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Mary Pyo

Mary Pyo

Proving Parallelograms

Slide Duration:

Table of Contents

I. Tools of Geometry
Coordinate Plane

16m 41s

Intro
0:00
The Coordinate System
0:12
Coordinate Plane: X-axis and Y-axis
0:15
Quadrants
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
Segment Addition Postulate
5:02
Example and Definition of Segment Addition Postulate
5:03
Segment Addition Postulate
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
Angle Addition Postulate
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
Adjacent Angles
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
II. 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
III. 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
IV. 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
V. 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
VI. Quadrilaterals
Parallelograms

29m 11s

Intro
0:00
Quadrilaterals
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
Quadrilaterals Summary
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
VII. 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
VIII. 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
IX. 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
Radius
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
Arc Addition Postulate
8:25
Arc Addition Postulate
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
Inscribed Quadrilateral
5:50
Inscribed Quadrilateral
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
X. 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
Quadrilateral
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
XI. 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
Radius
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
XII. 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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Proving Parallelograms

  • Parallelogram Theorems:
    • If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram
    • If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram
    • If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram
    • If one pair of opposite sides of a quadrilateral is both parallel and congruent, then the quadrilateral is a parallelogram
  • A quadrilateral is a parallelogram if:
    • Both pairs of opposite sides are parallel
    • Both pairs of opposite sides are congruent
    • Both pairs of opposite angles are congruent
    • Diagonals bisect each other
    • A pair of opposite sides is both parallel and congruent

Proving Parallelograms

Determine whether the following statement is true or false.

If AB ≅ CD and AD ≅ BC , then quadrilateral ABCD is a parallelogram.
True.
Fill in the blank in the statement with always, sometimes or never.
If both sides of opposite sides of a quadrilateral are parallel, then it is _____ a parallelogram.
Always.
Determine whether the following statement is true or false.
If the diagonals of a quadrilateral bisect each other, then it is a parallelogram.
True.
Determine whether the following statement is true or false.
If four angles of a quadrilateral are all congruent to each other, then the quadrilateral is a parallelogram.
True.

AE = 15, AD = 25, find the measurement of AC and BC .
  • AC = 2AE
  • AC = 30
  • BC = AD
BC = 25.

Parallelogram ABCD, AB = 4x − 6, CD = 3x + 8, find x.
  • AB ≅ CD
  • 4x − 6 = 3x + 8
x = 14.

Parallelogram ABCD, m∠B = 2x + 10, m∠A = 28, find x.
  • m∠A + m∠B = 180
  • 28 + 2x + 10 = 180
  • 2x = 142
x = 71.
Determine if the quadrilateral ABCD is a parallelogram.
A(3, − 2), B(2, 3), C( − 4, 4), D( − 3, − 3).
  • AB = √{(2 − 3)2 + (3 − ( − 2))2} = √{1 + 25} = √{26}
  • CD = √{( − 3 − ( − 4))2 + ( − 3 − 4)2} = √{1 + 49} = √{50}
  • AB ≠ CD
Quadrilateral ABCD is not a parallelgram.

Determine whether the following statement is true or false.
If quadrilateral ABCD is a parallelogram, then ∆ ABE ≅ ∆ CDE.
True.

Given: AD ||BC , ∠BAC ≅ ∠DCA
Prove: Quadrilateral ABCD is a parallelogram.
  • Statements; Reasons
  • AD ||BC; Given
  • ∠DAC ≅ ∠BCA ; Alternate interior angles
  • ∠BAC ≅ ∠DCA ; Given
  • AC ≅ AC ; Reflexive prop ( = )
  • ∆ BAC ≅ ∆ DCA ; ASA
  • AD ≅ BC ; Definition of congrent triangle
  • AD ||BC ; Given
  • Quadrilateral ABCD is a parallelogram; Parallelograms theorem.
S
tatements; Reasons AD ||BC; Given ∠DAC ≅ ∠BCA ; Alternate interior angles BAC ≅ ∠DCA ; Given AC ≅ AC ; Reflexive prop ( = ) ∆ BAC ≅ ∆ DCA ; ASA AD ≅ BC ; Definition of congrent triangle AD ||BC ; Given Quadrilateral ABCD is a parallelogram; Parallelograms 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.

