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Start learning today, and be successful in your academic & professional career. Start Today!  Mary Pyo

Midpoints and Segment Congruence

Slide Duration:

Section 1: Tools of Geometry
Coordinate Plane

16m 41s

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

17m 17s

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

31m 31s

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

42m 26s

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

42m 34s

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

19m

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

42m 47s

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

17m 24s

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

36m 3s

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

44m 31s

Intro
0:00
Properties of Equality
0:10
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
12:01
Multiplication and Division Properties
13:07
Exterior Angle Inequality Theorem
14:12
Example: Exterior Angle Inequality Theorem
14:13
Extra Example 1: Draw a Diagram for the Statement
18:32
Extra Example 2: Name the Property for Each Statement
19:56
Extra Example 3: State the Assumption
21:22
Extra Example 4: Write an Indirect Proof
25:39
Inequalities for Sides and Angles of a Triangle

17m 26s

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

28m 11s

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

29m 36s

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

29m 11s

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

42m 43s

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

29m 47s

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

39m 14s

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

30m 48s

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

20m 10s

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

27m 53s

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

34m 10s

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

24m 7s

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

27m 6s

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

21m 14s

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

40m 59s

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

37m 57s

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

40m 37s

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

21m 4s

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

35m 25s

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

52m 43s

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

22m 43s

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

35m 24s

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

21m 51s

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

27m 53s

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

26m 16s

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

27m 50s

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

23m 8s

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

27m 1s

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

27m 24s

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

17m 46s

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

20m 31s

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

36m 43s

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

18m 17s

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

38m 40s

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

23m 39s

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

38m 50s

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

26m 10s

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

21m 59s

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

22m 2s

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

14m 46s

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

16m 6s

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

14m 12s

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

23m 17s

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

18m 43s

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

21m 26s

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

37m 6s

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

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 0 answersPost by Hui Lu on August 31, 2018Why didn't instructor use statement / reasoning format in theorem proof? And I do feel some of the statement made by the instructor was not clear thought of. Makes the audience confused from time to time. 1 answerLast reply by: Ginger ChengFri May 17, 2019 9:36 PMPost by Mirza Baig on December 6, 2013nice video 0 answersPost by Shahram Ahmadi N. Emran on July 13, 2013Thanks 0 answersPost by saloni bhurke on February 15, 2012A good one. 2 answersLast reply by: Mark SimWed Mar 29, 2017 4:27 AMPost by Prakash Gopinathan on January 26, 2012Best video ever. 0 answersPost by Ahmed Shiran on June 4, 2011明白了！

### Midpoints and Segment Congruence

• The midpoint M of PQ is the point between P and Q such that PM = MQ
• On a number line, the coordinates of the midpoint of a segment whose endpoints have coordinates a and b is • In a coordinate plane, the coordinates of the midpoint of a segment whose endpoints have coordinates (x1, y1) and (x2, y2) are ### Midpoints and Segment Congruence

