Home » Mathematics » Geometry » Special Segments in Triangles

Mary Pyo

Mary Pyo

Special Segments in Triangles

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Table of Contents

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
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
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
Section 6: 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
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
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
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
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
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
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
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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Lecture Comments (4)

1 answer

Last reply by: Taylor Wright
Wed Jun 12, 2013 10:41 PM

Post by Jose Gonzalez-Gigato on February 3, 2012

As always, great lesson, Ms. Pyo. I do have one note: in Example III, I believe that the computation of the midpoint of segment RS is incorrect. For the y coordinate it should be: (6+(-2))/2 resulting them in (0,2) and not (0,4).

1 answer

Last reply by: Mary Pyo
Fri Feb 3, 2012 11:31 PM

Post by Dro Mahmoudi on November 13, 2011

you are the best, you and educator really saved me from a big nightmare

Special Segments in Triangles

  • Perpendicular Bisector: A line or line segment that passes through the midpoint and is perpendicular to that side
  • Perpendicular Bisector Theorems:
    • Any point on the perpendicular bisector of a segment is equidistant from the endpoints of the segment
    • Any point equidistant from the endpoints of a segment lies on the perpendicular bisector of the segment
  • Median: A segment that connects a vertex of a triangle to the midpoint of the side opposite the vertex
  • Altitude: A segment with one endpoint at the vertex and the other on the side opposite that vertex so that the segment is perpendicular to the side
  • Angle Bisector: A segment with one endpoint on the vertex and the other on the side opposite so that it divides the angle into two congruent angles
  • Angle Bisector Theorems:
    • Any point on the bisector of an angle is equidistant from the sides of the angle
    • Any point on or in the interior of an angle and equidistant from the sides of an angle lies on the bisector of the angle

Special Segments in Triangles

Find the point D on ―AB , so that ―CD is the median to ―AB .
  • A(3, 4), B( − 3, 2)
  • the midpoint of AB is point D ([(3 − 3)/2], [(4 + 2)/2])
D(0, 3).
, any point on ―AD is equidistant from ―AB and ―AC , determin what kind of segment is ―AD .
  
Angle bisector.
Draw and label to a figure to illustrate ―AM is the altitude of ∆ABC , M is on ―BC .
  
Draw the median ―AD of ∆ABC on the coordinate plane.
  
Determine whether the following statement is true or false.
Any point on the median of a segment is equidistant from the endpoint of the segment.
  
False.
, ―DE is the perpendicular bisector of ―BC , ―EC = 3x + 2, ―EB = 5x + 6, find x.
  • ―EC = ―EB
  • 3x + 2 = 5x + 6
  • − 2x = 4
x = − 2

∆ABC, A(3, 4), B(0, − 1), C(4, − 3), ―AD is the altitude of ―BC , find the line passes through points A and D.
  • the slope of BC is : [( − 3 − ( − 1))/(4 − 0)] = − [1/2]
  • the slope of AD is 2
  • then line AD is: y = 2x + b
  • line AD passes through A(3, 4)
  • 4 = 2*3 + b
  • b = − 2
line AD: y = 2x − 2.
Fill in the blank of the statement with always, sometimes or never.
Any point equidistant from the endpoints of a segment _____ lies on the perpendicular bisector of the segment.
Always