Answer

Proving Parallelograms

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
  • Parallelogram Theorems 0:09
    • Theorem 1
    • Theorem 2
  • Parallelogram Theorems, Cont. 3:10
    • Theorem 3
    • Theorem 4
  • Proving Parallelogram 6:21
    • Example: Determine if Quadrilateral ABCD is a Parallelogram
  • Summary 14:01
    • Both Pairs of Opposite Sides are Parallel
    • Both Pairs of Opposite Sides are Congruent
    • Both Pairs of Opposite Angles are Congruent
    • Diagonals Bisect Each Other
    • A Pair of Opposite Sides is Both Parallel and Congruent
  • 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

Transcription: Proving Parallelograms

Welcome back to Educator.com.0000

For this lesson, we are going to use the theorems and the properties you learned in the previous lesson to prove parallelograms.0002

Turning the properties that we learned into actual theorems, if/then statements:0012

the first one: If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.0020

Now, these theorems have no name; we have no name for the actual theorem, so we actually have to write it all out.0030

If I say, "If opposite sides are congruent, then it is a parallelogram," you can shorten it in that way.0038

So, if you ever have to use this theorem on a proof, then you can just shorten this as your reason,0060

instead of having to write this whole thing out; "if opposite sides are congruent, then it is a parallelogram."0066

Do something like that; you can just shorten words and phrases.0071

Then, our conditional statement: as long as we have opposite sides being congruent...if this, then parallelogram.0076

And this just means "parallelogram"; or actually, I can write it all out; maybe that will not be as confusing: "then parallelogram."0097

The second one: "If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram."0113

As long as we have (just like the property we learned in the previous lesson) a parallelogram, then we know that both pairs of opposite angles are congruent.0122

In the same way, the converse would be, "If both pairs of opposite angles are congruent, then it is a parallelogram."0134

It is just basically saying that if the opposite angles are congruent, then it is a parallelogram.0162

So, as long as we can prove this or this, then we can prove that it is a parallelogram.0180

Now, we have other options, too; there are actually more theorems.0186

The third theorem that we can use to prove quadrilaterals parallelograms is on their diagonals.0191

If we can prove that the diagonals (you can just say "if diagonals") bisect each other, then it is a parallelogram.0202

You can shorten it in that way; if you can just prove that the diagonals bisect each other,0223

in that way, then you have proven that the quadrilateral is a parallelogram.0235

Oh, I had it right...parallelogram.0250

And the fourth one, the last one: "If one pair of opposite sides of a quadrilateral is both parallel and congruent, then the quadrilateral is a parallelogram."0255

This is one that wasn't on the previous lesson; this is actually not a property of a parallelogram.0267

This is just an extra theorem that says that if you can prove that only one pair of opposite sides is both,0279

parallel and congruent, then you can prove that it is a parallelogram.0295

Now, again, this is not a property of a parallelogram; it is just that you have to prove that one pair of opposite sides is both parallel and congruent.0310

That is one way that you can prove that it is a parallelogram.0320

With other theorems, you have to prove two pairs: the first one was two pairs of opposite sides being congruent;0325

the second one was two pairs of opposite angles being congruent; for this one, you have to prove that both diagonals bisect each other.0332

But for this one, this is the only theorem where it has one pair, but it just has to be two things about that one pair of sides.0341

So then, you can just shorten it by saying, "If one pair of opposite sides is parallel and congruent, then it is a parallelogram."0353

Maybe you can say something like that--just shorten it like that, in that way.0375

This right here--we are just determining if this quadrilateral is a parallelogram.0383

