Home » Mathematics » Geometry » Angles and Arc

Mary Pyo

Mary Pyo

Angles and Arc

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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 (2)

1 answer

Last reply by: Stephen Yang
Thu Feb 28, 2019 5:34 PM

Post by Matthew Zhang on August 23, 2017

What if the arc is 180 degrees, and the other arc is also 180 degrees? Which is major and which is minor?

Angles and Arc

  • Central angle: An angle whose vertex is at the center of a circle
  • The sum of the measures of the central angles of a circle all adjacent to each other is 360 degrees
  • Know the difference between a minor and major arc
  • Arc measure: The measure of a minor arc is the measure of its central angle. The measure of a major arc is 360 degrees minus the measure of its central angle
  • The measure of a semicircle is 180 degrees
  • Arc Addition Postulate: The measures of an arc formed by two adjacent arcs is the sum of the measures of the two arcs
  • Arc Length: A part of the circumference proportional to the measure of the arc compared to the entire circle
  • Concentric Circles: Circles that lie in the same plane with the same center, but have different radii
  • Concentric circles, along with all other circles, are similar
  • Circles that have the same radii are congruent
  • Congruent circles are also similar circles
  • Two arcs of one circle with the same measure are congruent arcs
  • Congruent arcs also have the same arc length

Angles and Arc

Draw a central angle in circle A and name it.

∠BAC is the central angle.
Determine whether the following statement is true or false.
The sum of the measures of the central angles of a circle all adjacent to each other can be more than 360o.
False.
Write the minor arc and the major arc in the circle.
  • Minor arc: ⁀BC
  • Major arc: ⁀BEC
Minor arc: ⁀BC
Major arc: ⁀BEC
Find the measures of the minor arc and the major arc in circle A.
  • Minor arc: m⁀BC = 110o
  • Major arc: m⁀BEC = 250o
Minor arc: m⁀BC = 110o
Major arc: m⁀BEC = 250o
Determine whether the following statement is true or false.
If O is a point on ⁀MN, then m⁀MO + m⁀ON = m⁀MN.
True.
Find the length of arc BC, r = 8,
.
  • [120/360] = [x/(2πr)]
x = [120*2*3.14*8/360] = 17.2
Fill in the blank in the following statement with sometimes, never or always.
Concentric circles are ______ congruent.
Sometimes.
Determine whether the following statement is true or false:
Congruent arcs have same length.
True.

Circle A and circle C are congruent, AB = 12, find the length of ⁀DFE.
  • The radius of circle C is 12
  • m⁀DFE = 255o
  • [255/360] = [x/2p*12]
x = [255*2*3.14*12/360] = 53.4.

Circle A and circle C are congruent, determine whether ⁀BD is congruent to ⁀EF .
  • m⁀BD = 360 - 85 - 150 = 125
  • m⁀EF = 125
  • The radius of the two circles are the same
⁀BD is congruent to ⁀EF

*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

Angles and Arc

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
  • Central Angle 0:06
    • Definition of Central Angle
  • Sum of Central Angles 1:17
    • Sum of Central Angles
  • Arcs 2:27
    • Minor Arc
    • Major Arc
  • Arc Measure 5:24
    • Measure of Minor Arc
    • Measure of Major Arc
    • Measure of a Semicircle
  • Arc Addition Postulate 8:25
    • Arc Addition Postulate
  • Arc Length 9:43
    • Arc Length and Example
  • Concentric Circles 16:05
    • Concentric Circles
  • Congruent Circles and Arcs 17:50
    • Congruent Circles
    • Congruent Arcs
  • 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

Transcription: Angles and Arc

Welcome back to Educator.com.0000

For the next lesson, we are going to go over angles and arcs of circles.0002

The first angle we are going to go over is called the central angle.0009

A central angle is an angle within a circle whose vertex is at the center of the circle.0013

Here is an arrow to show you that this is a central angle.0020

This arc right here that the angle is hugging is called the intercepted arc, because it is intercepting the circle.0026