Given: ―AB = ―AC , ―AD is the median to ―BC
Prove: ―AD is also the altitude of ―BC .
  • Statements; Reasons
  • ―AB = ―AC; Given
  • ―AD is the median of ―BC; given
  • ―BD ≅ ―CD ; definition of median
  • ―AD ≅ ―AD ; refelxive prop ( = )
  • ∆ABD ≅∆ACD ; SSS
  • ∆ ADB ≅ ∆ ADC ; CPCTC
  • m∠ADB = m∠ADC ; definition of ≅∠s
  • m∠ADB + m∠ADC = 180o ; definition of linear angles
  • 2 m∠ADB = 180o ; substitution prop of ( = )
  • m∠ADB = 90o ; division prop of ( = )
  • ―AD ⊥―BC ; definition of perpendicular
  • ―AD is also the altitude of ―BC ; definition of altitude.
Statements; Reasons
―AB = ―AC; Given
―AD is the median of ―BC; Given
―BD ≅ ―CD ; definition of median
―AD ≅ ―AD ; refelxive prop ( = )
∆ABD ≅∆ACD ; SSS
∆ ADB ≅ ∆ ADC ; CPCTC
m∠ADB = m∠ADC ; definition of ≅∠s
m∠ADB + m∠ADC = 180o ; definition of linear angles
2 m∠ADB = 180o ; substitution prop of ( = )
m∠ADB = 90o ; division prop of ( = )
―AD ⊥―BC ; definition of perpendicular
―AD is also the altitude of ―BC ; definition of altitude.
Determine the following statement is true or false.
The median to the base of an isosceles triangle also bisects the vertex angle and is perpendicular to the base of the triangle.
True

*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

Special Segments in Triangles

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
  • Perpendicular Bisector 0:06
    • Perpendicular Bisector
  • Perpendicular Bisector 4:07
    • Perpendicular Bisector Theorems
  • Median 6:30
    • Definition of Median
  • Median 9:41
    • Example: Median
  • Altitude 12:22
    • Definition of Altitude
  • Angle Bisector 14:33
    • Definition of Angle Bisector
  • Angle Bisector 16:41
    • Angle Bisector Theorems
  • Special Segments Overview 18:57
    • Perpendicular Bisector
    • Median
    • Altitude
    • Angle Bisector
    • Examples: Special Segments
  • 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

Transcription: Special Segments in Triangles

Welcome back to Educator.com.0000

This next lesson is on special segments within triangles.0002

The first one (there are a few) is the perpendicular bisector.0007

Now, we know that "perpendicular" means that it is going to form a right angle.0014

And a "bisector" is a segment or an angle that cuts a segment or an angle in half.0021

So, a perpendicular bisector is going to be a line or a line segment that passes through the midpoint0030

(that is the bisector part of it) and is perpendicular to that side.0039

Two things: the segment has to pass through the side so that it is going to be perpendicular, and it is going to bisect that side.0045

So, if I want to draw the perpendicular bisector of the side AC, I have to draw a line or a line segment0055

that is going to be perpendicular to the side and is going to bisect it--it is going to cut it in half at its midpoint.0065

Let's say that that is the midpoint right there; that means that this is cut in half.0073

And then, I have to draw a segment that is going to be perpendicular and is going to bisect it--something like that,0079

where it is perpendicular and it bisects; so this right here is going to be the perpendicular bisector.0090

So again, the perpendicular bisector is a line or a line segment that is going to make it perpendicular and bisect the side.0098

Now, I could draw a perpendicular bisector for each side of the triangle; this is just for side AC.0106

But since I have three sides, I can draw three perpendicular bisectors: one for each side.0114

If you are only required to draw one, then you can just draw it like this.0121

You don't have to draw it all the way through; it can be a line or a line segment, so you can draw a line,0126

or you can just cut it right here and just make it a segment.0131

But that would just be for one of the sides; you can draw three, for each of the sides.0136

If you wanted to draw a perpendicular bisector for side AB, it might be helpful for you to just turn your paper so that this is the horizontal side.0142

So again, we want to have the midpoint, and then I am going to do that to show that that is the midpoint.0154

And then, I need to draw a segment that is going to be perpendicular, like that.0164

That is two; the third one is going to be for side BC.0179

So again, ignore this line right here; so then, that would be 1, 2, 3, to show that those two parts are congruent.0183

And then, draw...like that.0200

What should happen is that all three perpendicular bisectors, the ones that you draw for each of the sides, should all meet at one point.0217

Again, we have a perpendicular bisector; it is a line or line segment that is perpendicular to the side and bisects the side.0229