In the previous lesson, we did a couple of these; in that case, the problems before in the last lesson,0390

you knew that it was a parallelogram, but then you just had to show that the slopes are the same, show that the sides were congruent...0399

For this problem, we have to determine if it is a parallelogram.0409

We don't know that it is a parallelogram; so then, using the same methods, using the distance formula,0416

we have to see if it is going to come out to be the same.0421

If these two are the same, and these two are the same, then we have to say that it is a parallelogram.0424

So, it is the same thing; you are using the same methods.0434

Before, all you were doing was just showing the numbers of the parallelogram, showing that this is 5, and this is 5, too, and so on.0437

And that is it--just verifying; you were just giving the measurements of them.0448

But for this, we are actually proving that it is a parallelogram by finding distance or finding slope and seeing whether or not they are the same.0452

Again, you can use the distance formula, or you can use slope.0464

If you are going to use the distance formula to show that these opposite sides are congruent,0469

and that these opposite sides are congruent, then you are going to be using the first theorem we went over,0474

saying that if two pairs of opposite sides are congruent, then it is a parallelogram.0479

If I use slope and find the slope of AB, find the slope of CD, and they are the same, that is showing that they are parallel.0485

And then, I find the slope of AD and the slope of BC, and say that they are the same--they have the same slope, which means that they are parallel.0495

I am not using one of the theorems, because remember: we said that if you state that two pairs of opposite sides are parallel,0504

that is just the definition of a parallelogram; so by definition, we can say that it is a parallelogram, if we use slope,0516

because then we are showing that opposite sides are parallel.0523

We are not using one of the theorems; we are actually just using the definition of a parallelogram.0526

It doesn't matter which one you use; you can just use one of the theorems, or you can use the definition of parallelogram to show that they are parallel--whichever.0531

And then, the distance formula, if you wanted to use that, is the square root of0541

the first x minus the second x, squared, plus the first y minus the second y, squared.0548

Slope is y2 - y1, over x2 - x1, or rise over run.0558

Rise measures up/down; run measures left/right.0573

In this case, slope will probably be a little bit easier, because for slope, all you have to do is count.0580

You can just count how many units you are going up, down, left, and right, whereas with distance, you have to calculate each thing out.0587

This also: if you have the points written out for you, then this can be pretty easy.0597

But we are just going to use the rise and run to find the slope by counting.0606

When you move up, that is a positive number, and that is going to go on the top, in the numerator.0614

When you go to the right, it is a positive; when you go down, it is a negative; and when you go to the left, it is a negative.0622

So then, that is because when you go up, you are going towards the positive y-axis.0628

If you to the right, you are going towards the positive x-axis.0634

If you go down, then you are going towards the negative y-axis; you are going towards the negative numbers, so if you go down, it is a negative number.0637

If you move left, you are going towards the negative x numbers, so that is also a negative number.0644

From A to B: now, it doesn't matter if you travel from A to B, or if you go from B to A--it does not matter.0652

So, if we go from A to B, we are going to count up 3; remember: going up is positive, so that is positive 3, over...0659

we go to the right 1, so the slope is 3/1, or just 3. The slope of AB is 3.0670

For BC, I am going to count from B to C; so I am going to count up/down first, the rise; do that one first.0683

From B to C, I have to go down; I am going to go 1, 2, 3, 4; I have to go down 4; so the slope of BC is -40692

(because going down is negative)...then from here, I am going to go 1, 2, 3, 4.0704

So, I went to the right 4, and that is a positive, because I went to the right, which makes this slope -1.0710

From C to D (it doesn't matter if you go from D to C or C to D), if I want to go from C to D,0720

then I am going to count 1, 2, 3, down 3; so the slope of CD is down 3, which is -3, over...0726

from here, I am going to go left 1; left 1 is -1; so then, -3/-1 is 3.0737

And then, from D to A, I can go...the slope of AD is 1, 2, 3, 4; that is a positive 4, because I am going up 4;0749

then 1, 2, 3, 4...that is a negative 4; I am going to the left 4.0764

And that makes this a negative 1; so since AB and CD have the same slope, I know that AB is parallel to CD.0770

