the Logic
17:02
Using Laws of Logic, cont.
18:47
Example 3: Determine the Logic
19:03
Example 4: Determine the Logic
20:56
Extra Example 1: Determine the Conclusion and Law
22:12
Extra Example 2: Determine the Conclusion and Law
25:39
Extra Example 3: Determine the Logic and Law
29:50
Extra Example 4: Determine the Logic and Law
31:27
Proofs in Algebra: Properties of Equality

44m 31s

Intro
0:00
Properties of Equality
0:10
Addition Property of Equality
0:28
Subtraction Property of Equality
1:10
Multiplication Property of Equality
1:41
Division Property of Equality
1:55
Addition Property of Equality Using Angles
2:46
Properties of Equality, cont.
4:10
Reflexive Property of Equality
4:11
Symmetric Property of Equality
5:24
Transitive Property of Equality
6:10
Properties of Equality, cont.
7:04
Substitution Property of Equality
7:05
Distributive Property of Equality
8:34
Two Column Proof
9:40
Example: Two Column Proof
9:46
Proof Example 1
16:13
Proof Example 2
23:49
Proof Example 3
30:33
Extra Example 1: Name the Property of Equality
38:07
Extra Example 2: Name the Property of Equality
40:16
Extra Example 3: Name the Property of Equality
41:35
Extra Example 4: Name the Property of Equality
43:02
Proving Segment Relationship

41m 2s

Intro
0:00
Good Proofs
0:12
Five Essential Parts
0:13
Proof Reasons
1:38
Undefined
1:40
Definitions
2:06
Postulates
2:42
Previously Proven Theorems
3:24
Congruence of Segments
4:10
Theorem: Congruence of Segments
4:12
Proof Example
10:16
Proof: Congruence of Segments
10:17
Setting Up Proofs
19:13
Example: Two Segments with Equal Measures
19:15
Setting Up Proofs
21:48
Example: Vertical Angles are Congruent
21:50
Setting Up Proofs
23:59
Example: Segment of a Triangle
24:00
Extra Example 1: Congruence of Segments
27:03
Extra Example 2: Setting Up Proofs
28:50
Extra Example 3: Setting Up Proofs
30:55
Extra Example 4: Two-Column Proof
33:11
Proving Angle Relationships

33m 37s

Intro
0:00
Supplement Theorem
0:05
Supplementary Angles
0:06
Congruence of Angles
2:37
Proof: Congruence of Angles
2:38
Angle Theorems
6:54
Angle Theorem 1: Supplementary Angles
6:55
Angle Theorem 2: Complementary Angles
10:25
Angle Theorems
11:32
Angle Theorem 3: Right Angles
11:35
Angle Theorem 4: Vertical Angles
12:09
Angle Theorem 5: Perpendicular Lines
12:57
Using Angle Theorems
13:45
Example 1: Always, Sometimes, or Never
13:50
Example 2: Always, Sometimes, or Never
14:28
Example 3: Always, Sometimes, or Never
16:21
Extra Example 1: Always, Sometimes, or Never
16:53
Extra Example 2: Find the Measure of Each Angle
18:55
Extra Example 3: Find the Measure of Each Angle
25:03
Extra Example 4: Two-Column Proof
27:08
Section 3: Perpendicular & Parallel Lines
Parallel Lines and Transversals

37m 35s

Intro
0:00
Lines
0:06
Parallel Lines
0:09
Skew Lines
2:02
Transversal
3:42
Angles Formed by a Transversal
4:28
Interior Angles
5:53
Exterior Angles
6:09
Consecutive Interior Angles
7:04
Alternate Exterior Angles
9:47
Alternate Interior Angles
11:22
Corresponding Angles
12:27
Angles Formed by a Transversal
15:29
Relationship Between Angles
15:30
Extra Example 1: Intersecting, Parallel, or Skew
19:26
Extra Example 2: Draw a Diagram
21:37
Extra Example 3: Name the Figures
24:12
Extra Example 4: Angles Formed by a Transversal
28:38
Angles and Parallel Lines

41m 53s

Intro
0:00
Corresponding Angles Postulate
0:05
Corresponding Angles Postulate
0:06
Alternate Interior Angles Theorem
3:05
Alternate Interior Angles Theorem
3:07
Consecutive Interior Angles Theorem
5:16
Consecutive Interior Angles Theorem
5:17
Alternate Exterior Angles Theorem
6:42
Alternate Exterior Angles Theorem
6:43
Parallel Lines Cut by a Transversal
7:18
Example: Parallel Lines Cut by a Transversal
7:19
Perpendicular Transversal Theorem
14:54
Perpendicular Transversal Theorem
14:55
Extra Example 1: State the Postulate or Theorem
16:37
Extra Example 2: Find the Measure of the Numbered Angle
18:53
Extra Example 3: Find the Measure of Each Angle
25:13
Extra Example 4: Find the Values of x, y, and z
36:26
Slope of Lines

44m 6s

Intro
0:00
Definition of Slope
0:06
Slope Equation
0:13
Slope of a Line
3:45
Example: Find the Slope of a Line
3:47
Slope of a Line
8:38
More Example: Find the Slope of a Line
8:40
Slope Postulates
12:32
Proving Slope Postulates
12:33
Parallel or Perpendicular Lines
17:23
Example: Parallel or Perpendicular Lines
17:24
Using Slope Formula
20:02
Example: Using Slope Formula
20:03
Extra Example 1: Slope of a Line
25:10
Extra Example 2: Slope of a Line
26:31
Extra Example 3: Graph the Line
34:11
Extra Example 4: Using the Slope Formula
38:50
Proving Lines Parallel

25m 55s

Intro
0:00
Postulates
0:06
Postulate 1: Parallel Lines
0:21
Postulate 2: Parallel Lines
2:16
Parallel Postulate
3:28
Definition and Example of Parallel Postulate
3:29
Theorems
4:29
Theorem 1: Parallel Lines
4:40
Theorem 2: Parallel Lines
5:37
Theorems, cont.
6:10
Theorem 3: Parallel Lines
6:11
Extra Example 1: Determine Parallel Lines
6:56
Extra Example 2: Find the Value of x
11:42
Extra Example 3: Opposite Sides are Parallel
14:48
Extra Example 4: Proving Parallel Lines
20:42
Parallels and Distance

19m 48s

Intro
0:00
Distance Between a Points and Line
0:07
Definition and Example
0:08
Distance Between Parallel Lines
1:51
Definition and Example
1:52
Extra Example 1: Drawing a Segment to Represent Distance
3:02
Extra Example 2: Drawing a Segment to Represent Distance
4:27
Extra Example 3: Graph, Plot, and Construct a Perpendicular Segment
5:13
Extra Example 4: Distance Between Two Parallel Lines
15:37
Section 4: Congruent Triangles
Classifying Triangles

28m 43s

Intro
0:00
Triangles
0:09
Triangle: A Three-Sided Polygon
0:10
Sides
1:00
Vertices
1:22
Angles
1:56
Classifying Triangles by Angles
2:59
Acute Triangle
3:19
Obtuse Triangle
4:08
Right Triangle
4:44
Equiangular Triangle
5:38
Definition and Example of an Equiangular Triangle
5:39
Classifying Triangles by Sides
6:57
Scalene Triangle
7:17
Isosceles Triangle
7:57
Equilateral Triangle
8:12
Isosceles Triangle
8:58
Labeling Isosceles Triangle
9:00
Labeling Right Triangle
10:44
Isosceles Triangle
11:10
Example: Find x, AB, BC, and AC
11:11
Extra Example 1: Classify Each Triangle
13:45
Extra Example 2: Always, Sometimes, or Never
16:28
Extra Example 3: Find All the Sides of the Isosceles Triangle
20:29
Extra Example 4: Distance Formula and Triangle
22:29
Measuring Angles in Triangles

44m 43s

Intro
0:00
Angle Sum Theorem
0:09
Angle Sum Theorem for Triangle
0:11
Using Angle Sum Theorem
4:06
Find the Measure of the Missing Angle
4:07
Third Angle Theorem
4:58
Example: Third Angle Theorem
4:59
Exterior Angle Theorem
7:58
Example: Exterior Angle Theorem
8:00
Flow Proof of Exterior Angle Theorem
15:14
Flow Proof of Exterior Angle Theorem
15:17
Triangle Corollaries
27:21
Triangle Corollary 1
27:50
Triangle Corollary 2
30:42
Extra Example 1: Find the Value of x
32:55
Extra Example 2: Find the Value of x
34:20
Extra Example 3: Find the Measure of the Angle
35:38
Extra Example 4: Find the Measure of Each Numbered Angle
39:00
Exploring Congruent Triangles

26m 46s

Intro
0:00
Congruent Triangles
0:15
Example of Congruent Triangles
0:17
Corresponding Parts
3:39
Corresponding Angles and Sides of Triangles
3:40
Definition of Congruent Triangles
11:24
Definition of Congruent Triangles
11:25
Triangle Congruence
16:37
Congruence of Triangles
16:38
Extra Example 1: Congruence Statement
18:24
Extra Example 2: Congruence Statement
21:26
Extra Example 3: Draw and Label the Figure
23:09
Extra Example 4: Drawing Triangles
24:04
Proving Triangles Congruent