That means that it cuts it in half; that is a perpendicular bisector.0242

And then, for the perpendicular bisector, a couple of theorems: Any point on the perpendicular bisector of a segment...0249

when you just draw another triangle, and let's say my perpendicular bisector is right there,0257

then what this theorem is saying is that any point on the perpendicular bisector0281

(now, this is the perpendicular bisector)--any point on this line is equidistant from the endpoints of the segment.0284

So, this is the segment, AB; I could pick any point on this perpendicular bisector, and the distance from this point to this endpoint,0294

and this point to this endpoint, is going to be the same.0307

"Equidistant" means that it is the same distance; from this point to point A and this point to point B is going to be the same; that is what it is saying.0310

Or I could pick any point, maybe here; again, from this point to point A, and this point to point B, this is going to be the same.0322

That is what this one is saying.0334

And then, the next one: Any point equidistant from the endpoints of a segment lies on the perpendicular bisector of the segment.0335

This is just the converse of this; it is saying that if, without knowing where my perpendicular bisector is,0342

I draw two segments from point A and from point B out to the same point,0349

so that the two distances are the same, then it will lie on the perpendicular bisector.0356

The first one is saying that, if the point lies on the perpendicular bisector, then it is equidistant from the two endpoints, point A and point B.0362

The second one is saying that, if I draw two segments to a point equidistant from point A and point B, then it lies on the perpendicular bisector.0373

OK, so then, that was the first one; that was the perpendicular bisector; the second one is a median.0392

Now, when you think of median, think of middle or midpoint.0398

It is kind of like a perpendicular bisector; with perpendicular bisectors, we worked with the midpoint, too.0406

But that had two conditions; it had to be perpendicular to the side, and the bisector had to go through the midpoint of the side.0413

Median is just through the midpoint--just that one condition.0422

So, think of median as "middle"; but the condition here is this--it is going to go to the middle of the side, but from the vertex opposite that side.0426

So, the vertex opposite this side is B; it can't be A, and it can't be C; it has to be B.0449

So, when I draw a segment from the vertex to that point, this would be the median.0458

If I label this as D, BD is the median of this triangle, of this side, AC.0473

Again, the difference between this and a perpendicular bisector: a perpendicular bisector has nothing about the vertex.0481

You don't care about where the vertex is.0489

You just have to draw the segment so that it is perpendicular to the side, and it is cutting at the midpoint.0492

The median is just the segment from the vertex to the midpoint of the side opposite.0498

So then, again, since we have three sides of a triangle, I can draw three medians in a triangle.0509

This is just one median; the next median I can draw from...let's say that is the midpoint--I'll do that.0514

So again, I am going to draw it from this point, the segment with two endpoints; one is there, and one is at this vertex.0523

And then again, from this one, let's say it is about here: 1, 2, 3, 1, 2, 3...to show that these parts are the same, or congruent.0540

And then again, I am going to draw from there all the way to the vertex opposite.0554

And again, for a median, if you draw all three medians of a triangle, then it should meet at one point, right there.0566

That is the median; the median is like the middle; so far, we did "perpendicular bisector," and we did "median."0575

OK, let's do this problem for median: Find the point S on segment AB so that CS is a median.0583

I want to find a point labeled S on AB, this side right here, so that from C to that S is going to be a median.0597

Now, remember: median has to do with the midpoint or middle.0608

Remember: the median has two endpoints: one is from the vertex, so that means it is going to come from C, vertex C;0615

and it is going to go to the midpoint of AB, and that point is going to be labeled S.0623

So, I need to find the midpoint of AB, because CS has to be a median; that means S has to be the midpoint of AB.0630

How do I find the midpoint when I am given two points?0641

A is at 1, 2, 3...(-3,2); B is (1,-4); so I am just going 1, 2, 3, 4; positive 1, negative 4.0644

To find the midpoint of this point to this point, it is going to be (x1 + x2) divided by 2;0666

so I am going to add up the x's and divide it by 2; and then I am going to add up my y-coordinates and divide it by 2.0676

So, it is like the average; to find the midpoint, you are going to find the average of the x's and the average of the y's.0684

