  Mary Pyo

Proofs in Algebra: Properties of Equality

Slide Duration:

Section 1: Tools of Geometry
Coordinate Plane

16m 41s

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

17m 17s

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

31m 31s

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

42m 26s

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

42m 34s

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

19m

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

42m 47s

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

17m 24s

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

36m 3s

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

44m 31s

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

41m 2s

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

33m 37s

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

37m 35s

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

41m 53s

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

44m 6s

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

25m 55s

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

19m 48s

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

28m 43s

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

44m 43s

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

26m 46s

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

47m 51s

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

27m 53s

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

43m 44s

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

26m 34s

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

33m 30s

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

17m 26s

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

28m 11s

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

29m 36s

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

29m 11s

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

42m 43s

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

29m 47s

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

39m 14s

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

30m 48s

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

20m 10s

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

27m 53s

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

34m 10s

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

24m 7s

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

27m 6s

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

21m 14s

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

40m 59s

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

37m 57s

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

40m 37s

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

21m 4s

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

35m 25s

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

52m 43s

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

22m 43s

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

35m 24s

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

21m 51s

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

27m 53s

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

26m 16s

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

27m 50s

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

23m 8s

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

27m 1s

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

27m 24s

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

17m 46s

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

20m 31s

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

36m 43s

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

18m 17s

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

38m 40s

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

23m 39s

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

38m 50s

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

26m 10s

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

21m 59s

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

22m 2s

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

14m 46s

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

16m 6s

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

14m 12s

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

23m 17s

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

18m 43s

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

21m 26s

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

37m 6s

Intro
0:00
Dilations
0:06
Dilations
0:07
Scale Factor
1:36
Scale Factor
1:37
Example 1
2:06
Example 2
6:22
Scale Factor
8:20
Positive Scale Factor
8:21
Negative Scale Factor
9:25
Enlargement
12:43
Reduction
13:52
Extra Example 1: Find the Scale Factor
16:39
Extra Example 2: Find the Measure of the Dilation Image
19:32
Extra Example 3: Find the Coordinates of the Image with Scale Factor and the Origin as the Center of Dilation
26:18
Extra Example 4: Graphing Polygon, Dilation, and Scale Factor
32:08
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• ## Related Books 0 answersPost by David Saver on March 12, 2015In Example 2, for reason number 4 are you saying that the reason could be either substitution or addition property of equality?Both answers would be correct in this situation?Or should we write both properties down as the reason? 0 answersPost by Khalid Khan on October 5, 2014In Proof Example 2, wouldn't the reason for statement three be "Addition Property of Equality?" In Proof Example 1, you did 2x+(3x+9)=21, making the next step 5x-9=21. You said that this was the addition property of equality. In Example 2, it would be the same reason, because you are just simplifying it, correct? 1 answerLast reply by: Kyle LettererSun Aug 5, 2018 3:12 PMPost by patrick guerin on September 25, 2014what are the other types of proofs 0 answersPost by Manfred Berger on June 1, 2013Shouldn't the division property of equality actually only be valid for all any nonzero c?

### Proofs in Algebra: Properties of Equality

• Additional Property of Equality: For all numbers a, b, and c, if a = b, then a + c = b + c
• Subtraction Property of Equality: For all numbers a, b, and c, if a = b, then a – c = b – c
• Multiplication Property of Equality: For all numbers a, b, and c, if a = b, then a × c = b × c
• Division Property of Equality: For all numbers a, b, and c, if a = b, then a/c = b/c
• Reflexive Property of Equality: For every number a, a = a
• Symmetric Property of Equality: For all numbers a and b, if a = b, then b = a
• Transitive Property of Equality: For all numbers a, b, and c, if a = b and b = c, then a = c
• Substitution Property of Equality: For all numbers a and b, if a = b, then a may be replaced by b in any equation or expression
• Distribute Property of Equality: For all numbers a, b, and c, a(b + c) = ab + ac
• One way to organize deductive reasoning is by using a two-column proof