47m 51s

Intro
0:00
SSS Postulate
0:18
Side-Side-Side Postulate
0:27
SAS Postulate
2:26
Side-Angle-Side Postulate
2:29
SAS Postulate
3:57
Proof Example
3:58
ASA Postulate
11:47
Angle-Side-Angle Postulate
11:53
AAS Theorem
14:13
Angle-Angle-Side Theorem
14:14
Methods Overview
16:16
Methods Overview
16:17
SSS
16:33
SAS
17:06
ASA
17:50
AAS
18:17
CPCTC
19:14
Extra Example 1:Proving Triangles are Congruent
21:29
Extra Example 2: Proof
25:40
Extra Example 3: Proof
30:41
Extra Example 4: Proof
38:41
Isosceles and Equilateral Triangles

27m 53s

Intro
0:00
Isosceles Triangle Theorem
0:07
Isosceles Triangle Theorem
0:09
Isosceles Triangle Theorem
2:26
Example: Using the Isosceles Triangle Theorem
2:27
Isosceles Triangle Theorem Converse
3:29
Isosceles Triangle Theorem Converse
3:30
Equilateral Triangle Theorem Corollaries
4:30
Equilateral Triangle Theorem Corollary 1
4:59
Equilateral Triangle Theorem Corollary 2
5:55
Extra Example 1: Find the Value of x
7:08
Extra Example 2: Find the Value of x
10:04
Extra Example 3: Proof
14:04
Extra Example 4: Proof
22:41
Section 5: Triangle Inequalities
Special Segments in Triangles

43m 44s

Intro
0:00
Perpendicular Bisector
0:06
Perpendicular Bisector
0:07
Perpendicular Bisector
4:07
Perpendicular Bisector Theorems
4:08
Median
6:30
Definition of Median
6:31
Median
9:41
Example: Median
9:42
Altitude
12:22
Definition of Altitude
12:23
Angle Bisector
14:33
Definition of Angle Bisector
14:34
Angle Bisector
16:41
Angle Bisector Theorems
16:42
Special Segments Overview
18:57
Perpendicular Bisector
19:04
Median
19:32
Altitude
19:49
Angle Bisector
20:02
Examples: Special Segments
20:18
Extra Example 1: Draw and Label
22:36
Extra Example 2: Draw the Altitudes for Each Triangle
24:37
Extra Example 3: Perpendicular Bisector
27:57
Extra Example 4: Draw, Label, and Write Proof
34:33
Right Triangles

26m 34s

Intro
0:00
LL Theorem
0:21
Leg-Leg Theorem
0:25
HA Theorem
2:23
Hypotenuse-Angle Theorem
2:24
LA Theorem
4:49
Leg-Angle Theorem
4:50
LA Theorem
6:18
Example: Find x and y
6:19
HL Postulate
8:22
Hypotenuse-Leg Postulate
8:23
Extra Example 1: LA Theorem & HL Postulate
10:57
Extra Example 2: Find x So That Each Pair of Triangles is Congruent
14:15
Extra Example 3: Two-column Proof
17:02
Extra Example 4: Two-column Proof
21:01
Indirect Proofs and Inequalities

33m 30s

Intro
0:00
Writing an Indirect Proof
0:09
Step 1
0:49
Step 2
2:32
Step 3
3:00
Indirect Proof
4:30
Example: 2 + 6 = 8
5:00
Example: The Suspect is Guilty
5:40
Example: Measure of Angle A < Measure of Angle B
6:06
Definition of Inequality
7:47
Definition of Inequality & Example
7:48
Properties of Inequality
9:55
Comparison Property
9:58
Transitive Property
10:33
Addition and Subtraction Properties
12:01
Multiplication and Division Properties
13:07
Exterior Angle Inequality Theorem
14:12
Example: Exterior Angle Inequality Theorem
14:13
Extra Example 1: Draw a Diagram for the Statement
18:32
Extra Example 2: Name the Property for Each Statement
19:56
Extra Example 3: State the Assumption
21:22
Extra Example 4: Write an Indirect Proof
25:39
Inequalities for Sides and Angles of a Triangle

17m 26s

Intro
0:00
Side to Angles
0:10
If One Side of a Triangle is Longer Than Another Side
0:11
Converse: Angles to Sides
1:57
If One Angle of a Triangle Has a Greater Measure Than Another Angle
1:58
Extra Example 1: Name the Angles in the Triangle From Least to Greatest
2:38
Extra Example 2: Find the Longest and Shortest Segment in the Triangle
3:47
Extra Example 3: Angles and Sides of a Triangle
4:51
Extra Example 4: Two-column Proof
9:08
Triangle Inequality

28m 11s

Intro
0:00
Triangle Inequality Theorem
0:05
Triangle Inequality Theorem
0:06
Triangle Inequality Theorem
4:22
Example 1: Triangle Inequality Theorem
4:23
Example 2: Triangle Inequality Theorem
9:40
Extra Example 1: Determine if the Three Numbers can Represent the Sides of a Triangle
12:00
Extra Example 2: Finding the Third Side of a Triangle
13:34
Extra Example 3: Always True, Sometimes True, or Never True
18:18
Extra Example 4: Triangle and Vertices
22:36
Inequalities Involving Two Triangles

29m 36s

Intro
0:00
SAS Inequality Theorem
0:06
SAS Inequality Theorem & Example
0:25
SSS Inequality Theorem
4:33
SSS Inequality Theorem & Example
4:34
Extra Example 1: Write an Inequality Comparing the Segments
6:08
Extra Example 2: Determine if the Statement is True
9:52
Extra Example 3: Write an Inequality for x
14:20
Extra Example 4: Two-column Proof
17:44
Parallelograms

29m 11s

Intro
0:00
0:06
Four-sided Polygons
0:08
Non Examples of Quadrilaterals
0:47
Parallelograms
1:35
Parallelograms
1:36
Properties of Parallelograms
4:28
Opposite Sides of a Parallelogram are Congruent
4:29
Opposite Angles of a Parallelogram are Congruent
5:49
Angles and Diagonals
6:24
Consecutive Angles in a Parallelogram are Supplementary
6:25
The Diagonals of a Parallelogram Bisect Each Other
8:42
Extra Example 1: Complete Each Statement About the Parallelogram
10:26
Extra Example 2: Find the Values of x, y, and z of the Parallelogram
13:21
Extra Example 3: Find the Distance of Each Side to Verify the Parallelogram
16:35
Extra Example 4: Slope of Parallelogram
23:15
Proving Parallelograms

42m 43s

Intro
0:00
Parallelogram Theorems
0:09
Theorem 1
0:20
Theorem 2
1:50
Parallelogram Theorems, Cont.
3:10
Theorem 3
3:11
Theorem 4
4:15
Proving Parallelogram
6:21
Example: Determine if Quadrilateral ABCD is a Parallelogram
6:22
Summary
14:01
Both Pairs of Opposite Sides are Parallel
14:14
Both Pairs of Opposite Sides are Congruent
15:09
Both Pairs of Opposite Angles are Congruent
15:24
Diagonals Bisect Each Other
15:44
A Pair of Opposite Sides is Both Parallel and Congruent
16:13
Extra Example 1: Determine if Each Quadrilateral is a Parallelogram
16:54
Extra Example 2: Find the Value of x and y
20:23
Extra Example 3: Determine if the Quadrilateral ABCD is a Parallelogram
24:05
Extra Example 4: Two-column Proof
30:28
Rectangles

29m 47s

Intro
0:00
Rectangles
0:03
Definition of Rectangles
0:04
Diagonals of Rectangles
2:52
Rectangles: Diagonals Property 1
2:53
Rectangles: Diagonals Property 2
3:30
Proving a Rectangle
4:40
Example: Determine Whether Parallelogram ABCD is a Rectangle
4:41
Rectangles Summary
9:22
Opposite Sides are Congruent and Parallel
9:40
Opposite Angles are Congruent
9:51
Consecutive Angles are Supplementary
9:58
Diagonals are Congruent and Bisect Each Other
10:05
All Four Angles are Right Angles
10:40
Extra Example 1: Find the Value of x
11:03
Extra Example 2: Name All Congruent Sides and Angles
13:52
Extra Example 3: Always, Sometimes, or Never True
19:39
Extra Example 4: Determine if ABCD is a Rectangle
26:45
Squares and Rhombi