So then, for the x's, it is (-3 + 1)/2; for the y's, it is (2 + -4)/2; so this would be -2/2 and -2/2.0690

Well, -2/2 is -1, and -2/2 here is -1; that means (-1,-1).0709

This point right here is where S is; that means that, if I draw a line from C all the way to S, like that, that is a median,0717

because S is the midpoint between A and B; so here is point S.0731

The third segment is the altitude: now, altitude--just think of it as being perpendicular to the side.0744

It is kind of like the perpendicular bisector, except that there is no bisector--just the "perpendicular."0763

But then, one of the endpoints also has to be at the vertex.0769

One point is at the vertex, and one point on the side opposite, so that it is perpendicular.0777

A median only takes the "midpoint" side, and the altitude only takes the "perpendicular" side.0783

And then, both of them together is like the perpendicular bisector.0788

So, I just have to draw a segment going from C down to this side so that it is perpendicular, not caring about midpoint.0792

All I care about is that it is from this endpoint at the vertex, and it is perpendicular to the side.0804

Let's say...that right there; now again, there are three sides, so I need to draw three altitudes.0814

Now, it kind of looks like this BC is already the altitude; I'll just draw it like that.0829

And again, it is from this vertex to this side, so that it is perpendicular.0840

And then, where are they meeting?--right there.0855

So again, an altitude is from the vertex to the opposite side so that the segment is perpendicular.0858

This is the fourth one, the angle bisector; now, the angle bisector is bisecting (cutting in half),0875

but it is the angle that is being cut in half, not the segment, like the perpendicular bisector.0886

An angle bisector is a segment with one endpoint on the vertex, and it is coming out;0891

but for this one, because it is the angle bisector, we don't care where it lands on the side.0901

It doesn't matter where it lands; it is not going to be perpendicular; it is not going to be the midpoint...0907

it could be, but that is not what it has to be.0912

All it has to be: the condition is that it is coming from the vertex, and it is cutting the angle in half.0916

"Angle bisector" means that the angle is being cut in half.0924

So, all I care about is making sure that I draw a segment so that this angle is going to be bisected.0928

OK, well, let's just say that that is cut in half.0939

And then, if I draw something like that, let's say that is cutting in half the angles.0943

And then, if I draw that, we can say that this angle is cut in half.0954

See how this has no regard for where it is touching the side (as long as it is touching it).0969

But I am not saying that this has to be perpendicular, or it has to be at the midpoint--none of that.0976

The only thing is that the angle has to be bisected.0983

And again, I do three because there are three sides; I have to do it to each of those angles.0986

And then again, they meet at one point.0992

That is the angle bisector, where the segment is bisecting the angle from the vertex.0995

Any point on the bisector of an angle (let me draw a triangle again; I'll say my angle bisector is right there) is equidistant from the sides of the angle.1004

If I just have a point that is any point on this angle bisector, it is equidistant to the sides, like that, or like that.1031

Remember: if we want to find a distance from a point to a side, it has to be perpendicular.1042

Remember: if you are standing in front of a wall, how do you find the distance--how far away you are from that wall?1048

You don't measure at an angle; you have to measure directly, so that you are perpendicular.1053

That distance from you to the wall has to be perpendicular to the wall.1058

You can't just turn your body at an angle and find your distance that way.1064

The same thing works here: if you want to find the distance between this point and this side, this is you; this is the wall.1070

It has to be perpendicular; so let me draw this out again so that it looks like it is perpendicular.1075

You are going to go straight out like this; it is just saying that, if this is the angle bisector,1086

then any point on the angle bisector is equidistant; that means that the distance to the sides of the angle,1092

which are these two sides, is going to be equidistant.1102

And then, this is the converse; any point on, or in the interior of, an angle, and equidistant from the sides of the angle, lies on the bisector of the angle.1106

It is the same thing; it is just saying that, if I just find the distance to a point from the sides,1116

so that it is equidistant, then it is going to lie on the angle bisector.1125

On the angle bisector, any point is going to be equidistant from the sides.1131

OK, let's go over the four that we went over, the special segments of a triangle.1139

