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Mary Pyo

Mary Pyo

Squares and Rhombi

Slide Duration:

Table of Contents

I. Tools of Geometry
Coordinate Plane

16m 41s

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

17m 17s

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

31m 31s

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

42m 26s

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

42m 34s

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

19m

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

42m 47s

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

17m 24s

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

36m 3s

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

44m 31s

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

41m 2s

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

33m 37s

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

37m 35s

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

41m 53s

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

44m 6s

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

25m 55s

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

19m 48s

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

28m 43s

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

44m 43s

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

26m 46s

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

47m 51s

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

27m 53s

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

43m 44s

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

26m 34s

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

33m 30s

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

17m 26s

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

28m 11s

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

29m 36s

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

29m 11s

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

42m 43s

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

29m 47s

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

39m 14s

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

30m 48s

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

20m 10s

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

27m 53s

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

34m 10s

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

24m 7s

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

27m 6s

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

21m 14s

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

40m 59s

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

37m 57s

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

40m 37s

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

21m 4s

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

35m 25s

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

52m 43s

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

22m 43s

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

35m 24s

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

21m 51s

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

27m 53s

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

26m 16s

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

27m 50s

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

23m 8s

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

27m 1s

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

27m 24s

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

17m 46s

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

20m 31s

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

36m 43s

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

18m 17s

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

38m 40s

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

23m 39s

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

38m 50s

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

26m 10s

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

21m 59s

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

22m 2s

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

14m 46s

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

16m 6s

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

14m 12s

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

23m 17s

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

18m 43s

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

21m 26s

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

37m 6s

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

0 answers

Post by Oscar Wang on December 7, 2014

I really love your handwriting. It's so symmetrical and neat! :)

0 answers

Post by cinzia zullian on December 29, 2011

I am little confuse, i think that in the extra example II should be find the m<BEC.

Squares and Rhombi

  • Rhombus: Quadrilateral with four congruent sides
  • Plural form of rhombus is rhombi
  • Diagonals of a Rhombus:
    • The diagonals of a rhombus are perpendicular
    • If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus
    • Each diagonal of a rhombus bisects a pair of opposite angles
  • Square: Quadrilateral with four right angles and four congruent sides

Squares and Rhombi

Determine whether the following statement is true or false.
If the four sides of a quadrilateral are congruent, then the quadrilateral is a rhombus.
True.
Fill in the blank with sometimes, never or always.
A rhombus is ______ a parallelogram.
Always.

Determine whether the following statement is true or false.
ABCD, if AC ⊥BD , then ABCD is a rhombus.
True.

Rhombus ABCD, m∠ADE = 2x + 8, m∠DAE = 3x + 2, find x.
  • m∠ADE + m∠DAE = 90o
  • 2x + 8 + 3x + 2 = 90o
  • 5x = 80
x = 16.
Fill in the blank in the statement with sometimes, never or always.
A square is ____ a rhombus.
Always.
Fill in the blank in the statement with sometimes, never or always.
A rectangle is _____ a square.
Sometimes.
In quadrilateral ABCD, if a pair of opposite sides are congruent and parallel, and the diagonals are perpendicular to each other, then quadrilateral ABCD is a
A. Parallelogram
B. Square
C. Rhombus
D. Rectangle
A and C.

Rhombus ABCD, m∠AEB = 3x + 6, m∠CDE = x + 8, find m∠ CDE.
  • m∠AEB = 90
  • 3x + 6 = 90
  • x = 28
m∠CDE = 28 + 8 = 36.

Quadrilateral ABCD, if AB = BC = CD = DA, and AC = BD, then quadrilateral ABCD is
A. Parallelogram
B. Square
C. Rectangle
D. Rhombus
A and C.
Determine whether quadrilateral ABCD is a parallelogram, a rectangle, a rhombus, or a square with the given vertices.
A( − 4, 1), B( − 2, − 3), C(2, − 1), D(0, 3).
  • AB = √{( − 2 − ( − 4))2 + ( − 3 − 1)2} = √{4 + 16} = 2√5
  • BC = √{(2 − ( − 2))2 + ( − 1 − ( − 3))2} = √{16 + 4} = 2√5
  • CD = √{(0 − 2)2 + (3 − ( − 1))2} = √{4 + 16} = 2√5
  • DA = √{( − 4 − 0)2 + (1 − 3)2} = √{16 + 4} = 2√5
  • So AB = BC = CD = DA
  • AB ≅ BC ≅ CD ≅ DA
  • AC = √{(2 − ( − 4))2 + ( − 1 − 1)2} = √{36 + 4} = 2√{10}
  • BD = √{(0 − ( − 2))2 + (3 − ( − 3))2} = √{4 + 36} = 2√{10}
  • AC = BD
  • AC ≅ BD
Quadrilateral ABCD is a square.