39m 14s

Intro
0:00
Rhombus
0:09
Definition of a Rhombus
0:10
Diagonals of a Rhombus
2:03
Rhombus: Diagonals Property 1
2:21
Rhombus: Diagonals Property 2
3:49
Rhombus: Diagonals Property 3
4:36
Rhombus
6:17
Example: Use the Rhombus to Find the Missing Value
6:18
Square
8:17
Definition of a Square
8:20
Summary Chart
11:06
Parallelogram
11:07
Rectangle
12:56
Rhombus
13:54
Square
14:44
Extra Example 1: Diagonal Property
15:44
Extra Example 2: Use Rhombus ABCD to Find the Missing Value
19:39
Extra Example 3: Always, Sometimes, or Never True
23:06
Extra Example 4: Determine the Quadrilateral
28:02
Trapezoids and Kites

30m 48s

Intro
0:00
Trapezoid
0:10
Definition of Trapezoid
0:12
Isosceles Trapezoid
2:57
Base Angles of an Isosceles Trapezoid
2:58
Diagonals of an Isosceles Trapezoid
4:05
Median of a Trapezoid
4:26
Median of a Trapezoid
4:27
Median of a Trapezoid
6:41
Median Formula
7:00
Kite
8:28
Definition of a Kite
8:29
11:19
A Quadrilateral with Two Pairs of Adjacent Congruent Sides
11:20
Extra Example 1: Isosceles Trapezoid
14:50
Extra Example 2: Median of Trapezoid
18:28
Extra Example 3: Always, Sometimes, or Never
24:13
Extra Example 4: Determine if the Figure is a Trapezoid
26:49
Section 7: Proportions and Similarity
Using Proportions and Ratios

20m 10s

Intro
0:00
Ratio
0:05
Definition and Examples of Writing Ratio
0:06
Proportion
2:05
Definition of Proportion
2:06
Examples of Proportion
2:29
Using Ratio
5:53
Example: Ratio
5:54
Extra Example 1: Find Three Ratios Equivalent to 2/5
9:28
Extra Example 2: Proportion and Cross Products
10:32
Extra Example 3: Express Each Ratio as a Fraction
13:18
Extra Example 4: Fin the Measure of a 3:4:5 Triangle
17:26
Similar Polygons

27m 53s

Intro
0:00
Similar Polygons
0:05
Definition of Similar Polygons
0:06
Example of Similar Polygons
2:32
Scale Factor
4:26
Scale Factor: Definition and Example
4:27
Extra Example 1: Determine if Each Pair of Figures is Similar
7:03
Extra Example 2: Find the Values of x and y
11:33
Extra Example 3: Similar Triangles
19:57
Extra Example 4: Draw Two Similar Figures
23:36
Similar Triangles

34m 10s

Intro
0:00
AA Similarity
0:10
Definition of AA Similarity
0:20
Example of AA Similarity
2:32
SSS Similarity
4:46
Definition of SSS Similarity
4:47
Example of SSS Similarity
6:00
SAS Similarity
8:04
Definition of SAS Similarity
8:05
Example of SAS Similarity
9:12
Extra Example 1: Determine Whether Each Pair of Triangles is Similar
10:59
Extra Example 2: Determine Which Triangles are Similar
16:08
Extra Example 3: Determine if the Statement is True or False
23:11
Extra Example 4: Write Two-Column Proof
26:25
Parallel Lines and Proportional Parts

24m 7s

Intro
0:00
Triangle Proportionality
0:07
Definition of Triangle Proportionality
0:08
Example of Triangle Proportionality
0:51
Triangle Proportionality Converse
2:19
Triangle Proportionality Converse
2:20
Triangle Mid-segment
3:42
Triangle Mid-segment: Definition and Example
3:43
Parallel Lines and Transversal
6:51
Parallel Lines and Transversal
6:52
Extra Example 1: Complete Each Statement
8:59
Extra Example 2: Determine if the Statement is True or False
12:28
Extra Example 3: Find the Value of x and y
15:35
Extra Example 4: Find Midpoints of a Triangle
20:43
Parts of Similar Triangles

27m 6s

Intro
0:00
Proportional Perimeters
0:09
Proportional Perimeters: Definition and Example
0:10
Similar Altitudes
2:23
Similar Altitudes: Definition and Example
2:24
Similar Angle Bisectors
4:50
Similar Angle Bisectors: Definition and Example
4:51
Similar Medians
6:05
Similar Medians: Definition and Example
6:06
Angle Bisector Theorem
7:33
Angle Bisector Theorem
7:34
Extra Example 1: Parts of Similar Triangles
10:52
Extra Example 2: Parts of Similar Triangles
14:57
Extra Example 3: Parts of Similar Triangles
19:27
Extra Example 4: Find the Perimeter of Triangle ABC
23:14
Section 8: Applying Right Triangles & Trigonometry
Pythagorean Theorem

21m 14s

Intro
0:00
Pythagorean Theorem
0:05
Pythagorean Theorem & Example
0:06
Pythagorean Converse
1:20
Pythagorean Converse & Example
1:21
Pythagorean Triple
2:42
Pythagorean Triple
2:43
Extra Example 1: Find the Missing Side
4:59
Extra Example 2: Determine Right Triangle
7:40
Extra Example 3: Determine Pythagorean Triple
11:30
Extra Example 4: Vertices and Right Triangle
14:29
Geometric Mean

40m 59s

Intro
0:00
Geometric Mean
0:04
Geometric Mean & Example
0:05
Similar Triangles
4:32
Similar Triangles
4:33
Geometric Mean-Altitude
11:10
Geometric Mean-Altitude & Example
11:11
Geometric Mean-Leg
14:47
Geometric Mean-Leg & Example
14:18
Extra Example 1: Geometric Mean Between Each Pair of Numbers
20:10
Extra Example 2: Similar Triangles
23:46
Extra Example 3: Geometric Mean of Triangles
28:30
Extra Example 4: Geometric Mean of Triangles
36:58
Special Right Triangles

37m 57s

Intro
0:00
45-45-90 Triangles
0:06
Definition of 45-45-90 Triangles
0:25
45-45-90 Triangles
5:51
Example: Find n
5:52
30-60-90 Triangles
8:59
Definition of 30-60-90 Triangles
9:00
30-60-90 Triangles
12:25
Example: Find n
12:26
Extra Example 1: Special Right Triangles
15:08
Extra Example 2: Special Right Triangles
18:22
Extra Example 3: Word Problems & Special Triangles
27:40
Extra Example 4: Hexagon & Special Triangles
33:51
Ratios in Right Triangles

40m 37s

Intro
0:00
Trigonometric Ratios
0:08
Definition of Trigonometry
0:13
Sine (sin), Cosine (cos), & Tangent (tan)
0:50
Trigonometric Ratios
3:04
Trig Functions
3:05
Inverse Trig Functions
5:02
SOHCAHTOA
8:16
sin x
9:07
cos x
10:00
tan x
10:32
Example: SOHCAHTOA & Triangle
12:10
Extra Example 1: Find the Value of Each Ratio or Angle Measure
14:36
Extra Example 2: Find Sin, Cos, and Tan
18:51
Extra Example 3: Find the Value of x Using SOHCAHTOA
22:55
Extra Example 4: Trigonometric Ratios in Right Triangles
32:13
Angles of Elevation and Depression

21m 4s

Intro
0:00
Angle of Elevation
0:10
Definition of Angle of Elevation & Example
0:11
Angle of Depression
1:19
Definition of Angle of Depression & Example
1:20
Extra Example 1: Name the Angle of Elevation and Depression
2:22
Extra Example 2: Word Problem & Angle of Depression
4:41
Extra Example 3: Word Problem & Angle of Elevation
14:02
Extra Example 4: Find the Missing Measure
18:10
Law of Sines

35m 25s

Intro
0:00
Law of Sines
0:20
Law of Sines
0:21
Law of Sines
3:34
Example: Find b
3:35
Solving the Triangle
9:19
Example: Using the Law of Sines to Solve Triangle
9:20
Extra Example 1: Law of Sines and Triangle
17:43
Extra Example 2: Law of Sines and Triangle
20:06
Extra Example 3: Law of Sines and Triangle
23:54
Extra Example 4: Law of Sines and Triangle
28:59
Law of Cosines

52m 43s

Intro
0:00
Law of Cosines
0:35
Law of Cosines
0:36
Law of Cosines
6:22
Use the Law of Cosines When Both are True
6:23
Law of Cosines
8:35
Example: Law of Cosines
8:36
Extra Example 1: Law of Sines or Law of Cosines?
13:35
Extra Example 2: Use the Law of Cosines to Find the Missing Measure
17:02
Extra Example 3: Solve the Triangle
30:49
Extra Example 4: Find the Measure of Each Diagonal of the Parallelogram
41:39
Section 9: Circles
Segments in a Circle

22m 43s

Intro
0:00
Segments in a Circle
0:10
Circle
0:11
Chord
0:59
Diameter
1:32
2:07
Secant
2:17
Tangent
3:10
Circumference
3:56
Introduction to Circumference
3:57
Example: Find the Circumference of the Circle
5:09
Circumference
6:40
Example: Find the Circumference of the Circle
6:41
Extra Example 1: Use the Circle to Answer the Following
9:10
Extra Example 2: Find the Missing Measure
12:53
Extra Example 3: Given the Circumference, Find the Perimeter of the Triangle
15:51
Extra Example 4: Find the Circumference of Each Circle
19:24
Angles and Arc