### Proofs in Algebra: Properties of Equality

Name the property of equality that justifies the statement.
If 4x + 5 = 9, then 4x = 4.
subtraction property of equality.
Name the property of equality that justifies the statement.
If a = 2y + 5 and a = b, then 2y + 5 = b.
Substitution property of equality.
Name the property of equality that justifies the statement.
2(m∠1 + m∠2) = 2m∠1 + 2m∠2.
Distributive property of equality.
Name the property of equality that justifies the statement.
If 4x + 5 = 6y + 4, then 6y + 4 = 4x + 5
Symetric property of equality.
Name the property of equality that justifies the statement.
If ∆ ABC ≅ ∆ DEF, and ∆ DEF ≅ ∆ MON, then ∆ ABC ≅ ∆ MON.
Transitive property of equality.
Name the property of equality that justifies the statement.
If 5m∠3 = 15, then m∠3 = 3.
Division property of equality.
Name the property of equality that justifies the statement.
If 2a + 3 = 5a + b, then 4a + 6 = 10a + 2b.
Multiplation property of equality.
Write the reason for each statement.
Given AB = 2x + 5, C is the midpoint of AB, BC = 4 Prove x = 1.5.
Statements
1.C is the midpoint of AB
2.AB = 2BC
3. 2x + 5 = 2*4 4. 2x = 3 5. x = 1.5
Reasons:
1. Given
2. definition of midpoint
3. substitution ( = )
4. subtraction property ( = )
5. division property ( = ).
Write the reason for each statement.
Given m∠ABC = 5x − 5, bisect ∠ABC, m∠ABD = 25o
Prove x = 11. Statements
1. bisect ∠ABC
2. m∠ABC = 2m∠ABD
3. m∠ABC = 2*25 = 50
4. 5x − 5 = 50
5. 5x = 55
6. x = 11
Reasons
1. Given
2. Bisector prostulate
3. Subst( = )
4. Subst( = )
6. Division property of equality.
Write a two - column proof.
Given ∠1 and ∠2 are complementary angles, ∠2 and ∠3 are supplementary angles.
Prove m∠3 − m∠1 = 90o.
 Statements 1. ∠1 and ∠2 are complementary angles 2. m∠1 + m∠2 = 90o 3. ∠2 and ∠3 are supplementary angles 4. m∠2 + m∠3 = 180o 5. m∠2 = 90o − m∠1, m∠2 = 180o − m∠3 6. 90o − m∠1 = 180o − m∠3 7. − m∠1 = 90o − m∠3 8. m∠3 − m∠1 = 90o.
 Reasons 1. Given 2. Definition of complementary angles 3. Given 4. Difinition of supplementary angles 5. Subtraction property of equality 6. Transitive property of equality 7. Subtraction property of equality 8. Addition property of equality

*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.

### Proofs in Algebra: Properties of Equality

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
• Properties of Equality 0:10
• Subtraction Property of Equality
• Multiplication Property of Equality
• Division Property of Equality
• Addition Property of Equality Using Angles
• Properties of Equality, cont. 4:10
• Reflexive Property of Equality
• Symmetric Property of Equality
• Transitive Property of Equality
• Properties of Equality, cont. 7:04
• Substitution Property of Equality
• Distributive Property of Equality
• Two Column Proof 9:40
• Example: Two Column Proof
• 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

### Transcription: Proofs in Algebra: Properties of Equality

Welcome back to Educator.com.0000

For this lesson, we are going to talk about some properties of equality, and we are going to work on some proofs.0002

Going over some properties first: these are all properties of equality,0013

meaning that they have something to do with them equaling each other, something to do with the word "equal."0017

Now, the first one, the addition property of equality, is when you have, let's say, numbers a, b, and c.0028

If a equaled b, if the number a is the same as b, then if you add c to a, then that is the same thing as adding c to b.0039

So, if a = b, then a + c = b + c; that is the addition property of equality, because you are adding the same number to a and b, since a and b are the same.0051

And the subtraction property of equality: again, you have numbers a, b, and c.0070

a is equal to b; then, a - c is equal to b - c, as long as you subtract the same number.0076

But when you are dealing with subtraction, then it is the subtraction property.0087

But as long as you are subtracting the same number from both sides, then it is still the same.0090

You still have an equation, with equal sides.0096

The multiplication property of equality: again, you have numbers a, b, and c.0101