Find the midpoint of AB and AC on the number line. Midpoint of AC: [( − 6 + 4)/2] = [( − 2)/2] = − 1.
Midpoint of AB: [( − 6 + ( − 2))/2] = [( − 8)/2] = − 4
Find the midpoint of AB on the coordinate plane. • A(4, 3), B( − 4, 0)
• the midpoint is: ([(4 + ( − 4))/2], [(3 + 0)/2])
(0, 1.5).
A, B and C are on the same coordinate plane, C is the midpoint of AB , A(2x + 1, y − 5), B(4, 2y − 2), C(3x + 4, 9), find A.
• XB = [(XA + XC)/2], YB = [(YA + YC)/2]
• 4 = [((2x + 1) + (3x + 4))/2], 2y − 2 = [((y − 5) + 9)/2]
• 8 = 5x + 5, 4y − 4 = y + 4
• x = [3/5], y = [8/3]
• A (2 ×[3/5] + 1, [8/3] − 5)
A ([11/5], − [7/3]).
Draw a diagram to show AB is bisected by andEF. Draw a diagram to show ML = [1/2]MN and MP = [1/2]ML. Draw a diagram to show AB ≅ BC ≅ CD . C is the midpoint of AB , D is the mid point of AC , A (2, 4), B (6, 8). Find D.
• C is ([(2 + 6)/2], [(4 + 8)/2])
• C (4, 6)
• D ([(2 + 4)/2], [(4 + 6)/2])
D (3, 5).
EF bisects at M,CD bisects MB at N, AM = 4x + 1, BN = x + 3, find the measure of MN . • AM = BM = 2BN
• 4x + 1 = 2(x + 3)
• x = 2.5
MN = BN = x + 3 = 5.5.
Line m passes through point C(3, 5) and the midpoint of segment AB , find whether m passes D (3, − 2). • A (0, − 1), B (6, − 1)
• the midpoint of AB is ([(0 + 6)/2], [( − 1 − 1)/2]), which is (3, − 1)
m passes through point D.
Draw a line that bisects AB on a coordinate plane, A (0, − 1), B (4, 3)
• midpoint of AB is ([(0 + 4)/2], [( − 1 + 3)/2]), which is (2, 1)
Draw a line passes that through point (2, 1) on the coordinate plane, *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.

### Midpoints and Segment Congruence

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
• Definition of Midpoint 0:07
• Midpoint
• Midpoint Formulas 1:30
• Midpoint Formula: On a Number Line
• Midpoint Formula: In a Coordinate Plane
• Midpoint 4:40
• Example: Midpoint on a Number Line
• Midpoint 6:05
• Example: Midpoint in a Coordinate Plane
• Midpoint 8:28
• Example 1
• Example 2
• Segment Bisector 15:14
• Definition and Example of Segment Bisector
• Proofs 17:27
• Theorem
• Proof
• Midpoint Theorem 19:37
• Example: Proof & Midpoint Theorem
• 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

### Transcription: Midpoints and Segment Congruence

Welcome back to Educator.com.0000

This lesson is on midpoints and segment congruence.0002

We are going to talk about more segments.0007

The definition of midpoint: this is very important, the definition of a midpoint.0010

Midpoint M of PQ is the point right between P and Q, such that PM = MQ; that means PM is equal to MQ.0017

So, if I say that M, the midpoint...it is the point right in the middle of P and Q, so it is the "midpoint," the middle point--this is M...0033

then I can say that PM, this right here, is equal to MQ, because you are just cutting it in half exactly, so it is two equal parts.0049

I can write little marks like that to show that this segment right here, PM, and QM are the same.0058

That is the definition of midpoint; that means that, if PQ, let's say, is 20, and M is the midpoint of PQ, then PM is going to be 10; this is 10, and this is 10.0069

So then, if it is the midpoint, then this part will have half the measure of the whole thing.0084

OK, some formulas: the first one: this is on a number line--that is very important.0093

Depending on where we are trying to find the midpoint, you are going to use different formulas.0101

On the number line, the coordinates of the midpoint of a segment whose endpoints have coordinates a and b is (a + b) divided by 2.0106

So, again, only on a number line: if I have a number line like this, say...this is 0; this is 1; 2, 3, 4, 5;0116

if I want to find the midpoint between 1 and 5--so then, this will be a, and this will be b--0132

then I just add up the two numbers, and I divide it by 2.0140

That is the same...just think of it as average: whenever you try to find the midpoint on a number line, you are finding the average.0145

You add them up, and you divide by 2: so it is just 1 + 5, divided by 2, which is 6/2, and that is 3; so the midpoint is right here.0151

Now, if you are trying to find the midpoint on a coordinate plane, then it is different, because you have points;0171

you don't have just numbers a and b--you have points.0181

So, to find the midpoint with the endpoints (x1,y1) and (x2,y2),0184

you are going to use this formula right here: remember from the last lesson: we also used (x1,y1)0193

and (x2,y2) for the distance formula--remember that, for this one,0200

it is not (x2,y2), because that is very different.0205

This right here, these numbers, 1...it is just saying that this is the first point.0208