35m 24s

Intro
0:00
Central Angle
0:06
Definition of Central Angle
0:07
Sum of Central Angles
1:17
Sum of Central Angles
1:18
Arcs
2:27
Minor Arc
2:30
Major Arc
3:47
Arc Measure
5:24
Measure of Minor Arc
5:24
Measure of Major Arc
6:53
Measure of a Semicircle
7:11
8:25
8:26
Arc Length
9:43
Arc Length and Example
9:44
Concentric Circles
16:05
Concentric Circles
16:06
Congruent Circles and Arcs
17:50
Congruent Circles
17:51
Congruent Arcs
18:47
Extra Example 1: Minor Arc, Major Arc, and Semicircle
20:14
Extra Example 2: Measure and Length of Arc
22:52
Extra Example 3: Congruent Arcs
25:48
Extra Example 4: Angles and Arcs
30:33
Arcs and Chords

21m 51s

Intro
0:00
Arcs and Chords
0:07
Arc of the Chord
0:08
Theorem 1: Congruent Minor Arcs
1:01
Inscribed Polygon
2:10
Inscribed Polygon
2:11
Arcs and Chords
3:18
Theorem 2: When a Diameter is Perpendicular to a Chord
3:19
Arcs and Chords
5:05
Theorem 3: Congruent Chords
5:06
Extra Example 1: Congruent Arcs
10:35
Extra Example 2: Length of Arc
13:50
Extra Example 3: Arcs and Chords
17:09
Extra Example 4: Arcs and Chords
19:45
Inscribed Angles

27m 53s

Intro
0:00
Inscribed Angles
0:07
Definition of Inscribed Angles
0:08
Inscribed Angles
0:58
Inscribed Angle Theorem 1
0:59
Inscribed Angles
3:29
Inscribed Angle Theorem 2
3:30
Inscribed Angles
4:38
Inscribed Angle Theorem 3
4:39
5:50
5:51
Extra Example 1: Central Angle, Inscribed Angle, and Intercepted Arc
7:02
Extra Example 2: Inscribed Angles
9:24
Extra Example 3: Inscribed Angles
14:00
Extra Example 4: Complete the Proof
17:58
Tangents

26m 16s

Intro
0:00
Tangent Theorems
0:04
Tangent Theorem 1
0:05
Tangent Theorem 1 Converse
0:55
Common Tangents
1:34
Common External Tangent
2:12
Common Internal Tangent
2:30
Tangent Segments
3:08
Tangent Segments
3:09
Circumscribed Polygons
4:11
Circumscribed Polygons
4:12
Extra Example 1: Tangents & Circumscribed Polygons
5:50
Extra Example 2: Tangents & Circumscribed Polygons
8:35
Extra Example 3: Tangents & Circumscribed Polygons
11:50
Extra Example 4: Tangents & Circumscribed Polygons
15:43
Secants, Tangents, & Angle Measures

27m 50s

Intro
0:00
Secant
0:08
Secant
0:09
Secant and Tangent
0:49
Secant and Tangent
0:50
Interior Angles
2:56
Secants & Interior Angles
2:57
Exterior Angles
7:21
Secants & Exterior Angles
7:22
Extra Example 1: Secants, Tangents, & Angle Measures
10:53
Extra Example 2: Secants, Tangents, & Angle Measures
13:31
Extra Example 3: Secants, Tangents, & Angle Measures
19:54
Extra Example 4: Secants, Tangents, & Angle Measures
22:29
Special Segments in a Circle

23m 8s

Intro
0:00
Chord Segments
0:05
Chord Segments
0:06
Secant Segments
1:36
Secant Segments
1:37
Tangent and Secant Segments
4:10
Tangent and Secant Segments
4:11
Extra Example 1: Special Segments in a Circle
5:53
Extra Example 2: Special Segments in a Circle
7:58
Extra Example 3: Special Segments in a Circle
11:24
Extra Example 4: Special Segments in a Circle
18:09
Equations of Circles

27m 1s

Intro
0:00
Equation of a Circle
0:06
Standard Equation of a Circle
0:07
Example 1: Equation of a Circle
0:57
Example 2: Equation of a Circle
1:36
Extra Example 1: Determine the Coordinates of the Center and the Radius
4:56
Extra Example 2: Write an Equation Based on the Given Information
7:53
Extra Example 3: Graph Each Circle
16:48
Extra Example 4: Write the Equation of Each Circle
19:17
Section 10: Polygons & Area
Polygons

27m 24s

Intro
0:00
Polygons
0:10
Polygon vs. Not Polygon
0:18
Convex and Concave
1:46
Convex vs. Concave Polygon
1:52
Regular Polygon
4:04
Regular Polygon
4:05
Interior Angle Sum Theorem
4:53
Triangle
5:03
6:05
Pentagon
6:38
Hexagon
7:59
20-Gon
9:36
Exterior Angle Sum Theorem
12:04
Exterior Angle Sum Theorem
12:05
Extra Example 1: Drawing Polygons
13:51
Extra Example 2: Convex Polygon
15:16
Extra Example 3: Exterior Angle Sum Theorem
18:21
Extra Example 4: Interior Angle Sum Theorem
22:20
Area of Parallelograms

17m 46s

Intro
0:00
Parallelograms
0:06
Definition and Area Formula
0:07
Area of Figure
2:00
Area of Figure
2:01
Extra Example 1:Find the Area of the Shaded Area
3:14
Extra Example 2: Find the Height and Area of the Parallelogram
6:00
Extra Example 3: Find the Area of the Parallelogram Given Coordinates and Vertices
10:11
Extra Example 4: Find the Area of the Figure
14:31
Area of Triangles Rhombi, & Trapezoids

20m 31s

Intro
0:00
Area of a Triangle
0:06
Area of a Triangle: Formula and Example
0:07
Area of a Trapezoid
2:31
Area of a Trapezoid: Formula
2:32
Area of a Trapezoid: Example
6:55
Area of a Rhombus
8:05
Area of a Rhombus: Formula and Example
8:06
Extra Example 1: Find the Area of the Polygon
9:51
Extra Example 2: Find the Area of the Figure
11:19
Extra Example 3: Find the Area of the Figure
14:16
Extra Example 4: Find the Height of the Trapezoid
18:10
Area of Regular Polygons & Circles

36m 43s

Intro
0:00
Regular Polygon
0:08
SOHCAHTOA
0:54
30-60-90 Triangle
1:52
45-45-90 Triangle
2:40
Area of a Regular Polygon
3:39
Area of a Regular Polygon
3:40
Are of a Circle
7:55
Are of a Circle
7:56
Extra Example 1: Find the Area of the Regular Polygon
8:22
Extra Example 2: Find the Area of the Regular Polygon
16:48
Extra Example 3: Find the Area of the Shaded Region
24:11
Extra Example 4: Find the Area of the Shaded Region
32:24
Perimeter & Area of Similar Figures

18m 17s

Intro
0:00
Perimeter of Similar Figures
0:08
Example: Scale Factor & Perimeter of Similar Figures
0:09
Area of Similar Figures
2:44
Example:Scale Factor & Area of Similar Figures
2:55
Extra Example 1: Complete the Table
6:09
Extra Example 2: Find the Ratios of the Perimeter and Area of the Similar Figures
8:56
Extra Example 3: Find the Unknown Area
12:04
Extra Example 4: Use the Given Area to Find AB
14:26
Geometric Probability

38m 40s

Intro
0:00
Length Probability Postulate
0:05
Length Probability Postulate
0:06
Are Probability Postulate
2:34
Are Probability Postulate
2:35
Are of a Sector of a Circle
4:11
Are of a Sector of a Circle Formula
4:12
Are of a Sector of a Circle Example
7:51
Extra Example 1: Length Probability
11:07
Extra Example 2: Area Probability
12:14
Extra Example 3: Area Probability
17:17
Extra Example 4: Area of a Sector of a Circle
26:23
Section 11: Solids
Three-Dimensional Figures

23m 39s

Intro
0:00
Polyhedrons
0:05
Polyhedrons: Definition and Examples
0:06
Faces
1:08
Edges
1:55
Vertices
2:23
Solids
2:51
Pyramid
2:54
Cylinder
3:45
Cone
4:09
Sphere
4:23
Prisms
5:00
Rectangular, Regular, and Cube Prisms
5:02
Platonic Solids
9:48
Five Types of Regular Polyhedra
9:49
Slices and Cross Sections
12:07
Slices
12:08
Cross Sections
12:47
Extra Example 1: Name the Edges, Faces, and Vertices of the Polyhedron
14:23
Extra Example 2: Determine if the Figure is a Polyhedron and Explain Why
17:37
Extra Example 3: Describe the Slice Resulting from the Cut
19:12
Extra Example 4: Describe the Shape of the Intersection
21:25
Surface Area of Prisms and Cylinders