*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

Squares and Rhombi

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
  • Rhombus 0:09
    • Definition of a Rhombus
  • Diagonals of a Rhombus 2:03
    • Rhombus: Diagonals Property 1
    • Rhombus: Diagonals Property 2
    • Rhombus: Diagonals Property 3
  • Rhombus 6:17
    • Example: Use the Rhombus to Find the Missing Value
  • Square 8:17
    • Definition of a Square
  • Summary Chart 11:06
    • Parallelogram
    • Rectangle
    • Rhombus
    • Square
  • 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

Transcription: Squares and Rhombi

Welcome back to Educator.com.0000

In the next lesson, we are going to continue on with parallelograms.0002

And we are going to go over, more specifically, squares and rhombi.0005

First, the rhombus: now, rhombus and rhombi are actually the same thing.0011

Rhombus is singular, so when you have only one, then it is a rhombus; when it is plural--you have more than one--then it is actually called rhombi.0018

So then, if you hear "rhombus" or "rhombi," then you are talking about the same thing.0028

Now, what is a rhombus? A rhombus is a quadrilateral with four congruent sides.0034

Here is an example of a rhombus: now, more specifically, a rhombus is a special type of parallelogram.0043

You can say that a rhombus is a parallelogram with four congruent sides.0050

This right here, this property, is very specific to the rhombus.0055

Now, to continue our flowchart, if we have a quadrilateral, quadrilateral goes down to parallelogram;0063

and then, parallelogram...we went over the rectangle; that was a special type of parallelogram;0079

and then now, we are going to go over the rhombus: it is another special type of parallelogram.0089

Now, because the rhombus is a special type of parallelogram, all of the properties of a parallelogram now apply to the rhombus,0100

just like when we went over rectangles: since a rectangle is a special type of parallelogram,0110

all of the properties of parallelograms apply to the rectangle.0115

The same happens with a rhombus; all of the properties apply to the rhombus, as well.0118

We went over one property that is very specific to the rhombus; it is all of the properties of the parallelogram, plus four congruent sides.0127

Now, there are also a couple of properties that have to do with the diagonals of a rhombus.0136

The first one: the diagonals of a rhombus are perpendicular.0142

Now, the property on diagonals for a parallelogram is that they just bisect each other.0147

They bisect each other; so we know that diagonals bisect each other--that is the very general property of the parallelogram.0153

Then, for the rectangle, it became that they bisect each other, and they are congruent.0173

And then now, the diagonals of a rhombus bisect each other and are perpendicular.0181

Let's say I have a rhombus that...here are my diagonals; we know that diagonals bisect each other.0193

That means that this diagonal and this part are congruent.0205

And then, the other two halves are congruent to each other.0209

And then, this one, the one that is a little more specific to the rhombus, is that they are also perpendicular.0213

So, if it is a rhombus, then the diagonals are perpendicular.0223

And this is the theorem that goes with that; so if the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus.0229

This property is very specific to the rhombus, meaning that this property only applies to the rhombus--0241

so much so that you can actually use it to prove that it is a parallelogram.0248

So, if you can prove that the diagonals are perpendicular, then you can prove that that parallelogram is a rhombus.0253

Any time you see anything that is perpendicular--diagonals being perpendicular--you know automatically that that is a rhombus--0263

of course, as long as it is a parallelogram (it is a parallelogram with diagonals perpendicular--then, a rhombus).0269

The next property that has to do with the diagonals of a rhombus is that they bisect opposite angles.0277

There is my rhombus; there are my diagonals; we know that it is perpendicular; we know that the sides are congruent.0296

But then, this one now says that the diagonals bisect opposite angles.0308

So then, it is bisecting those angles, meaning that this angle is now cut into half.0316