And then, (x2,y2): it is the second point, because all points are (x,y).0214

So, they are just saying, "OK, well, then, if this is (x,y) and this is (x,y)..." we are just saying that that is the first (x,y) and these are the second (x,y).0221

You are given two points, and you have to find the midpoint.0231

Then, you just take the average, so it is the same as this formula on the number line--0235

you are just adding up the x's, dividing it by 2, and that becomes your x-coordinate;0242

and then you add up the y's, and you divide by 2.0248

You are just taking the average of the x's and the average of the y's.0251

Remember: to find the average, you have to add them up and then divide by however many of them there are.0254

So, in this case, we have two x's, so you add them up and divide by 2.0260

For the y's, to find the average, you are going to add them up and divide by 2; and that is going to be your midpoint.0266

So, to find the average, you are finding what is exactly in between them.0272

Let's do a few examples: Use a number line to find the midpoint of AB.0279

Here is A at -2, and B is at 8; and this is supposed to have a 7 above it.0287

AB: to find the midpoint, to find the point that is right between A and B, I am going to add them up and divide by 2: so -2 + 8, divided by 2.0298

Again, just think of midpoint as average; so add them up and divide by 2.0317

It is going to be 6; -2 + 8 is 6, over 2; and then 3; so right here--that is the midpoint.0324

You can kind of tell if it is going to be the right answer; that is the midpoint, the point right in the middle of those two.0337

If I got 0 as my answer, well, 0 is too close to A and far away from B, so you know that that is not the right answer.0345

The same thing if you get 5 or 6 or even 7--you know that that is the wrong answer.0353

So, it should be right in the middle of those two points.0358

OK, to find the midpoint of AB here--well, we know that we are not going to use the first one, (a + b)/2,0363

because this is in the coordinate plane, and we have to use the second one, where we have (x1,y1) and (x2,y2).0377

So, in this case, since they don't give us the points, we have to find the points ourselves.0390

So, A is at (1,2); this is 1, 2, 3; and then, B is at (-1,-3).0394

For the midpoint, it is the same concept as finding it on the coordinate plane.0412

When you are finding the average, you are just finding the average of the x's; and then you find the average of the y's.0418

You just have two steps: all I do is take...0424

Now, it doesn't matter which one you label (x1,y1) and which is one is (x2,y2).0429

So then, we could just make this (x1,y1), and this could be (x2,y2); it does not matter.0436

You have to have this thing x and this thing y...both of these thing x's and both of these thing y's.0446

Let's see: x2 + x1, or x1 + x2, is 2, plus -1, divided by 2;0456

that is the average of the x's; comma; the average of the y's is going to be 3 + -3, over 2.0467

Here we have 1/2; and 3 + -3 (that is 3 - 3) is 0; our midpoint is going to be at (1/2,0); so this right here is our midpoint.0479

After you find the midpoint, kind of look at it and see if it looks like the midpoint.0499

A couple more problems: If C is at (2,-1), that is the midpoint of AB, and point A is on the origin, find the coordinates of B.0511

C is the midpoint of AB; so if I have AB there, and C is the midpoint right there (there is C), point A is on the origin; find the coordinates of B.0529

Then C is (2,-1), and then A is (0,0): now this is obviously not what it looks like; it is on the coordinate plane, since we are dealing with points.0543

But this is just so I get an idea of what I am supposed to be doing, because in this type of problem, they are not asking us to find the midpoint.0555

They are giving us the midpoint, and they want us to find B--they want us to find one of the endpoints.0562

So, we know the midpoint; so how do we solve this?0570

The midpoint is (2,-1): well, the formula I know is...how did you get (2,-1)?--how did you get the midpoint?0573

You do x1 + x2; you add up the x's and divide it by 2;0584

and then to find the y-coordinate, you add up the y's, and you divide it by 2.0590