This right here...we are going to go over arcs in a little bit, but this is an arc--it is part of the circle.0037

The arc that the angle is essentially hugging is called the intercepted arc, so this is the intercepted arc.0045

Again, remember that this side of the central angle is a radius of the circle, and so is this side; both sides have to be radii of the circle.0062

The vertex is at the center; and this is called the central angle.0072

Here, we have a whole bunch of central angles; there is the center; and if we add up all of these central angles0079

(remember that no central angles can be overlapping with each other--they have to be adjacent;0092

they have to be side-by-side), all of these central angles together are going to add up to 360.0098

And we know that, if we go all the way around, a full circle is going to give us 360 degrees.0109

It is all of these central angles added up together--that forms a circle; all of these together are going to be 360 degrees.0119

Remember that, if you are missing one of the angles from here, but you are given three out of these four different central angles,0129

then you just have to add them up and subtract that from 360--that is the sum of central angles.0137

So now, arcs: remember: arcs are part of the circle--they are on the circle.0149

And these central angles--we have the same central angle for this circle and this circle--they are the same central angle.0159

But there are two different arcs that we can talk about.0166

Now, for this one, if we just look at this intercepted arc from here all the way to here, the arc right here, this is called the minor arc.0190

It is less than 180 degrees, so it is less than half; and it is the intercepted arc from the central angle.0191

So then, here we can call this arc AB; and the way you do that--you write AB, and you have to write a little arc above it.0202

We know that, if it is a segment, then we do a little flat line above it.0212

But because it is an arc, it has to be a little curve.0218

Now, that is a minor arc; now, for this one, the other arc, right here, from here all the way around,0225

like this, it is called the major arc, because it is bigger; it is greater than 180 degrees.0237

This is the major arc; the same central angle divides the circle up into two different arcs, the minor arc and the major arc.0251

Now, the major arc is a little bit different, because, if I say arc AB, then whoever I am talking to has to know which arc I am talking about.0261

If I just say arc AB, how are they going to know if I am talking about this arc or this big arc, the major arc?0273

When it comes to the major arc, there has to be another point here, like this; and you have to give it three variables.0282

You have to say arc ACB, with the same little arc over it.0292

If you just say arc AB, using two points, then you are talking about the minor arc.0302

If you are going to talk about the major arc, you have to name it with three variables: arc ACB.0308

And that is how you know if you are talking about the minor arc or the major arc.0317

Arc measure: the measure of a minor arc is the measure of its central angle.0327

So again, we know that the central angle is the angle whose vertex is on the center.0335

This central angle, this angle right here, and the intercepted arc have the same exact measure.0344

And it is important to know the difference between measure and length.0352

Now, we are going to go over length in a little bit; but the measure is the angle measure.0357

Here, whatever angle measure this is--let's say this is 100--then the measure of the intercepted arc is also going to be 100 degrees.0365

If I label this as AB, and this as D, then I can say that the measure of angle ADB is 100, and the measure of arc AB is also 100.0379

Keep in mind: the central angle and the intercepted arc have the same exact measure.0405

Now, that is the minor arc; if you want to look for the major arc, which is all this right here...0410

well, we know that all of the angles within the circle have to add up to 360.0418

So, you can just do 360 minus this 100, and then that will give you the measure of the major arc.0423

And we know that the measure of a semicircle (oh, that is a bad circle; I will do that again...let's say a circle like that,0432

but a little bit darker)...there is the center, and I am going to draw a segment (chord) through the center, which makes it a diameter;0452

the measure of this angle right here--even though it doesn't look like an angle, it is still an angle;0465

this still has a measure of 180 degrees--that is what it is saying that the measure of a semicircle is.0477

If this angle is 180 degrees, then all of this right here, the arc--that would be the intercepted arc of this angle, so this is also 180.0485

So, that is what it is saying: a semicircle, half the circle, has the measure of 180 degrees.0498

Arc Addition Postulate: The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs.0507

If you remember, from the beginning of the course, you went over the Segment Addition Postulate and the Angle Addition Postulate.0517