38m 50s

Intro
0:00
Prisms
0:06
Bases
0:07
Lateral Faces
0:52
Lateral Edges
1:19
Altitude
1:58
Prisms
2:24
Right Prism
2:25
Oblique Prism
2:56
Classifying Prisms
3:27
Right Rectangular Prism
3:28
4:55
Oblique Pentagonal Prism
6:26
Right Hexagonal Prism
7:14
Lateral Area of a Prism
7:42
Lateral Area of a Prism
7:43
Surface Area of a Prism
13:44
Surface Area of a Prism
13:45
Cylinder
16:18
Cylinder: Right and Oblique
16:19
Lateral Area of a Cylinder
18:02
Lateral Area of a Cylinder
18:03
Surface Area of a Cylinder
20:54
Surface Area of a Cylinder
20:55
Extra Example 1: Find the Lateral Area and Surface Are of the Prism
21:51
Extra Example 2: Find the Lateral Area of the Prism
28:15
Extra Example 3: Find the Surface Area of the Prism
31:57
Extra Example 4: Find the Lateral Area and Surface Area of the Cylinder
34:17
Surface Area of Pyramids and Cones

26m 10s

Intro
0:00
Pyramids
0:07
Pyramids
0:08
Regular Pyramids
1:52
Regular Pyramids
1:53
Lateral Area of a Pyramid
4:33
Lateral Area of a Pyramid
4:34
Surface Area of a Pyramid
9:19
Surface Area of a Pyramid
9:20
Cone
10:09
Right and Oblique Cone
10:10
Lateral Area and Surface Area of a Right Cone
11:20
Lateral Area and Surface Are of a Right Cone
11:21
Extra Example 1: Pyramid and Prism
13:11
Extra Example 2: Find the Lateral Area of the Regular Pyramid
15:00
Extra Example 3: Find the Surface Area of the Pyramid
18:29
Extra Example 4: Find the Lateral Area and Surface Area of the Cone
22:08
Volume of Prisms and Cylinders

21m 59s

Intro
0:00
Volume of Prism
0:08
Volume of Prism
0:10
Volume of Cylinder
3:38
Volume of Cylinder
3:39
Extra Example 1: Find the Volume of the Prism
5:10
Extra Example 2: Find the Volume of the Cylinder
8:03
Extra Example 3: Find the Volume of the Prism
9:35
Extra Example 4: Find the Volume of the Solid
19:06
Volume of Pyramids and Cones

22m 2s

Intro
0:00
Volume of a Cone
0:08
Volume of a Cone: Example
0:10
Volume of a Pyramid
3:02
Volume of a Pyramid: Example
3:03
Extra Example 1: Find the Volume of the Pyramid
4:56
Extra Example 2: Find the Volume of the Solid
6:01
Extra Example 3: Find the Volume of the Pyramid
10:28
Extra Example 4: Find the Volume of the Octahedron
16:23
Surface Area and Volume of Spheres

14m 46s

Intro
0:00
Special Segments
0:06
0:07
Chord
0:31
Diameter
0:55
Tangent
1:20
Sphere
1:43
Plane & Sphere
1:44
Hemisphere
2:56
Surface Area of a Sphere
3:25
Surface Area of a Sphere
3:26
Volume of a Sphere
4:08
Volume of a Sphere
4:09
Extra Example 1: Determine Whether Each Statement is True or False
4:24
Extra Example 2: Find the Surface Area of the Sphere
6:17
Extra Example 3: Find the Volume of the Sphere with a Diameter of 20 Meters
7:25
Extra Example 4: Find the Surface Area and Volume of the Solid
9:17
Congruent and Similar Solids

16m 6s

Intro
0:00
Scale Factor
0:06
Scale Factor: Definition and Example
0:08
Congruent Solids
1:09
Congruent Solids
1:10
Similar Solids
2:17
Similar Solids
2:18
Extra Example 1: Determine if Each Pair of Solids is Similar, Congruent, or Neither
3:35
Extra Example 2: Determine if Each Statement is True or False
7:47
Extra Example 3: Find the Scale Factor and the Ratio of the Surface Areas and Volume
10:14
Extra Example 4: Find the Volume of the Larger Prism
12:14
Section 12: Transformational Geometry
Mapping

14m 12s

Intro
0:00
Transformation
0:04
Rotation
0:32
Translation
1:03
Reflection
1:17
Dilation
1:24
Transformations
1:45
Examples
1:46
Congruence Transformation
2:51
Congruence Transformation
2:52
Extra Example 1: Describe the Transformation that Occurred in the Mappings
3:37
Extra Example 2: Determine if the Transformation is an Isometry
5:16
Extra Example 3: Isometry
8:16
Reflections

23m 17s

Intro
0:00
Reflection
0:05
Definition of Reflection
0:06
Line of Reflection
0:35
Point of Reflection
1:22
Symmetry
1:59
Line of Symmetry
2:00
Point of Symmetry
2:48
Extra Example 1: Draw the Image over the Line of Reflection and the Point of Reflection
3:45
Extra Example 2: Determine Lines and Point of Symmetry
6:59
Extra Example 3: Graph the Reflection of the Polygon
11:15
Extra Example 4: Graph the Coordinates
16:07
Translations

18m 43s

Intro
0:00
Translation
0:05
Translation: Preimage & Image
0:06
Example
0:56
Composite of Reflections
6:28
Composite of Reflections
6:29
Extra Example 1: Translation
7:48
Extra Example 2: Image, Preimage, and Translation
12:38
Extra Example 3: Find the Translation Image Using a Composite of Reflections
15:08
Extra Example 4: Find the Value of Each Variable in the Translation
17:18
Rotations

21m 26s

Intro
0:00
Rotations
0:04
Rotations
0:05
Performing Rotations
2:13
Composite of Two Successive Reflections over Two Intersecting Lines
2:14
Angle of Rotation: Angle Formed by Intersecting Lines
4:29
Angle of Rotation
5:30
Rotation Postulate
5:31
Extra Example 1: Find the Rotated Image
7:32
Extra Example 2: Rotations and Coordinate Plane
10:33
Extra Example 3: Find the Value of Each Variable in the Rotation
14:29
Extra Example 4: Draw the Polygon Rotated 90 Degree Clockwise about P
16:13
Dilation

37m 6s

Intro
0:00
Dilations
0:06
Dilations
0:07
Scale Factor
1:36
Scale Factor
1:37
Example 1
2:06
Example 2
6:22
Scale Factor
8:20
Positive Scale Factor
8:21
Negative Scale Factor
9:25
Enlargement
12:43
Reduction
13:52
Extra Example 1: Find the Scale Factor
16:39
Extra Example 2: Find the Measure of the Dilation Image
19:32
Extra Example 3: Find the Coordinates of the Image with Scale Factor and the Origin as the Center of Dilation
26:18
Extra Example 4: Graphing Polygon, Dilation, and Scale Factor
32:08
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• ## Related Books

### Proving Triangles Congruent

• SSS (Side-Side-Side) Postulate: If the sides of one triangle are congruent to the sides of a second triangle, then the triangles are congruent
• SAS (Side-Angle-Side) Postulate: If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent
• ASA (Angle-Side-Angle) Postulate: If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent
• AAS (Angle-Angle-Side) Theorem: If two angles and a non-included side of one triangle are congruent to the corresponding two angles and side of a second triangle, then the two triangles are congruent.
• CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

### Proving Triangles Congruent

Determin whether the following statement is true or false.
If two sides and an angle of one triangle are congruent to two sides and an angle of another triangle, then the two triangles are congruent.
False
Determin whether the following statement is true or false.

VABC and ∆ DEF, if AB ≅ DE , BC ≅ EF , AC ≅ DF , then ∆ ABC ≅ ∆ DEF
True
Write a proof.

Given: ABC and DBC are right angles, AB ≅ BD
Prove: ∆ ABC ≅ ∆ DBC
• Statements ; Reasons
• ABC and DBC are right angles ; Given
• mABC = mDBC = 90o ; Definition of right angles
• ABC ≅ DBC ; Definition of congruent angles
• BC ≅ BC ; Reflexive prop ( = )
• AB ≅ BD ; Given
• ∆ ABC ≅ ∆ DBC ; SAS post.
Statements ; Reasons
ABC and DBC are right angles ; Given
mABC = mDBC = 90o ; Definition of right angles
ABC ≅ DBC ; Definition of congruent angles
BC ≅ BC ; Reflexive prop ( = )
AB ≅ BD ; Given
∆ ABC ≅ ∆ DBC ; SAS post.
BAC ≅ EDF, ABC ≅ DEF, AB ≅ DE , Determin which postulate or theorem can be used to prove the two triangles are congruent.
ASA post.
Fill in the blank in the statement with always, sometimes or never.
If two angles and a side of one triangle are congruent to the conrresponding two angles and a side of another triangle, then the two triangles are ____ congruent.
Always
Fill in the blank in the statement with always, sometimes or never.
If two sides and an angle of one angle are congruent to the conrresponding two sides and an angle of another triangle, then the two triangles are ____ congruent.
Sometimes
Determin whether the following statement is true or false.
If a triangle has a right angle, then its congruent triangle can have an obtuse angle.
False
Write SSS post in an if and then form.
If three sides of one triangle are congruent to the corresponding three angles of another triangle, then the two triangles are congruent.