And that is 2, and this is -1; so this is the formula to find the midpoint; this is the midpoint.0599

All of this equals this, and all of this equals that.0610

I can just say, if we make A (0,0)--if this is our (x1,y1)...then what is this going to be? (x2,y2), right?0617

It is as if our coordinates for B are going to be the x2 (this is what we are solving for)...0633

we are solving for x2, and we are solving for y2.0640

Those are the two points that we need.0644

Going back to this--well, we know what x1 was, and we know that this whole thing equals this whole thing.0649

So, I am going to make this a 0, plus x2/2; all of that equals 2.0657

When you found the average of the x's, you got 2; but you just don't know what this value is.0670

Then, if you solve for x2, you get 4; so x2 = 4.0681

Then, you have to do the same thing for the y's; so you add up the y's; this plus this, divided by 2, equals -1.0689

You just write that out; 0 + y2, divided by 2, equals -1.0703

And this is what I am solving for, again; so what is y2? y2 became -2, because you multiply the 2 over: -2.0714

That means that our coordinates for B are (4,-2).0728

So again, you can use the formula to find one of the endpoints, too.0741

If they give you the midpoint, then just use it; you have to use this formula to come up with these values, too.0748

So then, we know that the sum of the x's, divided by 2, is 2; so you do 0 + x2 = 2.0759

And then, the sum of the y's, divided by 2, is -1; so you do 0 + y2, divided by 2, equals -1.0768

OK, we will do another example later.0778

The second one: E is the midpoint of DF; let me draw DF; and E is the midpoint.0782

DE, this, is 4x - 1; and EF is 2x...this must be plus 9...find the value of x and the measure of DF.0799

They want us to find x and the value of the whole thing.0816

Since we know that E is the midpoint, we know that DE and EF are exactly the same.0821

So, I can just take these two and make them equal to each other: 4x - 1 = 2x + 9.0830

And then, you solve for x; subtract it over: 2x equals...you add 1...10...x equals 5.0838

Then that is one of the things they wanted us to find: x = 5.0847

And then, find the measure of DF; how do you find the measure of DF?0853

Well, I have x, so I am able to find DE, or I can find EF.0858

Let's plug it into DE: 4 times 5...you substitute 5 in for x...minus 1; there is 20 - 1, which is 19.0866

If this is 19, then what does this have to be? 19.0879

Just to double-check: 2 times 5 is 10, plus 9 is 19.0883

You can just do 19 + 19, or you can do 19 times 2; you are going to get DF = 38, because it is this times 2...19 + 19...it is the whole thing.0888

Segment bisector: Any segment, line, or plane that intersects a segment at its midpoint.0915

A segment bisector is anything that cuts the segment in half.0922

It bisects the segment, meaning that it cuts it in half; so bisecting just means that it cuts it in half; think of it that way.0931

Here, these little marks mean that this segment and this segment are the same.0939

CD is the segment bisector of AB, because CD is the one that cut AB in half.0948

Now, the one that is doing the cutting--the one that is bisecting, or the one that is cutting in half--is the segment bisector.0959

This bisected the segment AD; this is a line segment, CD; this segment bisector is a segment.0968

It doesn't have to be just a segment; it could be a segment; it could be a line; it could be a plane.0978

In this case, it is a segment; if you draw it out like this, it is still a segment bisector, because it is the segment that was bisected.0985

The bisector can be anything: it can be in the form of a line, too.0999

It can also be in the form of a plane; so if I have (I am a horrible draw-er, but) something like this, and it bisects it right there,1003

then a plane could be a segment bisector, because it intersects the segment at its midpoint, point D.1024

Again, a segment bisector is anything that cuts the segment in half, that bisects it, that intersects it at its midpoint.1035

So, we are going to just talk a little bit about proofs, because the next thing we are going to go over is a theorem.1051

And we have to actually prove theorems.1059

Remember: we talked about postulates--how postulates are statements that we can just assume to be true,1062

meaning that once they give us a postulate, then we can just go ahead and use it from there.1069