This is the exact same thing; we are just using arcs instead of segments and angles.0528

Arc Addition Postulate: If you have this arc right here, PR, and Q is between (it doesn't have to be the midpoint;0536

just anywhere between) P and R, then we know that the measure of arc PQ plus the measure of arc QR is going to be the measure of arc PR.0544

If Q is a point on PR, then the measure of PQ, plus the measure of QR, is going to equal the measure of arc PR.0563

That is the same exact thing as the Angle Addition Postulate and the Segment Addition Postulate.0573

Arc length: now, this is different from arc measure.0586

We went over arc measure already; we know that arc measure is the same measure as the central angle.0591

So then, if we are looking for the measure of this arc right here, it is the intercepted arc of the central angle, so they are going to have the same measure.0602

Length, however, is different: length is talking about the part of the circumference.0613

It is like if this whole thing is 2πr; then what is the length of just this part right here, this piece?0619

So, it is kind of like perimeter; you are looking for the part of the perimeter.0629

But you know that the perimeter for a circle is the circumference; it measures the outside, the length of the whole circle.0634

But the arc length...we are just trying to figure out the length of this small portion of it, so it is different than measure.0643

The way you do this: now, if you have the measure of the central angle here, θ,0652

and this is the radius, r, we are going to make this into proportions.0663

To find the length of arc AB, we are going to look for the measure of arc AB; that is what we are going to look for.0669

We are going to create a proportion, since we know that it is proportional; the arc with all of the measure is going to be proportional to the arc length.0680

We are going to create a proportion to help us solve for the arc length.0692

We are going to have to create a ratio for the measure, and then we are going to create a ratio for length.0700

And then, we are just going to make them equal to each other.0713

To find the ratio of the measure...here, from arc AB, this is the measure.0718

If this is θ, if the central angle has a measure of θ, then so will arc AB.0731

So then, that will be θ over the whole; it is kind of like part over whole, θ over...what is the measure of the whole thing? 360.0739

So, when it comes to measure, this is going to be our ratio.0756

And then, the length: for the length, the part is the one that we are looking for.0763

So then, this is going to be arc (I shouldn't put "measure"--just "the length of AB")...I am going to write x;0771

let's say that this is x right here; that is what we are looking for--that is the part.0782

And then, the whole thing: well, if you are trying to find the length of the whole circle, that is known as circumference,0789

so it would just be circumference, or you could write 2πr, which is the formula for circumference.0796

This is essentially the formula to figure out the length of the arc.0802

And all it is: it is just part and whole for measure, and the part and the whole for the length.0810

We know that the measure has to do with angles and degrees: the central angle is the part for the measure,0817

and then it is over the whole circle, which is 360; then with length, it is the length of the arc AB,0829

which is what we are trying to find, over the whole thing, 2πr.0837

Let's say, now, that θ (we are doing an example) is 50 degrees, and the radius is 10.0844

Let's solve for it: this is 50, so arc AB has an angle measure of 50, over 360, which is equal to x,0860

which is what we are looking for (we are looking for this, the arc length), over the whole length, which is 2π times 10.0878

So then, here we have a proportion; and we are going to just cross-multiply and solve for it.0890

If you have a calculator (I have a calculator here on my screen), you are going to do 2 times π times 10 times 50, which is going to equal 360x.0896

That is 2 times 10, times 50, times π; and then, from here, you can just divide the 360; and we get x as 8.73.0918

That is the arc length--that is the part of the circumference of the circle.0954

And we will do another example later on in this section.0962

Concentric circles are two or more circles that lie on the same plane (it is flat), and they share the same center.0967

This circle right here and this big circle right here--they are known as concentric circles, because they have the same center.0983

But concentric circles have to have a different radius, because the radius, obviously, for this one is going to be shorter than the radius for this big one.0992

They have different radii; and we know that they can't be congruent.1009

Concentric circles cannot be congruent, because they do have different radii.1019

And so, instead, they are going to be similar; and what makes them similar?1027

Remember: anything with the same shape, but a different size, is similar; so these two circles...1033