A perpendicular bisector: remember: it had to be perpendicular, and bisecting the side; that is the perpendicular bisector.1147

For a median, if this is the side of a triangle, then we don't care what this looks like, as long as this is bisected--the midpoint of the side.1173

The altitude: we don't care what these sides look like, as long as it is perpendicular.1190

The angle bisector: it is coming out from the angle of the triangle so that these are bisected.1204

A perpendicular bisector is going to be like this, like this, and like this; you can draw arrows or not; that is a perpendicular bisector.1221

For a median, it is from the vertex, so that it is congruent; from the vertex...that is the median.1256

The altitude is just drawn so that it is perpendicular, so it is like that.1286

And the angle bisector, the last one, is drawn so that it is bisecting the angles.1311

Now, this is supposed to be bisecting the angle.1331

Those are the four special segments.1338

Again, the perpendicular bisector has to be perpendicular and bisect the sides.1341

The median is just bisecting the side; the altitude is just perpendicular; the angle bisector is bisecting the angle.1346

Let's do our examples: Draw and label a figure to illustrate each: BD is a median of triangle ABC, and D is between A and C.1356

BD is the median, so I am going to draw triangle ABC; there is A, B, C; and BD--that means that it is coming out from here.1370

BD is the median of triangle ABC, and D is between A and C.1385

That means that, since we are dealing with BD as the median, D has to be the midpoint; there is D; there is BD.1391

And all you had to do is draw it and label it.1405

The next one: GH is an angle bisector of triangle EFG, and H is between E and F.1411

Triangle EFG: H is between E and F; GH is an angle bisector; so H is between E and F so that GH is an angle bisector.1421

Now, we don't care, as long as H is anywhere in between here; it is not going to be perpendicular;1444

it could be, but that is not the rule; the rule is that the angle is bisected.1453

This is like this, and then this point would be H.1461

So, GH is an angle bisector of that triangle.1471

Draw the altitudes for each triangle: I want to draw an altitude (remember: the altitude also has an endpoint on the vertex).1478

So, if I want to draw an altitude from A to side BC, it looks like this is already the altitude.1492

I can just say that this side right here would be the altitude of this side BC.1499

And then, B to AC is going to be like that, and then C to AB is going to be like...that is not right...let me erase part of my triangle...1503

so then, this BC would be the perpendicular bisector of AB, also.1527

This is going to be the altitude of BC; and then, CB is going to be the altitude of AB.1536

And then, for this one, this triangle is an obtuse triangle because angle B is greater than 90 degrees.1551

So, if I want to draw the altitude from this to the side, that is pretty easy; that just goes straight down.1559

But then, this one is a little bit different, because I obviously can't draw an altitude1570

from point A to somewhere between B and C, so that it will be 90 degrees.1577

So, what I have to do is extend this out a little bit (it is kind of hard to draw a straight line on this thing).1584

Let's say that that is my straight line; that is CB extending out.1595

Then, my altitude will have to go outside of this triangle, because, since it is an obtuse angle,1603

I would have to draw it on the outside so that it will be 90 degrees.1616

If I draw it on the inside, it is just going to be a bigger obtuse angle.1619

That is the altitude for that side; and then, if I want to draw the altitude from C to this side, AB, the same thing: it is an obtuse angle.1628

So, I am going to extend this out; I am going to draw from C all the way there so that it is perpendicular.1639

Those would be my three altitudes; now, if you want them to all meet, which they should, in this case, they all met right here.1651

For this one, you would just have to keep drawing this out, keep drawing this out,1659

and keep drawing this out, and then they would eventually meet right there.1664

But if you just have to draw the altitudes, then you would have to just draw that, draw that, and then draw this.1668

The coordinate points of triangle RST are those three points; AB is a perpendicular bisector, so it is this, through RS.1679

So now, I don't have to actually graph this out on a coordinate plane.1698

But if you are a visual person, and you like to see how it looks, then you can go ahead and plot them.1706