Theorems, however, have to be proved or justified using definitions, postulates, and previously-proven theorems.1074

So, whenever there is a theorem--some kind of statement--then it has to be backed up by something.1082

It has to show why that is true; and then you prove it.1090

And then, once it is proven, you can start using that theorem from there on, whenever you need to.1094

Now, a proof is a logical argument in which each statement you make is backed up by a statement that is accepted as true.1102

You can't just give a statement; you have to back it up with a reason: why is that statement true?1109

And that is what a proof is.1115

Now, there are different types of proofs; the one that is used the most is called a two-column proof.1116

But we are going to go over that, actually, in the next chapter.1125

The paragraph proof is one type of proof, in which you write a paragraph to explain why a conjecture for a given situation is true.1128

So, you are just explaining in words why that is true.1140

And a conjecture is an if/then statement: if something is true, then something else.1145

So, actually, you are going to go over that more in the next chapter, too.1153

But that is all you are doing; for a paragraph proof, you are just explaining it in words.1156

So, instead of a two-column proof, where you are listing out each statement, and you are giving the reasons for that,1162

to prove something, in a paragraph proof, you are just writing it as a paragraph to prove that a theorem is true.1168

The first proof that we are going to do is going to be a paragraph proof.1180

And that is to prove the midpoint theorem; so again, this is a theorem; it is not a postulate; so we can't just assume that this statement is true.1186

The midpoint theorem says that, if M is the midpoint of AB (here is AB), then AM is congruent to MB.1196

Now, we know, from the definition of midpoint, that if M is the midpoint of AB, then they have equal measures; that is the definition of midpoint.1208

And that is just talking about the measures of them.1223

But "congruent"--to show that AM is congruent to MB--that is the theorem, and we have to prove that first.1226

Given that M is the midpoint of AB, write a paragraph proof to show that AM is congruent to MB.1241

And then, the only purpose of this proof that we are going to do right now is to prove that this midpoint theorem is true.1250

And then, from there, we can just use it.1260

We are going to always start with the given: given that M is the midpoint of AB--that is the starting point.1265

Let's just write it out: From the definition of midpoint, we know that, since M is the midpoint of AB, AM is equal to MB.1278

Now, that means that AM and MB have the same measure--that AM has the same measure as MB.1306

Then, by the definition of congruence...the definition of congruence is just when you switch it from equal to congruent, or from congruent to equal.1333

So, by the definition of congruence, if AM is equal to MB, then (and I can just switch it over to congruent), AM is congruent to MB.1352

They are congruent segments--something like that.1379

You don't have to write exactly the same thing, but you are just kind of showing that we know1385

that we went over the definition of midpoint, and that is AM = MB.1389

And then, from there, you use the definition of congruence to show that AM is congruent to MB.1393

They are congruent segments--that is what the definition of congruence says.1401

Now that we have proven that the midpoint theorem is correct, or is true, then now, from now on,1410

for the remainder of the course, you can just use it whenever you need to--the midpoint theorem.1419

Extra Example 1: Use a number line to find the midpoint of each: BD.1427

We are going to find the midpoint of BD; again, to find the midpoint, you want to find the point that is right in the middle.1434

So, you are going to add up the two points and divide it by 2; so it is -2 + 9, divided by 2; that is 7/2.1442

You can leave it like that, or you can just write 3 and 1/2.1456

Between -2 and 9, it is going to be right there: three and a half.1465

CB: you know, I shouldn't write it like this, because it looks like BD...this looks like distance.1475

You can just write the midpoint of BD: so just be careful--don't write BD; don't write it like that.1496

Just write midpoint of BD, or...I am just going to write the number 2, just so that we know that that is the midpoint of CB.1505

CB is right here; now, you can go from B to C, or we can go from C to B.1514

It doesn't matter; if I go from C to B, 4 + -2, over 2, is going to be 2 divided by 2, which is 1.1520