And BC and AD have the same slope; that means that they are also parallel.0794

So, BC is also parallel to AD; I have two pairs of opposite sides parallel.0802

So, by the definition of parallelogram, this is a parallelogram, so yes, quadrilateral ABCD is a parallelogram.0813

OK, let's just summarize over the different theorems that we can use to prove parallelograms, before we actually start our examples.0843

A quadrilateral is a parallelogram if any one of these is true.0856

You don't have to prove all of these; just prove one of them.0863

If you prove one of these, then you can prove that the quadrilateral is a parallelogram.0867

The first one: a quadrilateral is a parallelogram if both pairs of opposite sides are parallel.0873

That is the definition of parallelogram; so as long as you can prove (this is the definition of parallelogram)--0882

as long as you can show--that this side is parallel to this side, and this side is parallel to this side,0892

then by the definition of parallelogram, the quadrilateral is a parallelogram.0902

The second one: If both pairs of opposite sides are congruent...as long as you show0909

that this side is congruent to that side and this side is congruent to that side, then you can state that this is a parallelogram.0916

Both pairs of opposite angles are congruent: that means that this angle is congruent to this angle, and this angle is congruent to this angle.0925

And remember: it has to be two pairs of opposite angles being congruent.0937

Then, that is a parallelogram.0940

Diagonals bisect each other--not "diagonals are congruent," but "they bisect each other."0945

That means that this diagonal is cut in half, and this diagonal is cut in half.0953

Those two halves are congruent; then this is a parallelogram.0958

And then, this is the one that is a little bit different; we have seen these as properties, but the last one is a special kind of theorem0966

that says, "Well, if you can prove that one pair of opposite sides (it doesn't matter if it is this pair or this pair,0980

as long as you can prove that that one pair of opposite sides) is both parallel and congruent, then this will be a parallelogram."0986

So, if you have to prove parallelograms, you can just use any one of these five--whichever one you can use, depending on what you are given.0997

Then, you can do that to prove parallelograms.1006

Let's actually go through some examples now: the first one: Let's determine if each quadrilateral is a parallelogram.1012

In this case, the first one, I have one pair of opposite sides being parallel, and I have the other pair of sides being congruent.1022

Now, if you remember, from the theorems and the definition of parallelogram that we went over, none of them say that this is a parallelogram.1034

So, if I see that one pair of opposite sides is parallel, and the other side is congruent, that is not a parallelogram.1045

This could be a parallelogram, but there is no theorem, and there is no definition, that says this.1056

The closest one...well, there are a few; one of them says that it has to be both pairs of opposite sides being parallel.1063

We have one pair being parallel; if these two sides were parallel, then we could use the definition of parallelogram.1073

If both pairs of opposite sides are congruent...we have one pair that is congruent; this pair is not congruent, so then we can't use that.1079

And then, the last one, the special one that we went over--that has to be the same pair.1089

So, one pair, the same pair of opposite sides, being both parallel and congruent--then it is a parallelogram.1095

So, if these sides are both parallel and congruent, then we have a parallelogram.1104

Or these sides--if they were both parallel and congruent, then we can use that one; but it is none of those.1111

So, this one is "no"; we cannot determine it.1121

It could be a parallelogram, but we can't prove it, because there is no theorem--nothing to use to state as a reason, so this is a "no."1125

The next one: all I have are four right angles--nothing else; just four right angles.1135

Now, for this one, the theorem that has to do with angles is "opposite angles are congruent."1143

Now, this angle and this angle--are they congruent? Yes, they are.1151

This angle and this angle--are they congruent? Yes, they are.1156

So, this, therefore, is a parallelogram, so this one is "yes."1161

We can use that one theorem that says that two pairs of opposite angles are congruent.1166