Given: AD ||BC , AB ||CD .
Prove: ∆ ABC ≅ VCDA
• Statements ; Reasons
• AD ||BC ; Given
• ∠CAD ≅ ∠ACB ; alternate interior angles
• AB ||CD ; Given
• ∠DCA ≅ ∠BAC ; alternate interior angles
• AC ≅ AC; reflexive prop ( = )
• ∆ ABC ≅ VCDA ; ASA post.
Statements ; Reasons
AD ||BC ; Given
∠CAD ≅ ∠ACB ; alternate interior angles
AB ||CD ; Given
∠DCA ≅ ∠BAC ; alternate interior angles
AC ≅ AC; reflexive prop ( = )
∆ ABC ≅ VCDA ; ASA post.

Given: ∠ABC ≅ ∠DCB, ∠BAC ≅ ∠CDB Prove: ∆ ABC ≅ ∆ DCB
• Statements; Reasons
• ∠ABC ≅ ∠DCB, ∠BAC ≅ ∠CDB ; Given
• BC ≅ BC ; reflexive prop ( = )
• ∆ ABC ≅ ∆ DCB; AAS theorem
Statements; Reasons
∠ABC ≅ ∠DCB, ∠BAC ≅ ∠CDB ; Given
BC ≅ BC ; reflexive prop ( = )
∆ ABC ≅ ∆ DCB; AAS theorem

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

### Proving Triangles Congruent

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

• Intro 0:00
• SSS Postulate 0:18
• Side-Side-Side Postulate
• SAS Postulate 2:26
• Side-Angle-Side Postulate
• SAS Postulate 3:57
• Proof Example
• ASA Postulate 11:47
• Angle-Side-Angle Postulate
• AAS Theorem 14:13
• Angle-Angle-Side Theorem
• Methods Overview 16:16
• Methods Overview
• SSS
• SAS
• ASA
• AAS
• CPCTC
• Extra Example 1:Proving Triangles are Congruent 21:29
• Extra Example 2: Proof 25:40
• Extra Example 3: Proof 30:41
• Extra Example 4: Proof 38:41

### Transcription: Proving Triangles Congruent

Welcome back to Educator.com.0000

This lesson, we are going to prove triangles congruent.0003

In the previous lesson, remember, we talked about the definition of congruent triangles,0007

and how, if we have two triangles, their corresponding parts are going to be congruent.0013

We are going to take a closer look at some of the methods we can use to prove that two triangles are congruent.0019

The first method to prove that two triangles are congruent...0028

Now, according to the definition of congruent triangles, if all of the corresponding parts of the two triangles are congruent, then the two triangles are congruent.0033

But that is kind of a lot to do; that is a lot of work, because then, you would have to prove0047

that six parts are going to be congruent in order to prove that the triangles are congruent.0052

Instead, we have postulates and theorems that make it easier.0058

The first one is the SSS Postulate, or you could just call it SSS, which stands for Side-Side-Side Postulate.0065

And that is just saying that, if all of the sides of one triangle are congruent to the sides of the second triangle, then the triangles are congruent.0077

This is one method: instead of having to show that all six corresponding parts are congruent,0086

in order to prove that the triangles are congruent, you only have to do the three.0095

So, if AC is congruent to DF, AB is congruent to DE, and BC is congruent to EF,0100

once you have shown that those three sides are congruent to the three sides of the other triangle,0110

then you have proven that the two triangles are congruent.0115

So, the Side-Side-Side Postulate is that, if three sides of one triangle are congruent0118

to three sides of the other triangle, then triangle ABC is congruent to triangle DEF.0123

The next one: now, there are going to be a few different methods--the next method is SAS, which is Side-Angle-Side.0146

Again, as long as you can prove that a side is congruent to a side of the other triangle,0159

an angle is congruent to the angle, and another side; then you can prove that those two triangles are congruent.0167

But there is a condition: the angle, this angle right here, has to be an included angle.0177

"Included" means that the angle has to be in between the two sides--0188

a side, and then the angle, and then the side; side, side, side, angle, angle, and then side, side.0198

It can't be an angle that is one of the two outside angles; it has to be the included angle, between the two sides that you are showing are congruent.0208

That is side-angle-side--very important.0220

If two sides of one triangle are congruent to two sides of the other triangle, and the included angle, then the triangles are congruent.0225

Let's do one proof on the SAS Postulate.0239

Now, this is not actually a proof to prove this postulate; it is just a proof to show that the two triangles are congruent, using the SAS Postulate.0245

Given that E, this point right here, is the midpoint of BD, and then E is the midpoint of AC,0259

I know that if E is the midpoint, then that means that the two segments are congruent.0273

And then, I want to prove that triangle AEB is congruent to triangle CED.0279

I want to prove that this triangle is congruent to this triangle.0288

Whenever you get a proof where you have to prove triangles congruent, you can use any of the methods,0299

depending on what information you have--what is congruent to what--what you can show to be congruent.0306

But this one is going to be SAS, because we are just going to practice using this one.0312

And so far, I have a side of one of the triangles congruent to a side of the other triangle; so I have an S.0318

Then, I have another side congruent to the side of another triangle; so I have the other S.0329

Now, I need an A; but the A, the angle, has to be the included angle, so it has to be in between the two sides.0337

That means that that angle is going to be this angle right here--this angle, and then this angle.0345

But do I have a reason--can I just say that they are congruent?0353

No, the reason would be that they are vertical, and we know that vertical angles are congruent.0357

So, now I have all three parts, and I can go ahead and write out my proof.0363

Let's do a two-column proof, where I have statements and reasons.0370

Statements/reasons: I am just going to write it out like this.0378

#1: The given is always number 1.0383

Now, sometimes, if you look at examples of proofs in your book, the first one might not always be the given.0391

You might have the given as #1, and then maybe #3, or later on in the proof.0397

Sometimes, if you have two different given statements, you don't have to write both for step 1.0403

You can write one of them that you are going to use at a time, that you are going to use first.0411

And then, if you don't need the other given statement until later on in the proof,0417

then you can just wait until that step where you need it and then write it in; and your reason will still be "Given."0420

Or you can just write the whole thing under step 1.0428

E is the midpoint of BD, and E is the midpoint of AC; my reason is "Given."0434

OK, so then, from here, I have to prove each of these; and then I can prove that the triangles are congruent,0457

because the postulate says that if the side with the corresponding side and the angle with the corresponding angle0466

and the side with the side are congruent, then the triangles are congruent.0473

So then, I have to prove all of these first.0477

I can say that, since E is the midpoint of BD, BE is congruent to DE.0482

Make sure that, if you are going to say BE, then you have to say DE, because it has to be corresponding--the way you write it in order.0494

So, if I decide to write it as EB, that is fine; then you would have to write it as ED next.0502

So, BE is congruent to DE; what is the reason?0510

It is "definition of midpoint," because the definition of midpoint says that, if you have a midpoint,0515

then the midpoint will cut the segment into two equal parts.0527

So then, that is why these parts will be congruent--the definition of midpoint.0537

And then, I can say that angle AEB (I am looking at this angle right here for the second one) is congruent to angle CED.0544

And that is because they are vertical, and we know that vertical angles are congruent.0565

So, any time you have vertical angles, you can say that they are congruent; the reason is that vertical angles are congruent.0573

Now, my next step: I have to say that the other sides, these sides now, are congruent.0580

AE, this side of one of the triangles, is congruent to side CE; that is also because of the definition of midpoint.0585

So then, now can I say that the triangles are congruent?0603

This step right here, BE is congruent to DE: see how that is one of the triangle being congruent to one side of the triangle.0607

So, I have a side; then my next step was to show that the angles are congruent--I did that;0615

angle...and then another side of each of the triangles; did I meet all of the conditions?0623

Yes, I showed corresponding sides congruent, corresponding angles congruent, and corresponding sides congruent.0632

Now, I can say that the triangles are congruent: so triangle AEB, this triangle right here, is congruent to triangle CED.0642

And the reason is the SAS Postulate.0663

Again, these are just methods--SAS, SSS, and then we are going to go over a couple more.0674

Those are just methods to prove triangles congruent.0679

Otherwise, if we didn't have these methods, it would be a lot harder, because you would have to prove that each part, all six corresponding parts, are congruent.0686

These actually make it easier; we only have to prove three things congruent, three parts;0697

and then we can say that the triangles are congruent.0701

The next postulate is Angle-Side-Angle (ASA); here is one angle, angle A, congruent to angle F;0707

if angle A is congruent to angle F, and then AC is congruent to (remember, if I am going to say AC, then I have to say) FD0724