That means the midpoint from here, C, to B, is 1, right there.1535

You can also check; you can count; this is three units, and then this is three units, so it has to be the same.1545

And then, number 3: AD: A is -5; D is 9; divided by 2...this is 4, divided by 2, which is 2.1553

Here is -5, and here is 9; the midpoint is right here, between those two...from here to here, and from here to here.1568

OK, draw a diagram to show each: AC bisects BD.1586

That means that this is the one that is doing the bisecting; this is the segment bisector.1591

This is the segment that is getting bisected: so BD...here is BD.1597

Now, it doesn't have to look like mine, just as long as BD is getting bisected, or AC is intersecting BD at BD's midpoint.1606

You can just...if I say that this is the midpoint, then AC is the one that is cutting it like this; it is going to be A, and then C (it cuts it)...1619

OK, the next one: HI is congruent to IJ.1638

HI: see how this is congruent...this is part of the midpoint theorem...HI and IJ.1644

Here is HI; I has to be that in the middle; and J; so HI is congruent to IJ, and that is how you would want to show it; OK.1657

The next one: RT equals half of PT--let me just draw another segment; RT is equal to half of PT.1671

Now, look at what they have in common; here is T, and here is T.1683

Now, if RT is equal to half of PT, that means that you have to divide PT by 2 to get RT.1689

That means that the whole thing is going to be PT, because, if you have to divide it by 2, you cut it in half, meaning at its midpoint.1701

And that is going to be RT; that means R is right here--the midpoint.1712

Again, here is PT; so for example, if PT is 12 (say this whole thing, PT, is 12),1720

then you divide it by 2, or you multiply it by 1/2; then you get 6; that means RT has to equal 6.1728

And you don't have to write the numbers; just draw the diagram.1742

You could have just left it at P-R-T, or just showed it, maybe, like this.1747

The next example: Q is the midpoint of RS; if two points are given, find the coordinates of the third point.1754

RS's midpoint is Q; I'll show that; if two points are given, find the coordinates of the third point.1768

R is here; R is at (4,3); S is at (2,1); you have to find the midpoint.1785

To find the midpoint, you are going to take the sum of the x's, divided by 2, and that is going to be your x-coordinate.1799

So, x1 + x2, divided by 2, equals your x-coordinate for the midpoint.1808

4 + 2, divided by 2...and then you are going to take the y's, 3 + 1, divided by 2;1817

that is going to be 6 divided by 2, which is 3; so Q is 3, comma...4 divided by 2 is 2;1831

it is going to be at (3,2); there is the midpoint for that one.1844

The next one: let me just redraw R-Q-S...Q...on this one, they give you the midpoint, and they give you this.1851

This right here is going to be...you could make this (x1,y1), or (x2,y2).1872

And then, when you write it out, it is going to be -5 + x2 over 2, and then 2 + y2, divided by 2.1882

Now, we know that all of this equals...the midpoint is (-2,-1).1904

That means that these equal each other and these equal each other.1912

This is going to give you -2; and this is what we are solving for.1915

I can just make this thing equal to this thing: -5 + x2, over 2.1927

Now, this is not x2; be careful of that.1933

That equals -2; this is going to be -5 + x2 equals...you multiply the 2 over to the other side...-4.1936

You add the 5 over; so x2 = 1.1948

I am going to do the y over here; then you make this equal to -1; it is 2 + y2, over 2, equals -1.1956

You multiply the 2 over; 2 + y2 = -2; y2 = -4.1967

All right, S is going to be (1,-4); that is to find this right here.1977

With this one, we had to find Q, the midpoint; and with this one, we had to find S.1995

The last example: EC bisects AD; that means that EC is the one is doing the bisecting, and AD is the one that got bisected.2002

That means that AD is the one that got cut in half at C.2018

That means that this whole thing, AC, and CD are congruent.2022

EF bisects...this is supposed to be a line...and let's make this F, right here; that means that line EF bisects AC at B.2030