Now, some of you are probably looking at this and thinking, "But that is a rectangle!"1173

Yes, it is a rectangle; we are actually going to go over rectangles next lesson.1177

But a rectangle is a special type of parallelogram, so rectangles are parallelograms.1183

So, is this a parallelogram? Yes, it is.1193

So, if we have a rectangle, then we have a parallelogram.1198

But then, without even thinking of rectangles, with this alone, just looking at the angles, opposite angles are congruent;1204

so we have two pairs of opposite angles being congruent.1212

By that theorem, we have a parallelogram; so this is "yes."1218

All right, the next one: Find the value of x and y to ensure that each is a parallelogram.1225

ABCD: now, we have to be able to find the value of x and y so that these two sides will be congruent, and then these two sides will be congruent.1233

If I want these two sides to be congruent, and find a number for y that will make these congruent, then I have to solve them being congruent.1246

5y is equal to y + 24; here, I am going to subtract the y; so that way, this will be 4y is equal to 24.1257

Then, I divide the 4 from each side, and y is equal to 6.1273

If y is 6, then this will be 30; if y is 6, then this will be 30; so then, that is the value for y.1280

And then, for x, again, I have to do the same thing: so, 2x + 3 is equal to 3x - 4.1289

I am going to subtract the 2x here; you can add the 4 to this, so 7 is equal to x.1299

If x is 7, then this will be 14; this will be 17; here, this is 21 - 4, is 17.1317

The next one: now, this looks like it would be a square or rectangle, but you can't assume that.1329

I don't have anything that tells me that these are right angles; I don't have anything that tells me anything, really.1338

I have to find x and y so that these diagonals will be bisected, because that is what I am working with.1346

Then, this and this have to be congruent; this and this have to be congruent.1353

I am going to make x + 1 equal to 2x - 3, and then subtract the x here; 1 = x - 3; add the 3; then, this is 4 = x.1361

Let's see, the y's: this y and then this...this one is y + 4 = 20 - 3y.1388

So, if I add 3y, this is 4y + 4 = 20; subtract the 4; 4y = 16; divide the 4, and y = 4.1405

So then, now, if x is 4, and y is 4, then these parts of the diagonals will be congruent.1426

Therefore, the diagonals bisect each other, and then as a result, this is a parallelogram.1437

The next example: Determine if the quadrilateral ABCD is a parallelogram.1446

We are given the coordinates of all of the vertices of the quadrilateral.1453

And then, we have to determine if it is a parallelogram.1460

I can just draw it out here; it doesn't matter how you draw it, as long as, remember, when we label this out,1467

it has to be ABCD, or vertices have to be next to each other; it can't be jumping over, so it can't be ACDB--none of that.1475

It has to be in the order, consecutive.1488

And I am drawing this just to show which coordinates are next to each other, which ones are consecutive.1493

Again, you can use slope, or you can use the distance formula.1502

Since we used slope last time, let's use the distance formula this time.1506

I am going to find the distance of AB, compare that to the distance of CD, and see if they are congruent.1510

And then, before you move on, why don't you just try those two and see if they are congruent,1518

because if they are not, then you don't have to do any more work; you can just automatically say, "No, it is not a parallelogram."1523

So, just do one pair of sides first; and then, if they are congruent, then move on to the next pair, and then see if they are congruent.1531

The distance formula is (x1 - x2)2 + (y1 - y2)2.1544

The distance of AB is the square root of 5 - 9, squared, plus 6 - 0, squared.1561

5 - 9 is -4, squared; plus 6 squared...this is 16 + 36, which is 52.1576

Now, you can go ahead and simplify it if you want.1596

Your teacher might want you to simplify it.1599

But since all we are doing is just comparing to see if AB and CD are going to be the same,1602

I can just leave it like that, and then see if CD is going to come out to be the same thing.1608