(because they are corresponding in that order), and angle C is congruent to angle D0743

(so then, I have: this is an angle; the corresponding angle is congruent; the corresponding side is congruent;0753

and the corresponding angle is congruent), then triangle ABC is congruent to triangle...0762

A is corresponding with F, so triangle F...B with E, and then C with D.0774

So, to prove that these two triangles are congruent, I can use this postulate.0784

Now again, the other thing to mention here is "included side"; if it is ASA, make sure that the side is in between the two angles.0789

So then, if I am going to use angle A and angle F and angle C and angle D, the side has to be in between.0801

So, if I have, let's say, two triangles, and I did it with the angle here and another angle, the angle here, and side--this side here,0809

this is not Angle-Side-Angle, even though I have two angles and I have one side.0821

This would not be Angle-Side-Angle, because the side is not included, meaning it is not in between the two angles.0827

This is actually going to be Angle-Angle-Side; this is not ASA.0837

This one is AAS, Angle-Angle-Side, when we have two angles congruent to the corresponding angles of the second triangle,0855

and then a side congruent to the other side, but not included (meaning it is not in between).0867

See how this is angle, and then angle, and then side; so it goes in that order: Angle-Angle-Side.0875

That is when you can state the Angle-Angle-Side theorem; this is a theorem.0882

Make sure that, if you are proving triangles this way, using this theorem, you state that it is Angle-Angle-Side, and not Angle-Side-Angle.0891

They both have two angles and one side, but if the side is in between the two angles, then it is Angle-Side-Angle;0904

and if the side is not in between--it is not included--then it would be Angle-Angle-Side.0913

If angle A is congruent to angle F, and angle B is congruent to angle E, and BC is congruent to ED, then I can say,0922

because again, this is an angle, and the next is an angle, and the next is a side--then I can say that triangle ABC0946

is congruent to triangle...what is corresponding with A? F; with B, E; and then D.0957

So, once you prove that those three parts are congruent, then you can prove that those two triangles are congruent.0967

So, just to go over the methods again: these four, again, are methods to prove that two triangles are congruent.0977

SSS means Side-Side-Side; if I have two triangles that I want to prove congruent, then this is one method, one way to do it,0994

by showing that all three sides are congruent; so if this, then the triangles are congruent.1012

This is Side-Angle-Side; and that is saying that a side with an included angle, and then a side, like that;1027

the angle has to be in between the two sides; if it is this angle or this angle, then that is not it.1048

And then, if this is true--if I prove that these parts are congruent--then the triangles are congruent.1060

This is Angle-Side-Angle; if you can prove that those three parts are congruent, then you can prove that the triangles are congruent.1072

Angle-Angle-Side: now, when you use these in your proofs, you don't have to actually write out "Side-Side-Side,"1099

"Side-Angle-Side"...you can just write SSS; and these three are postulates, and this one is a theorem.1113

The last one, Angle-Angle-Side, is a theorem; you can just write SSS, though; that should be OK.1121

Or you can just write these.1125

This is Angle-Angle-Side: see how there is a difference between this and this.1128

In this one, the side is in between the two angles; and in this one, the side is not in between the two angles.1142

So, if you prove that these parts are congruent, then you can say that the triangles are congruent.1146

Now, this last one is from the last section, the last lesson: CPCTC is actually not used to prove that triangles are congruent.1155

This one stands for Corresponding Parts of Congruent Triangles are Congruent.1170

That means that the triangles have to be congruent first.1189

So, if I want to prove that...let's say I have this, and I have to prove that maybe this side is congruent to this side;1193

now, if there is no simple way to just prove that those two sides are congruent, then what I can do is:1209

using one of these methods, first prove that the triangles are congruent.1215

So, with just Side-Side-Side, or whatever it is, prove that these two triangles are congruent.1222

Once those triangles are congruent, then I can say that any corresponding parts are congruent.1231

Let's say it is given to me; if it is given to me, whether it is given or whether you proved it, once you say1241

that triangle ABC is congruent to triangle DEF, once this statement is written, then I can say AB is congruent to DE.1250

And the reason would be CPCTC.1268

Corresponding parts of congruent triangles are congruent.1274

These four are used to actually prove that the triangles are congruent; this one is used after the triangles are congruent.1277

So, let's go over our examples: Determine which postulate or theorem can be used to prove that the triangles are congruent.1289

Now, for this one, I have a side; here is the first triangle, and here is the other triangle.1298

Side and side--that is one part that is congruent; and then you have another side.1313

But then, you only have two; you need three; so I know that vertical angles,1321

these angles (even if it is not given to me, just from the fact that they are vertical, I can say that they) are congruent.1325

So then, this one...which one is this going to be?--Side-Side-Angle.1333

This is not Side-Angle-Side, because the congruent angles are not included; it is not in between the two congruent sides.1347

This is not Side-Angle-Side; this would be Side-Side-Angle.1358

Now, there is no such thing as Side-Side-Angle; that doesn't exist.1361

If you spell this backwards, the other way around, then it spells out a bad word.1368

If it spells out a bad word, it doesn't exist; just think of it that way.1374

You cannot use this to prove that the triangles are congruent;1382

there is no postulate or theorem that says that you can prove these, and then the triangles are congruent.1386

In this case, for this problem, we can't prove that they are congruent, because this is not a rule.1392

Now, if I said that angle D is congruent to angle B, then you can say Side-Angle-Side, and that is a rule.1403

So then, you can prove that the triangles are congruent that way.1412

But for this one, this is all that is given--this side with this side, this side with this side, and then vertical angles; so that is not one.1417

The next one: Now, again, I only have a side; I am trying to prove that this triangle is congruent to this triangle.1426

I have a side congruent to a side; I have a side.1436

I have another side congruent to this side; but then I am missing one more thing.1440

So, I have to try to see if there is a given--if there is something here that, even if they didn't give it to me, I know automatically that it is congruent.1448

And what that is, is this side right here.1458

So, if I split up these triangles, it is going to be like this and like that.1461

So, I am trying to prove that these two triangles are congruent; this side is congruent, and this side is congruent.1472

This right here is the same as this right here; they are sharing that side.1480

Since they are sharing that side, it has to be the same; it has to be congruent, automatically.1490

And that is the reflexive property, because AC is going to be congruent to itself.1499

So, AC is the same for this triangle, and it is the same for this triangle.1507

So, in this case, this is Side-Side-Side; even though this side is not given to you, we know that it is still congruent,1510

because it is reflexive; it is equaling itself; this is the same side for this, and it is the same side for this.1520

Automatically, it is congruent; it would be the Side-Side-Side Postulate.1526

OK, we are going to do a few proofs now.1541

So then, let's take a look at our given: angle A is congruent to angle E.1546

I am just going to do that; C is the midpoint of AE; that means that, if this is the midpoint, then these two parts are congruent.1555

That is given; and then I have to prove that this triangle, triangle ABC, is congruent to triangle EDC.1570

Since I am proving that two triangles are congruent, I have to use one of the methods: SSS, SAS, ASA, or AAS--one of those four methods.1577

Since I only have two parts--I have an angle, and I have a side--I need to look at this and see if anything else is given.1590

And I see that this angle right here is congruent to this angle right here, automatically, because they are vertical.1599

So, statements and reasons, right here: the #1 statement is going to be that angle A is congruent to angle E,1608

and that C is the midpoint of AE; the reason is "Given."1631

Now, since I know that my destination, my last step, my point B, is going to be that these two triangles are congruent,1644

do I know already what method I am going to be using to prove that those two triangles are congruent?1655

I have an angle, a side, and an angle, so I know that I am going to be using the ASA Postulate.1660

Now, I just have to state out each of these; I have to state out the angle; I have to state out the sides; and then I have to state the angles.1669

And then, once all three of these are stated, then I can say that these two triangles are congruent.1678

The next step: now, see how angle A is congruent to angle E--that angle is already stated, so that here, this is an angle.1688

Since C is the midpoint of AE, I am going to say that AC is congruent to EC.1700

The reason for that is "definition of midpoint."1709

And then, that is my side; and then, I am missing an angle, which is this right here.1722

Now, I can't say angle C, because angle C represents so many different angles.1736

So, I have to say angle ACB, or BCA; so ACB is congruent to...and if I am going to say angle ACB, then I have to say angle ECD,1742

because A and D are corresponding, because they are congruent.1755

ECD--see how that is another angle; what is my reason for that?--that vertical angles are congruent.1761

Now, see how I stated all of them now: I stated my angle; I stated my sides; and I stated my angle.1780

So then, I can say that triangle ABC is congruent to triangle EDC.1786

And what is my reason--what method did I use?--Angle-Side-Angle Postulate.1799

You are going to do a few of these kinds of proofs, where you are proving the triangles using one of the methods.1813

Always look at what is given; look at your diagram.1820

Your diagram is going to be very valuable when it comes to proofs, because you want to mark it up1824

and see what you have, what you have to work with, what more you have to do, and how you are going to get to this step right here.1828