That means that AC is bisected; B is the midpoint.2050

For each, find the value of x and the measure of the segment.2056

That means AC and CD are the same, and then AB and BC are the same.2060

AB equals 3x + 6; BC equals 2x + 14; they want you to find AC.2074

Again, AB and BC have the same measure; so I can just make them equal to each other...it equals 2x + 14.2087

Subtract the 2x over here; subtract the 6 over there; you get 8.2101

And then, find the value of x and the measure of the segment.2109

The segment right here, AC, is AB + BC; or it is just AB times 2, because it is doubled.2113

So, AC...here is my x, and AC equals...we will have to find AB first--or we can find BC; it doesn't matter.2126

AB is 3 times 8, plus 6; that is 24; that is 30.2142

24 plus 6 is 30; and then, AC is double that, so AC is 60.2152

AB is 30; that means that AC is 60.2161

AD, the whole thing, is 6x - 4; AC is 4x - 3; and you have to find CD.2172

Now, remember how EC bisected AD; that means that AD is cut in half; C is the midpoint.2184

AD, the whole thing, is 6x - 4; and then, AC is 4x - 3.2192

I can do this two ways: AC is 4x - 3--that means DC, or CD, is 4x - 3.2198

So, I can just do 4x - 3 + 4x - 3, or I can do (4x - 3) times 2; or we can do the whole thing, AD, 6x - 4, minus this one.2208

I am just going to do 2 times (4x - 3), because AC is 4x - 3, and AD is double that.2224

Now, even though this is 60, so you might assume that this whole thing is 120; this is a different problem.2238

So then, it is not going to have the same measure.2244

2 times AC equals AD, which is 6x - 4.2251

If we continue it here, it is going to be 8x (and this is the distributive property), minus 6, equals 6x minus 4.2261

This becomes 2x; if you add the 6 over there, it equals 2; x = 1.2275

So then, here is the x-value; and then, they want you to find CD, this right here.2283

Since CD is the same thing as AC, I can just find AC.2291

AC is 4 times 1, minus 3; so AC is 4 minus 3, is 1; so CD is 1; AC is 1; CD is 1.2298

OK, the last one: AD, the whole thing, is 5x + 2; and BC is 7 - 2x; find CD.2325

Now, this one is a little bit harder, because they give you AD, the whole thing, and they give you BC.2341

Now, remember: if C is the midpoint of AD, that means that this whole thing and this whole thing are the same.2353

Let's look at this number right here; if this is 60, then this will also be 60.2367

That means that the whole thing together is going to be 120.2374

Then this right here is 30; this right here is 30.2380

It is as if you take a piece of paper and you fold it in half; if you fold it in half, that is like getting your C, your midpoint.2389

Then, you take that paper, and you fold it in half again; so then, that is when you get point B.2399

So, now your paper is folded into how many parts? 1, 2, 3, 4.2405

One of these, AB...if you compare AB or, let's say, BC, to the whole thing, AD, then this is a fourth of the whole thing.2418

So, this into four parts becomes the whole thing; so you can take BC and multiply it by 4, because this is one part, 2, 3, 4.2429

BC times 4: 7 - 2x...multiply it by 4, and you get the whole thing, AD.2446

This is 28 - 8x = 5x + 2; I am going to subtract the 2 over; then I get 26; I am going to add the 8x over, and I get 13x; so x is 2.2456

And then, find CD: CD was 2 times BC, because BC with another BC is going to equal CD.2478

I am just going to find BC: 7 - 2(2)...just plug in x...so 7 - 4 is 3.2492

If BC is 3...now, these numbers right here; that was actually for number 1;2504

those are the values for number 1, and then I just used it as an example for number 3.2511

But don't think that these are the actual values (30 for BC, and then 120 for the whole thing); this is a different problem.2516

BC was 3; then what is CD? CD is 2 times BC--it is double BC, so CD is 6.2526

OK, well, that is it for this lesson; thank you for watching Educator.com.2542

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