And then, these angles are bisected, and these angles are bisected, so that, when they bisect a pair of opposite angles,0325

the opposite angles are also congruent, because we know that opposite angles are congruent,0336

because that is a property of a parallelogram: Opposite angles are congruent.0340

This whole angle is congruent to this whole angle; so this is cut in half, and each of these halves are also congruent to each other.0345

Again, there are two more properties of the rhombus, in addition to all of the properties of a parallelogram.0355

#1: Diagonals are perpendicular; #2: Diagonals bisect opposite angles.0362

Using those properties, let's talk about finding the missing value.0379

Again, here is our rhombus; and you know that angle 1 is congruent to angle 2,0387

because this diagonal bisects the angles, because it is a rhombus.0403

That means that I can just make the measure of angle 1 equal to the measure of angle 2: 2x + 6 = 3x - 19.0410

I am going to solve for x by subtracting the 2x over to the other side; I am going to get 6 = x - 19.0425

I am going to add 19 to the other side, and I get 25 = x; so there is my x.0433

Now, if I need to find the measure of angle 1, I am just going to plug it in; so the measure of angle 1 equals 2(25) + 6.0442

So, the measure of angle 1 is...this is 50, plus 6 is 56.0456

Now, you know that the measure of angle 1 is 56; the measure of angle 2 is also going to be 56.0462

But just to check your answer, you can find the measure of angle 2: 3 times...x is 25...minus 19.0469

3 times 25 is 75, minus 19...that is 56.0482

So then, we know that our answer is correct.0492

Now, next is the square: squares are actually very, very special.0498

How are they special? If you look at the definition, a quadrilateral with four right angles and four congruent sides,0505

we already heard these properties--we have already talked about these properties before.0516

If we break it down, four right angles--what has that property of four right angles?0524

We know that that belongs to the rectangle; this is the rectangle's property, four right angles.0534

And then, a quadrilateral with four congruent sides--that one belongs to something else, too, and that is the rhombus.0544

A rhombus is a quadrilateral with four congruent sides.0555

So, that means that a square is made up of the rectangle and the rhombus, which is why a square is a special type of both.0559

It has the properties of both the rectangle and the rhombus.0570

Continuing with our little flowchart: a quadrilateral goes down to a parallelogram, and then a parallelogram...0575

we went over two types, the rectangle and...we just went over...the rhombus;0595

and we know that a square is a special type of rhombus and rectangle; and that is the square.0606

Remember that a rectangle has all the properties of a parallelogram, plus its own.0622

And then, a rhombus has all of the properties of a parallelogram, plus its own properties.0633

So then, the square has all of the properties of a rectangle (since it is a type of rectangle);0642

it has all of the properties of a rhombus, which means that it also has all of the properties of a parallelogram.0650

So, a square is made up of all of these above; a square is just a special type of everything.0656

Now, here is that chart again; but let's actually go over each of the properties.0667

We know that a parallelogram is two types, rectangle and rhombus; and then, a square is a type of rectangle and rhombus.0674

But what about their properties?--let's go over their properties again.0683

A parallelogram, we know, has...the definition of a parallelogram says...two pairs (let me write that out) of opposite sides parallel.0686

And this is more of the definition of a parallelogram: two pairs of opposite sides parallel.0711

The properties: two pairs of opposite sides are congruent; two pairs of opposite angles are congruent;0717

and the diagonals bisect each other; and then, one more--consecutive angles are supplementary.0738

That is everything that has to do with a parallelogram.0766

Now, the rectangle: we know that a rectangle has...0772

Instead of listing all of these out, because we know that rectangles have all of the properties of a parallelogram,0793

instead of writing all of these out, we will just say "all properties of parallelogram."0798

That means that it includes all of this.0805

We know that it has four right angles, and then, what about their diagonals? Diagonals are congruent.0809

Those are the properties of a rectangle.0831

For the rhombus, again, it has all properties of a parallelogram, and then, four congruent sides;0835

and then, their diagonals are perpendicular, and the diagonals bisect the angles.0862

Those are the properties of a rhombus; now, what about a square?0882

I know that a square has all of the properties of a parallelogram, so I am going to write that out: "all of parallelogram;0888

all of the properties of a rectangle"; and then, "all properties of the rhombus."0899