I am just going to do a little sketch of what it will look like.1715

So, if I have -2 right there, let's say this is R.1721

S, let's say, is -2; (2,-2) is S; and T is (5,4), so here is T.1727

So, my triangle is going to look something like this.1741

AB is a perpendicular bisector through RS; that means that my perpendicular bisector is through this side.1749

That means that this side is going to be the one that is perpendicular to it and that is bisected.1758

So, that is the perpendicular bisector, which means that it is going to look something like that; it is perpendicular.1765

And then, this is going to be A and B, this point and this point.1784

Find the point of intersection of AB and RS.1796

I want to find the point of intersection; now, first of all, we have to find the point of intersection between AB and RS.1805

And then, I want to find the slope of AB and RS.1817

So, the point of intersection, we know, is right there; we also know that the same point, that point A, is the midpoint of RS.1822

So, as long as I find the midpoint of RS, that would be the point of intersection between the two segments,1833

because again: this segment and this segment meet at point A, which is the midpoint of RS.1839

The point of intersection is going to be the midpoint.1846

#1: the midpoint of RS--I am going to use these two points; remember, to find the midpoint, I am going to add up my x's,1852

divide it by 2, and add up my y's...6 - -2, divided by 2; it is going to be 0, comma...minus a negative becomes a plus,1864

so it is plus 2, is 8; 8 divided by 2 is 4; so then, #1: the point of intersection would be (0,4).1881

And again, the reason why I did midpoint is because that is where they intersect: they intersect at the midpoint of RS.1892

And then, that is my answer; that is the point of intersection.1900

Then, #2: Find the slope of AB and RS.1903

I have points R and S; so to find the slope, it is (y2 - y1)/(x2 - x1); this is slope.1908

Again, using the same points, let's see: if I label this (x1,y1),1924

and label this as (x2,y2)...again, the x2 and y21933

is not talking about "squared"; make sure you write this 1 and these 2's below, not above it like an exponent.1942

And this is just saying the first and the first y; the second x and second y, because we know that this point is (x,y), and this is also (x,y).1958

So, these are the first (x,y)'s, and these are the second (x,y)'s.1967

So, y2 - y1 is -2 - 6 (and this is the slope of RS);1970

and then, x2 is 2, minus -2, which is -8 over...this becomes plus, so it is 4; this is -2.1984

The slope of RS is -2; now, I have to also find the slope of AB.1998

I don't know the point for B, but I don't have to know, because, if you have two lines,2009

and they are perpendicular to each other, then, remember: their slopes are negative reciprocals of each other.2020

So, now that I know the slope of RS, to find the slope of AB, it is just the negative reciprocal of it.2026

So, this is the slope of RS, and then the slope of AB is going to be the negative reciprocal; so that is the negative of (-1/2).2033

So then, this is going to be positive 1/2.2053

One is -2, and the other one is positive 1/2.2065

OK, the fourth example: Draw and label the figure for the statement; then write a proof.2074

The median to the base of an isosceles triangle bisects the vertex angle.2079

I am going to draw and label a figure, so I need an isosceles triangle.2084

Let's say that this is my isosceles triangle; the median to the base of it...we know that these are congruent, because it is the median...2097

of an isosceles triangle...bisects the vertex angle; that means that we want to prove...2116

Let me just label this; now, since this is how you want to draw and label it, you can just draw and label it however you want.2123

So, if I label this ABC, I can label this point as D.2130

So, my given statement--what do I know?--what is given?2138

I have an isosceles triangle, so I can say that triangle ABC is isosceles.2147

And then, I can say that BD is a median, because those are parts of the information that I have to use.2162

Now, I want to prove that it bisects the vertex angle; I am going to prove that angle ABD is congruent to angle...2176

if I said ABD, then I have to say angle CBD, because A and C are corresponding; so if I say ABD, then I have to say CBD.2197