It is like your map.1839

BD bisects AE; now, remember "bisector": when something bisects something else, it is cutting it in half.1843

So, if BD is bisecting AE, that means that BD is cutting AE in half--think of it that way--it is cutting it in half.1858

Don't get confused by what is cut in half, which one is going to be cut in half.1869

Is BD cut in half, or is AE cut in half?1879

Whichever one is doing the bisecting is the one that is doing the cutting.1884

Since BD is bisecting AE, BD is cutting AE in half; if BD is bisecting AE, that means that BD cut AE in half, so AE is cut in half.1888

And then, what else is given? Angle B is congruent to angle D.1907

I know that angle 1 is congruent to angle 2, because they are vertical angles; I am just going to mark that.1919

So, how would I be able to prove that these two triangles are congruent?1925

Angle-Angle-Side: it is not Angle-Side-Angle; it is Angle-Angle-Side.1929

Now, but then, look at my "prove" statement; it is not asking me to prove that the triangles are congruent.1939

It is asking me to prove that AB, this side, is congruent to this side.1943

But is there any way for me to be able to prove that those two sides are congruent?1951

I don't see a way to say that these two sides are congruent; there is nothing here that allows me to prove that AB is congruent to ED, except for CPCTC.1956

Remember: it is a random side that I have to prove congruent; whenever it is just a random side,1973

a random part, that you have to prove congruent, then you have to first prove that the triangles are congruent,1983

which we can do by Angle-Angle-Side; and then, once the triangles are congruent, you can say that corresponding parts are congruent.1991

So then, once the triangles are congruent, I can say that AB is congruent to ED,2002

because corresponding parts of congruent triangles are congruent.2008

I know (wrong one!)...statements/reasons...1: BD bisects AE; angle B is congruent to angle D; that is given.2013

OK, so one of the parts is already stated out: angle B is congruent to angle D--that is an angle.2048

Now, if you are going to do this like how I am doing it, how you are writing out what you are showing on the side,2058

that is good; but just be careful when it comes to your included angle or your included side,2066

because if you are not doing the right order--let's say I mention the angle--see how this angle is mentioned first,2072

and then maybe the next step--what if I mention the sides?2079

Then, be careful so that it is not going to be in that order.2084

I will show you when we get to that.2089

Here, the next step: I am going to say that AC is congruent to...now, if I am going to say AC,2094

I can't say CE; I have to say EC; remember corresponding parts--with AC, what is congruent to A?2103

E is, so then, if I say AC, then I have to say EC.2112

What is the reason--why are those two sides congruent?--because this is "definition of segment bisector."2118

If it was an angle that was bisected, then it would be "angle bisector."2133

But since this is a segment, it is "segment bisector."2139

OK, so then, this one is the side that is mentioned; and then, angle 1 is congruent to angle 2.2144

What is the reason for that?--"vertical angles are congruent," so that is my angle.2157

I have three parts mentioned, so now I can say that my triangles are congruent; triangle ABC is congruent to...2169

A is corresponding to E, so it has to be triangle E...what is corresponding with B?--D; C.2183

What is the reason? Now, ASA is one method, but we didn't use ASA--we used Angle-Angle-Side.2194

Now, that is what I was talking about earlier.2210

Be careful, because I didn't mention it in the order of AAS; I mentioned an angle, and then I mentioned the side, and then I mentioned the angle.2213

It is OK if this is out of order, but just be careful if you are going to write it out on the side like this, like how I am doing.2226

Then, you don't put it in that order, ASA; you have to look at the diagram and see what the order is.2233

It is Angle-Angle-Side--it is not Angle-Side-Angle.2239

I just mentioned it in this order, but it is not the actual order of the diagram.2242

Just be careful with that; it is OK to list them out like this, but then, when it comes to the order,2249

look back and say, "Is it ASA? No, it is AAS."2254

And then, I am done with the proof, right?2262

The whole point of proving these two triangles congruent is so that I can prove that parts of the triangles are congruent.2267

So then, now that it is stated that the triangles are congruent, I can now state any of the corresponding parts congruent.2279

Now, I can say that AB is congruent to ED, because these are parts of these congruent triangles.2291

What is my reason?--"corresponding parts of congruent triangles are congruent."2301

The corresponding parts are congruent, as long as they are from congruent triangles.2310

OK, we are going to do one more proof on this.2316

Let's see what we have: AB is parallel to DC, and then, AD is parallel to BC.2328

Now, just like those slash marks, I have to write this out twice.2340

And then, I want to prove that angle A is congruent to angle C.2349

Is there any way that I can prove that those two angles are congruent?2354

No, I don't think that there is anything; how would you prove that those two angles are congruent?2363

Well, then, can I do it in two steps, where I can prove that these two triangles are congruent,2371

and then use CPCTC to say that these parts are congruent?2379

Let's say if I can prove the triangles congruent: well, if these two are parallel (remember parallel lines?),2385

here is my transversal; see, extending it out makes it easier to see.2393

Then, alternate interior angles, that angle with this angle of this triangle, are going to be congruent.2400

And then, since these two lines are parallel, the same thing here: angle 1 is going to be congruent to angle 4.2407

If you want to see that again, these are the two parallel lines; this is AB, and this is DC.2417

Here is the transversal; this is 3, and this is 2; see if they are parallel--then the alternate interior angles are going to be congruent.2425

The same thing is going this way: my transversal...here is angle 4; here is angle 1;2438

as long as they are parallel, then these two alternate interior angles are congruent.2447

So then, I have two angles; I have Angle-Angle, but then I need one more; I need three.2453

Remember: earlier, we looked at a diagram similar to this, where we have two triangles, and they share a side.2461

If they share a side, then automatically, I can say that this side to this triangle is congruent to this side to this triangle.2471

That is another one of those automatic things: you have vertical angles that are automatically congruent, and you have a shared side that is automatically congruent.2479

Now I have three parts: I have an angle; I have a side; and I have angles.2491

Now, in order to prove that this angle is congruent to this angle, I can first say that this whole triangle is congruent to this whole triangle.2501

And then, these parts of those congruent triangles are going to be congruent.2512

Statements/reasons: 1: AB is parallel to DC, and AD is parallel to BC; "Given."2521

Step 2: Angle 1 is congruent to angle 4; and then, my reason for angle 1 being congruent to angle 4,2544

and angle 2 being congruent to angle 3, is going to be the same.2562

My reason is going to be the same, so I can just include it in the same step.2566

I don't have to separate it: angle 2 is congruent to angle 3.2571

And then, both of those are going to be...you can say "alternate interior angles theorem,"2577

or you can just write it out: "If lines are parallel, then alternate interior angles are congruent"--you could just write it like that.2589

Step 3: What do I have so far? I have my angle listed, an angle stated, and another angle stated; and now I have to state my side.2607

DB is congruent to BD; now, notice how I didn't write BD and BD; I wrote DB, and then I wrote BD.2620

If I draw this out again, if I separate out the two triangles, this is D, and this B; this is D, and this is B.2638

This angle right here is actually corresponding with this angle right here.2656

See how, if I flip it around, this angle and this angle are congruent; this angle and this angle are congruent,2661

because this is angle 1, and then, this is angle 4; and then, we know that angle 1 and angle 4 are congruent.2678

So then, this and this are corresponding; so then, I have to say B and D.2685

So, if it is DB, then I have to say BD; does that make sense?2692

Here is my side that I am sharing; that is this side right here.2698

Since angle 1 is congruent to angle 4 here, this angle and this angle are corresponding parts.2705

So, if I say DB, then I have to say BD, because this is corresponding to this, and this one is corresponding to this one.2715

This one is congruent to this.2723

Even though the letters are the same, DB and DB here, look at the angles: this one is corresponding to this angle,2727

so then, if you mention D here first, you have to mention B first for the next one.2736

Step 3: This is the reflexive property--any time something equals itself, this side equals the same side, then it is the reflexive property.2744

Then, did I say all that I needed to say?--yes, so now I can say, since I have all three parts, that the triangles are congruent.2761

Triangle ABD is congruent to triangle...what is corresponding with A? C; what is corresponding to B?--remember the angle, D; B.2775

What is my reason? Is my reason Angle-Angle-Side?2793

No, I have to look at the diagram: I used the Angle-Side-Angle Postulate.2796

I used the angle, an angle, and a side, but not in that order; it is in this order.2806

But that is not it; the whole point wasn't to just prove the triangles congruent.2812

The whole point was to prove them congruent so that these parts would be congruent.2819

So, angle A is congruent to angle C, and the reason is CPCTC.2826

Now, remember again: if I want to use this CPCTC rule, first the triangles must be congruent.2836

So, here this has to be stated somewhere before you use CPCTC.2844

And once it is stated, then you can use it, saying that any corresponding parts will be congruent.2851

That is it for this lesson; we will do a little more of this.2860

We are going to go over more triangle stuff in the next lesson, so we will see you then.2865

Thank you for watching Educator.com.2870

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