A square doesn't really have anything that is specific to its own; it just takes on all of the properties of everything else; that is a square.0916

That is why it is a little bit special; it is just a mixture of everything.0925

That is a summary chart; now, above this, we know that it is a quadrilateral, just a four-sided figure;0932

and then, it is more specific to the parallelogram, and so on down.0938

Let's now go into our examples: for our first one, we have this table, and all of the properties of diagonals.0946

So, you are going to write "yes" or "no" in each of the boxes, depending on if the diagonal property applies to that type of quadrilateral.0956

We have our parallelogram, rectangle, rhombus, and square.0971

The first property: The diagonals bisect each other.0975

Now, try to remember what property that is for--"the diagonals bisect each other."0982

That is a property of a parallelogram; so this one would be "yes."0989

Now, if that one is "yes," that means that this one applies to all of the other ones, because rectangles have all of the properties0998

of a parallelogram; so does a rhombus, and so does a square; this would be "yes," "yes," and "yes."1006

Now, the next one: Diagonals are congruent.1018

Well, I know that not all parallelograms have congruent diagonals; so then, this would be "no."1023

Or you can just leave it blank, just so you can see which ones are actually "yes"; it is just easier to see.1031

Let's look at this: what about rectangles--do rectangles have congruent diagonals?--and this one is "yes."1040

Does the rhombus have congruent diagonals?--this one is "no"; it does not.1050

The diagonals of a rhombus are perpendicular to each other, and they bisect the angles, but they are not congruent to each other.1057

That is not a property of a rhombus.1067

Now, if you ever get confused, just draw it out; and you know that, if you had to walk from here to here,1069

that just seems a lot further than walking from here to here; the distance looks a lot longer this way than it does this way.1080

So, they don't seem congruent, and the same thing with a parallelogram; so then, this would be "no."1086

But what about a square? Squares have all of the properties of the rectangle.1093

So, whatever applies to the rectangle also applies to the square, so this one is "yes."1099

Then, the next one: Each diagonal bisects a pair of opposite angles, meaning it does this.1106

Now, we have only gone over this one time, because it is very specific to one thing, and that is the rhombus; it is a property of the rhombus.1123

Now, again, squares have all of the properties of the rhombus; so then, this will also be "yes."1135

And then, the last one: The diagonals are perpendicular--which one is this for?1144

This one is also for the rhombus; the rhombus had both of these two diagonal properties, so this one is "yes";1151

and again, squares have all of the properties of the rhombus, so this one is also "yes."1158

And then, all of the blanks will just be "no"; so you can just fill it in with "no" if you have it written down.1164

Otherwise, this way, you can just see what is "yes" and what is "no."1170

OK, the next example: we are going to use the rhombus ABCD to find the missing value.1177

We have two problems; this is a problem, and this is another problem.1186

The first one: The measure of angle ABC is 120; find the measure of angle ACB.1193

We are given that this one is 120, and we don't know this one right here; this is what we are looking for, right here--this little part, angle ACB.1205

Let's see: we know that this is 120; now, we can do this a couple of ways.1219

We can say that, since this is 120, we can use a property of the parallelogram, because a rhombus has all of the properties of a parallelogram.1229

We can use the property of a parallelogram that says that consecutive angles are supplementary.1241

So then, if angle ABC is 120, then that would mean that angle BCD is the supplement to that; so that would be 60.1250

This whole thing is 60; and then, since it is a rhombus, we know that the diagonals are perpendicular, and that it bisects the angle.1262

These are cut into two equal parts; so if this whole thing is 60, then this has to be 30--it has to be half.1274

The measure of angle ACB is 30 degrees.1283

Now, another way to find that is to say that, if this whole thing is 120, then this would be 60;1288

this is a right angle; then, this is a triangle, so this would be a 30-60-90 degree triangle,1298

because the three angles of a triangle add up to 180.1308

A 30-60-90 triangle is a special right triangle; you just know that this is 90, so that means that these two angles have to be 90.1313

That is another way to find this angle measure.1323

Now, the next one: The measure of angle AED, this one right here, is 4x + 10; find x.1327