And all circles are actually similar; concentric circles are going to be similar...it doesn't matter if I draw another circle here;1041

I can draw a tiny little circle; it is still going to be similar to both of those, because it is the same shape.1049

All circles have the same shape; they just can be different sizes--some could be big; some could be small.1054

But since they do have the same size, they are all going to be similar; so just keep that in mind.1061

A few more things to go over: circles that have the same radii are congruent.1074

So, if you have a circle here, and that is r, and a circle here, and that is also r--the same variable--then they are going to be congruent.1081

And if you were to take one of these circles and place it on top of this one, then they would be exactly the same, because they have the same radius.1096

That is why they would be congruent.1106

Congruent circles are also similar circles; they are the same size and the same shape.1110

We know that, any time they have the same shape, whether or not they have the same size, they are similar.1118

So, congruent circles are also similar.1124

Two arcs of one circle with the same measure are congruent arcs.1128

Now, they have to be from the same circle, because we know that angle measures don't have to do with size.1135

We can have something really big, and we can have something really small, but they will have the same angle measure, or the same measure.1144

Within the same circle, two arcs with the same measure will be congruent.1153

If you have, let's say, from here to here, and from here to here--let's say that this arc right here is 90 degrees,1164

and this arc right here is also 90 degrees--then they will be congruent arcs.1178

Congruent arcs also have the same arc length: well, if they have the same measure (not angle measure, because they are arcs),1185

then they will also have the same length, because they are the same proportion of the whole circumference, and that is what length is.1198

So, that means that this is as long as this will be, because they have the same measure.1207

Going over examples: Name a minor arc, a major arc, and a semicircle.1216

A minor arc (and we have a few, so you can just name whichever one) can be arc AB, arc CD, arc DC, arc DA or AD.1223

Any of those can be minor arcs; so we can just say that a minor arc is arc AB.1241

A major arc: make sure that the major arc is greater than 180--that is what makes it major.1250

And you have to use three variables to name a major arc.1260

So, if we want to talk about this all the way to D, that would be a major arc.1265

And we have A, B, C, D, but you are not going to use all four; you can just use A...1276

you have to start with A, and you can name either B or C, just one of those; and then D would be the last one.1282

Or if you want to go the other way, then it would be D, then CA; or you could do DBA.1290

It doesn't matter, as long as you name one of the points that is in between those two.1298

A major arc would be arc ABD; if you said arc ACD, that would be the exact same thing.1306

And a semicircle: we know that a semicircle is exactly half, so this is a semicircle.1317

And for this one, you don't have to name three; you can just name two, because,1328

whether you go this way or whether you go that way, it will be the same measure, so I can just say arc AC.1334

You can also, if you want to specify which semicircle you are talking about, name the variable in the middle.1342

For our semicircle, we can say arc ABC, because AC can be this one or this one.1349

Sometimes, it doesn't matter; they have the same measure, and we know that a semicircle has a measure of 180.1359

But if I want to talk about this one, then it would be arc ABC.1366

The next one: Find the measure and the length of arc AB.1372

Here, that is arc AB; when it comes to measure, we know that the measure of arc AB (I can write it like this--1379

the measure of arc AB) is going to be the same as the central angle; this is the central angle; and that will be 96 degrees.1395

Now, for arc length, remember: we have to create that proportion.1408

I am going to create a proportion comparing the measure and the length.1414

For the measure, the part and the whole right here are going to be from the arc and from the whole circle.1426

From the arc, the measure is 96, over the whole circle, 360.1435

If you want, I can just write arc and circle.1444

And then, the arc length is what I am looking for; that is x, over the length of the circle (is the circumference, so)--that will be 2π times the radius, 2πr.1453

And that is 8; so then, from there, you are just going to solve this proportion.1469

This will be 16π times 96 equals 360x; and then, we divide the 360, so use your calculator; and x should be...1476

And your calculator should have a π button, but if it doesn't, then you could just use 3.14...and we get 13.40.1507

This is the arc length; now, if this was in, let's say, inches, then this will also be in inches; that is arc length.1529