If your teacher wants you to actually find the distance of each side and show the distance,1614

and make it simplified or round it to the nearest decimal, then you have to simplify that.1620

Or else, if it is just to determine if it is a parallelogram, then you can just leave it.1629

A way to simplify that, though, just to show you: we know that 52 is not a perfect square.1634

So, what you can do is a factor tree: 52...2 is a prime number, and 26; 2--circle it--and 13.1642

So, this is the same thing as the square root of 2 times 2 times 13; and then, we know that this can come out as a 2.1658

So then, this is 2√13.1668

CD next: CD is, using these two, 8 - 3 squared, plus -5 - 2, squared.1678

Oh, -5 minus a -2...that is a plus.1698

And then, the square root of...this is 5 squared, plus -3 squared; 25 + 9...this is 34.1702

We found AB and CD, and they are not the same; let me just double-check.1722

Let's double-check our work; this is 5 - 9, squared; 6 - 0, squared.1730

And then, for CD, it is 8 - 3, squared, and -5 - -2, squared.1740

We have 16 + 36, which is 52, so √52; the square root of 5 squared plus -3 squared is 25 + 9, which is √34.1752

So, I know that, since these are not congruent (this is √52, and this is √34), they are different.1767

I can stop here; I don't have to continue and show my other two sides (again, unless your teacher wants you to).1782

If all I have to determine is if this is a parallelogram or not, then I can just stop here and say, "No, it is not a parallelogram."1790

No, quadrilateral ABCD is not a parallelogram, because opposite sides are not congruent.1801

If it was congruent, if they were the same, then you would have to go ahead and find the distance of BC,1818

find the distance of AD, and then compare those two.1823

For the last example, we are going to complete a proof of showing that it is a parallelogram.1829

Always look at your given; using your given, you are going to go from point A1840

(this is your point A; this is your starting point, and then this is your ending point; that is point B) to point B.1844

How are we going to get there?1852

Right here, we know that AD is parallel to BC; oh, that is written incorrectly, so let's fix that; AD is parallel to BC; they were both wrong.1855

AD is parallel to BC, and AE is congruent to CE.1888

We know that those are true, and then we are going to prove that this whole thing is a parallelogram.1897

In order to prove that this is a parallelogram, we have to think back to one of those theorems1906

and see which one we can use to prove that this is a parallelogram.1913

The first one that we can use is the definition of parallelogram.1917

If we can say that both pairs of opposite sides are parallel, then it is a parallelogram.1921

All we have is one pair; we don't know that this pair is parallel, or can we somehow say that it is parallel?1928

I don't think so; the only way that we can prove that these two are parallel is if we have an angle,1937

some kind of special angle relationship with transversals--like if I say that alternate interior angles are congruent,1946

same-side/consecutive interior angles are supplementary...if I say that corresponding angles are congruent...1957

if something, then the lines are parallel; I could do that.1964

For this one, it would be alternate interior angles--if they were congruent,1969

if it somehow gave me that, then I could say that these two lines are parallel, AB and DC.1973

And then, I could say that the whole thing is a parallelogram, because I have proved that it has two pairs of opposite sides being parallel.1980

But I can't do that, because I don't have that information.1989

Can I say that both pairs of opposite sides are congruent, from what is given to me? No.1994

Can I say that opposite angles are congruent? No.2002

I could say that these angles are congruent; they are vertical.2009

Or I could say that this angle and this angle are congruent, because they are vertical; but that is all I have with the angles.2013

Can I say that diagonals bisect each other?2020

Well, I have one diagonal that is bisected.2024

Can I somehow say that this diagonal is bisected?2027

I don't think so, just by being given parallel, congruent, and these angles--no.2034

Can I say that the last one works (remember the special theorem?)--one pair being both parallel and congruent?2040

We have that this pair is parallel; can we say that this pair is also congruent?2049

Well, because I can say that this angle is congruent to this angle...2056

let me do that in red; that way, you know that that is not the given...2067

since I know that these lines are parallel, if this acts as my transversal, I can say that this angle is congruent to this angle.2072