That is all that they are going to give you--that it is 4x + 10.1345

But since we know that this is perpendicular (remember: the diagonals are perpendicular), this angle measure is 90.1350

So, make 4x + 10 equal to 90, since that is the angle measure here; and then, we know that it is equal to 90, so 4x = 80; x = 20.1357

The next example: Determine if each statement is always, sometimes, or never true.1387

If all sides of a quadrilateral are congruent, then it is a square.1394

Well, all of the sides being congruent--that is a property of a rhombus.1409

And a square is a special type of rhombus, but not all rhombi are squares.1421

Sometimes a rhombus is just a rhombus; it doesn't have to always be a square, so this one would be "sometimes."1427

If all of the sides of a quadrilateral are congruent, then it is always a rhombus; but a rhombus is sometimes a square.1439

If a quadrilateral is a parallelogram, then it is a rhombus.1450

Well, a parallelogram is not always a rhombus; it is sometimes a rhombus, because a parallelogram could also be a rectangle.1457

So, a parallelogram is only sometimes a rhombus.1470

The next one: If a quadrilateral is a rhombus, then it is a parallelogram.1480

Well, yes, because a rhombus is always a parallelogram--it is a special type of parallelogram.1485

Just like we said, if we have a dog, a type of dog is, let's say, Chihuahua.1489

Well, a Chihuahua is always a dog; since a Chihuahua is a type of dog, a Chihuahua is always going to be a dog.1495

Just like that, a rhombus is a special type of parallelogram; so a rhombus is always a parallelogram; so this is going to be "always."1506

The last one: If a quadrilateral is a square, then it is a rectangle.1517

Is a square always a rectangle? It is almost the same as #3: a rhombus is a type of parallelogram; therefore, it is always a parallelogram;1523

a square is a special type of rectangle; therefore, a square is always a rectangle.1532

Now, if this is still confusing, we can use that flowchart that we did to help us with this.1540

If I say, let's say, "parallelogram," instead of writing it all out--we know that that is a parallelogram--1549

a parallelogram became a rectangle, and it became a rhombus; then this and this became a square; on top of that is a quadrilateral.1559

Now, if it is going from "if" to "then" downwards, then the answer is going to be "sometimes."1574

If it is going from "if" to "then" upwards, then it is "always"; and if it is going side-to-side, then it is "never."1588

Let's look at these again: #1: If all sides of a quadrilateral are congruent, then it is a square.1598

We know that they are talking about the rhombus, basically saying that, if it is a rhombus, then it is a square.1605

Well, isn't this sometimes? A rhombus is only sometimes a square; a rhombus can just stay a rhombus, so that is "sometimes,"1613

because it is going downwards, from a rhombus to a square.1619

The next one: If a quadrilateral is a parallelogram, then it is a rhombus.1624

See how it is going downwards from a parallelogram to a rhombus; so if it is going down, it is "sometimes."1628

Down is sometimes; up is always; and side-to-side means never.1637

If a quadrilateral is a rhombus, then it is a parallelogram.1655

It is going upwards; then it is always, because this is a special type of that.1660

And then: If a quadrilateral is a square, then it is a rectangle.1666

See how it is going upwards? That is always; so you can use this to help you with this.1671

The next example: Determine whether quadrilateral ABCD is a parallelogram, a rectangle, a rhombus, or a square with the given vertices.1684

With these vertices, we are going to have to do a few things to see what the most specific type of quadrilateral is.1696

We know that, first of all, to figure out if it is a parallelogram, we have to find the slope.1709

Slope will help us with anything that has to do with being parallel or perpendicular, which has to do with both of these, this one and this one.1718

And then, for the rhombus, as long as we can show that it is a rectangle and a rhombus,1732

then we can just automatically say that it is a square, too, if it is both a rectangle and a rhombus.1742

Now, how do you show if it is a rhombus?1749

Well, you know that a rhombus has four congruent sides, so we would just have to use the distance formula to show the length of each side.1752

If parallelogram works, then we have to continue with rectangle; if rectangle works, we are going to continue with rhombus.1760

If this doesn't work, then it would be a rectangle; but let's say rectangle and rhombus worked--then it would be a square,1768

because "square" means that it is both rectangle and rhombus.1777

Oh, and then, also, for a parallelogram, if rectangle doesn't work after parallelogram, then we would move on to rhombus.1783