OK, from here, I am going to do my proof: so my statements and my reasons...#1: Triangle ABC is isosceles,2211

and then BD is the median; and the reason for that is because it is given.2240

2: From here, I can say that AB, this side, is congruent to this side.2250

Now, let's see what we have to do: I am trying to prove that this angle right here is congruent to this angle right here.2259

Now, in order for me to prove that those two angles are congruent,2273

I would probably have to first prove that these two triangles are congruent,2280

because there is no way that I could just say that this angle is congruent to this angle.2285

But if I prove that these two triangles are congruent, then I can say that any two corresponding parts are congruent.2289

So then, once these two triangles are congruent, then these two angles can be congruent.2297

As long as they are corresponding, any two parts of the triangles are congruent.2303

Then, I have to focus on how I am going to prove that these two triangles are congruent.2308

Well, I know that these two sides are congruent; I know that these two sides are congruent.2313

And I can say that this side of this triangle is congruent to this side of this triangle; that is the reflexive property.2319

I can prove that these two triangles are congruent by SSS; if you remember the rules, there is SSS, SAS, AAS, and then Angle-Side-Angle, ASA.2330

So, I could do that, or I have another option; I can say that, because (remember) in an isosceles triangle,2344

if I have the two legs being congruent, then these two angles are also congruent--remember: the base angles are also congruent.2352

So, I can say that, too, and then prove that these two triangles are congruent through SAS.2360

Either one works; the important thing is that we prove that these two triangles are congruent,2366

so that we can say that these two angles, those two parts, are congruent.2372

It is up to you--do it however you want to do it.2379

I am just going to use the reflexive property, and say that this side is congruent, and use SSS.2382

So, I am going to say that AB is congruent to CB, and that is my side; the reason would just be "definition of isosceles triangle,"2390

because the definition of isosceles triangle just says that two legs are congruent--"two or more sides of a triangle are congruent."2405

And then, I am going to say that these two parts are congruent.2419

So, even though I have it shown on my diagram, I have to write it as a step.2422

AD is congruent to CD, and the reason for that--I am going to say "definition of median,"2430

because it is the median that made those two parts congruent--so it is just "definition of median."2443

And then, that is another side that I have; and then I can say, "BD is congruent to BD," and this would be BD of this triangle,2451

and this would be BD of the other triangle; so I am saying that a side of one triangle is congruent to a side of another triangle.2463

And that would be the reflexive property.2469

Now, if you chose to say that angle A is congruent to angle C, then you can say "isosceles triangle theorem" as your reason--2476

"isosceles triangle theorem" or "base angles theorem," because,2484

since that is an isosceles triangle, automatically the base angles are congruent.2488

And then, the fifth step...that would be another side, so then, your reason,2493

if you say that the triangles are congruent here (the next step), wouldn't be SSS, like mine would be; it would be SAS.2499

Triangle ABD is congruent to triangle CBD, and again, my reason is SSS.2510

Then, from there, I can say that angle ABD is congruent to angle CBD; what is my reason?2525

Well, see how here you proved that the triangles are congruent.2536

Then, once this is stated, then you can say that any two parts are congruent by "corresponding parts of congruent triangles are congruent," CPCTC.2541

So, that would be my sixth step.2558

Again, in order to prove that these two angles are congruent, because there is no direct way to do it,2561

I have to prove that these two triangles are congruent so that I can say2568

that two corresponding parts, those two angles, are going to be congruent.2573

And then, you can do that by Side-Side-Side or Side-Angle-Side.2577

And then, once you prove that the triangles are congruent, then you can say that those two angles are congruent.2581

When you draw and label, if it doesn't give you a figure or a diagram for it, then just draw your own.2587

You can label it how you want, and then that would base your given and your prove statement.2594

But as long as you write a proof for this statement that they give you, "The median to the base of an isosceles triangle2600

bisects the vertex angle," this would be your conclusion (your "prove" statement).2608

Your hypothesis is going to be your given, and your conclusion of your statement is your "prove" statement.2612

Well, that is it for this lesson; I will see you next time.2622

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