Remember: it is just line, line, transversal; angle, angle; do you see that?2091

This is B; this is D; this is this angle right here; and this is this angle right here.2104

I can say that those angles are congruent, because the lines are parallel.2114

Well, I can now prove that these two triangles are congruent, because of Side-Angle-Angle, or Angle-Angle-Side.2120

Therefore, the triangles are congruent; and then, these sides will be congruent, because of CPCTC.2130

And then, I can say that it is parallelogram, because of that theorem of one pair being both parallel and congruent.2140

Let me just explain that again, one more time.2149

I need to prove that this is a parallelogram with the information that is given to me.2152

All I have that is given is that this side and this side are parallel, and this and this are congruent.2157

From what is given to me, I can say that these angles are congruent, because they are vertical;2167

and I can say that these angles are congruent, because alternate interior angles are congruent when the lines are parallel.2174

The whole point of me doing all of this is to show this using a theorem that says, if one pair of opposite sides2182

are both parallel and congruent, then it is a parallelogram.2194

I want to show that this side is both parallel (which is given) and congruent, so that I can say that this whole thing is a parallelogram.2200

But the only way to show that this side is congruent is to prove that this triangle and this triangle are congruent,2209

so that these sides of the triangle will be congruent, based on CPCTC.2223

If you are still a little confused--you are still a little lost--then just follow my steps of my proof.2231

And then, hopefully, you will be able to see, step-by-step, what we are trying to do.2236

Step 1: my statements and my reasons (just right here): #1: the statement is the given,2241

AD is parallel to BC, and AE is congruent to CE; what is my reason? "Given"--it was given to me.2258

Then, my next step: I am going to say...2277

Now, the angles that are in red--that is not the given statement; it is not anything that is given, so I have to state it and list it out.2281

I am going to say, "Angle AED" (I can't say angle E, because see how angle E can be any one of these;2291

so I have to say angle AED) "is congruent to angle CEB"; what is the reason for that?--"vertical angles are congruent."2302

My #3: Angle ADE is congruent to angle CBE; what is the reason for that?2324

"If parallel lines are cut by a transversal," (now again, you can write it all out, or actually,2353

we could probably just say "alternate interior angles theorem"; or if your book doesn't have a name for that,2370

then you can just write it out) "then alternate interior angles are congruent."2379

Now, I am just writing it out for those of you that don't have the name for it.2390

If you do, then you can just go ahead and write that out, and that would just be "alternate interior angles theorem."2392

The fourth step: now that we said that we have this side with this side from the triangle,2400

AE inside CE (that is the side), then we have this angle with this angle; there is an angle;2408

and this angle with this angle--these are all corresponding parts of the two triangles.2419

Now, we can say that the two triangles...triangle AED is congruent to triangle CEB.2427

And that means that this whole triangle, now, is congruent to this whole triangle.2442

What is the reason? Angle-Angle-Side.2446

If you are unsure what this is, then go back to the section on proving triangles congruent.2451

And then, we just proved that these two triangles are congruent by Angle-Angle-Side--that reason.2459

And then, now that the triangles are congruent, we can say that any corresponding parts of the two triangles are congruent.2467

Now I can say that AD is congruent to CB, and the reason is CPCTC; and that is "Corresponding Parts of Congruent Triangles are Congruent."2477

Corresponding parts of the congruent triangles are going to be congruent.2494

So now that we have stated that those two sides are congruent, now we can go ahead and say that quadrilateral ABCD is a parallelogram.2500

And the reason would be "if one pair is both parallel and congruent, then it is a parallelogram."2515

Remember: this is point A and point B; this is the starting point and ending point.2539

So, see how I have this statement right here, and my last statement.2543

All we did was prove that these two sides are congruent, so that we could use the theorem that we just went over.2551

That is it for this lesson; thank you for watching Educator.com.2560

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