We would have to actually show all three of these, but we don't have to show the square, because the square is just all of the above.1791

The slope of AB: again, it is the difference of the y's (so I am going to take the first y, which is -6,1802

and subtract the second y), and then take the first x, and subtract the second x.1812

A minus negative: this becomes a plus; this is -3 over 4.1827

Then, if you need help to determine which sides have to be parallel to what, and what sides have to be perpendicular to what,1837

just draw any quadrilateral, ABCD; you know it has to be in this order.1844

It doesn't matter if you do ABCD or if you do ABCD...it doesn't matter, as long as you know that it is in the order.1849

Then, the slope of BC: -3 minus -6, over 5 minus 1: this, again, becomes positive; this is 3/4.1860

Well, that is strange; let's see--let's double-check this.1888

For AB, -6 - -3 became -3; 1 - -3 became 4; then for this one, y2 is -3, minus y1, which is -6;1897

and then, x2, which is 5, minus x1, which is 4...this is 3/4; so it seems like it is correct.1923

Now, here, this slope is -3/4, and this slope is positive 3/4.1938

That means that they are not perpendicular to each other, because, in order for this side and this side to be perpendicular,1948

their slopes have to be the negative reciprocal of each other.1957

Then, if this is going to be -3/4, then this has to be positive 4/3, but it is not.1960

So, automatically, I can conclude that this AB and this are not perpendicular.1969

If it is not perpendicular, then I can cross out some of these: I can cross out the rectangle, and I can cross out the square,1978

because in order for it to be a square, this has to be true; rectangle has to be true, and rhombus has to be true.1986

Now, let's continue, because we still have two other options.1993

I am going to go for my next one, the slope of CD: 0 - -3 over 1 - 5.2000

So again, this becomes 3/-4; so that means that AB is parallel to CD.2013

Does that make that a parallelogram?--no, not yet: it has to be two pairs of opposite sides being parallel.2027

Or we can say that AB is congruent to DC, if you remember from the section on the parallelogram--2033

on the properties, or the theorems to prove a parallelogram.2040

There is one theorem; it is not a property of a parallelogram--it is a theorem that says that,2045

if you can prove that one pair is both parallel and congruent, then it is a parallelogram; that is another way you can do it.2051

Now, the last one, CB and then AD: 0 - -3, over 1 - -3, becomes 3/4.2065

So then, BC and AD are parallel, because they have the same slope.2084

So automatically, you know that that is a parallelogram; but then, a rhombus is also considered a parallelogram.2091

So then, we have to find the distance of each one, the distance of AB (let me do a lowercase d)...2096

We know that the distance formula is (x2 - x1), the difference of the x's, squared,2113

with the difference of the y's, squared, all under a square root.2125

For AB: 1 - -3, squared, plus -6 - -3, squared; this is going to be the square root of...2133

here is 4 squared; that is 16, plus...this is -3, squared, which is 9; that is going to equal √25, which is 5.2156

And then, let's find BC: the distance of BC: this is 5 + 6, squared, plus -3 + 6;2173

and I am just making them automatically plus for minus a negative.2200

Oh, wait: 5 - 1...I did the y instead of the x...the 5 minus the 1 is going to be 4 squared, which is 16,2207

plus 3 squared, which is 9, which is equal to √25, and that is 5.2226

And then, we know that this AB is going to be congruent to BC.2233

Let's try the other ones: the distance of...what is next?...CD: the square root of 5 - 1, squared...2245

Now, notice how I did this one minus this one before I did this one minus this one--it is the same thing,2263

as long as, for the next part, for the y's, you do the same order.2269

If you are going to do 5 - 1, then for the y's, you have to do -3 - 0, squared.2276

That is going to be right here...42 is 16, plus...(-3)2 is 9.2285

So again, that is the square root of 25, which is 5.2293

And then, the distance of the last one, AD: -3 - 1, squared, plus -3 - 0, squared: here we have -4 squared, which is 16,2298

plus -3 squared, which is 9; so then, this is the square root of 25, which is 5.2321

So, we know that all of the sides are congruent; that means that we have a rhombus, so this is a rhombus.2329

That is it for this lesson; thank you for watching Educator.com.2350