The measure is this, and the length is that.1541

The next one: Tell whether the given arcs are congruent.1550

Here are the arcs: arc AB and arc EF; we are trying to see if they are congruent; and for this one, it is arc AC and arc BD.1555

Here, we know that arc AB has a measure of 75; so the measure of arc AB is 75.1573

And because they are from two different circles, I am just going to say that they are congruent,1592

because these two circles have to be congruent for us to compare their arcs.1599

So then, these two circles are congruent; this is arc AB, 75; and now, we can look to see if this arc is going to be congruent to that arc,1603

because otherwise, I don't know the radius; and if they have the same radius, then I can say that they are congruent.1614

And then, I can compare their arcs; but since I don't know their radius, you just have to know that those circles are congruent.1622

Now, for this one, I don't know what the measure is for EF, so I want to find...1629

since I know that this central angle has the same measure as the arc, as long as I find this angle measure, I can find the measure of that arc.1639

So, how do you do it? Right here, this is 110, 165, and then that is unknown.1648

All three angles together are going to add up to 360; so this can be 165 + 110 is going to be 275;1655

I can subtract that from 360, and that is going to be 85 degrees right here.1667

If this is 85, then this will be 85 also; so this one is "no"; the measure of arc EF is 85 degrees.1693

This one is 75; this one is 85; so this one is "no"--those arcs are not congruent.1711

And this one: here, this is 55, so the measure of arc AC is 55; and then, for BD, we have to find that one.1720

Now, this is 55, and 70...we are missing this one and this one here.1741

But remember that this arc CD is a semicircle; so this one has a measure of 180.1747

So then, to find this arc measure, you can just do 360 - (70 + 55 + 180 + x).1755

Now, you can also, because this is 180, know that this is also 180; you can do that, also.1778

If you want to use the whole circle, you are just going to do it this way; or you can just do 55 + 70 + x and that will be also 180.1789

Here, we are going to get x as 55; so then, this arc BD is going to also be 55; the measure of arc BD is 55.1798

Therefore, these two arcs are congruent; so then, "yes," they are congruent.1822

And the last one: Use the circle to find the following.1834

Here, we are going to look for a few different things; the first one is going to be x.1839

Here, the center is P, so that is circle P; and then, we have BC, and we have CD being congruent; we have this thing, a semicircle...1851

Here, how do we find x? Well, this one right here, arc BAE, is 180 degrees, because it is a semicircle.1870

So, I can just take this x, add it to 5x, and make that equal to 180.1881

5x + x = 180; 6x = 180; divide the 6; x is equal to 30.1888

The next one: The measure of angle CPD: now, to find this one, we know that this is 30;1907

this is 150; so for the remaining angle measures, the central angles, we want to look for this one.1925

Now, since we know that this is 30, this one and this one are congruent, because the arcs are congruent.1940

Well, here, we know that this is also a semicircle; so if I label this, let's say (what variables shall I use?), as y, then this angle will also be y.1948

That means that this one, plus this, plus that, is going to equal 180; so I can do 180 = 30 + y + y.1963

180 = 30 + 2y; that is 150, since I subtract the 30; that equals 2y; I am going to divide the 2; I am going to get 75 = y.1984

y, we know, is the measure of angle CPD; so this will be 75 degrees; this is 75; this is also 75; and this one will be 30.2001

Now, this is 30, and I know that because these are vertical angles.2023

Since this is a diameter, and EB is a diameter, these will be vertical angles.2031

Or you can also say that this BCE, this whole thing right here, is a semicircle; it is going to add up to 180. So then, that is 30.2037

The measure of arc ED: well, the central angle right here is 30, so then this has to be 30 degrees.2050

And the measure of arc EAC is all of this together, this right here: 150 and 30 and 75.2064

So, this measure of arc EAC is going to equal 150 + 30 + 75.2079

This is 180 + 75, which is going to be 255; so that is the answer there: x is 30.2093

That is it for this lesson; thank you for watching Educator.com.2120

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