If a is equal to b, then a times c is equal to b times c; so again, you are multiplying the same number.0106

And the division property: for the numbers a, b, and c, if a is equal to b, then a/c is equal to b/c.0115

Now, here you have to look at this c, because you are dealing with division; so this can also be a/c = b/c.0132

This, even though it is a fraction, also means a divided by c; and when you are dealing with that,0143

since c is now the denominator, we have to keep in mind that c cannot be 0, because we can't have a 0 in the denominator.0150

So, be careful with that.0161

Now, I want to go back over these again; and since the next couple of lessons, we are going to be talking about segments0163

and angles, if I have, let's say, the measure of angle 1, the measure of angle 1 equals the measure of angle 2.0172

So then, if the measure of angle 1 is representing a, and the measure of angle 2 is representing b,0185

then the measure of angle 1, plus the measure of angle 3 (c is a new one) equals...what is b?...0191

the measure of angle 2, plus the measure of angle 3.0200

So, this is also the addition property of equality, but just using angles now.0204

The measure of angle 1, plus the measure of angle 3, equals the measure of angle 2, plus the measure of angle 3.0209

Then, you are adding the same angle measure to both of these sides.0214

The subtraction property is the same thing: if I have, let's say, the measure of angle 1,0220

minus the measure of angle 3, then that is the same thing as the measure of angle 2, minus the measure of angle 3.0230

The multiplication property does the same thing, and the division property would also be the same thing.0240

The reflexive property of equality: this one is when you have one number, a;0252

for every number a, then a equals a--it equals itself; a = a is the reflexive property.0261

You can have a segment AB equaling itself, AB; this is also the reflexive property; measure of angle 1 = measure of angle 1--reflexive property.0271

When you write this, you can write this like "reflexive"...we can write "property"...0286

and for the equality properties, even the ones that we just went over, the addition property,0296

subtraction, multiplication, and division--since they are all properties of equality,0301

you can write "reflexive property of," and then you can write an equals sign next to it, like that: "reflexive property," and then an equals sign.0306

And that equals sign represents the type of property that it is.0314

So, it is the reflexive property of equality.0321

The symmetric property is different than the reflexive property, because you are given two numbers,0326

a and b; you are saying that a equals something else; if a = b, then...and then, you are just going to flip it; and you say b will then equal a.0333

So, if AB = 10, then you can say 10 = AB; and that is the symmetric property.0348

For the symmetric property, you can just write "symmetric property of equality" like that, too.0361

The transitive property of equality: For all numbers a, b, and c, if a = b, and b = c, then a = c.0370

So, let's use angles: if the measure of angle 1 equals the measure of angle 2,0382

and the measure of angle 2 equals the measure of angle 3, then since this and this are the same,0390

the measure of angle 1 equals the measure of angle 3.0401

If this equals that, and that equals something else, then these two will equal each other; and that is the transitive property of equality.0404

This one you can write as "trans. property of equality" for short.0414

A couple more: the substitution property of equality: whenever you replace something in for something that is of the same value,0427

then you are using the substitution property; so if you have numbers a and b, and if a = b,0441

then a may be replaced by b in any equation or expression.0447

If I tell you that x = 4, and x + 5 = 9, then I can take this; since x is equal to 4, I see an x here;0452

so since this and this are the same, I can just replace the 4 in for x...plus 5, equals 9.0475

I am substituting in this for this; and that is the substitution property.0485

For the substitution property, be careful not to just write "sub.," because this can also be the subtraction property.0495

You can just maybe write it like that, or maybe you can write the whole thing out: "substitution property of equality."0504

The distributive property of equality: for all numbers a, b, and c, a times the sum of b and c is equal to ab + ac.0515

Remember: you take this value right here; you multiply it to all the values inside.0527

So, it is going to be a times b, and then plus a times c; and that is the distributive property.0536

You can also go the other way; you can take it from here; you can factor out the a.0548

We have an a in both terms; factor it out; in this term, I have a b, plus...and in this term, I have a c left; that is also considered the distributive property.0553

For the distributive property, you can write it like that; you can write "distributive of equality"; you can write "prop."0565

These are all properties of equality; we are going to be using them pretty often in what is called a proof.0578

And there are a couple of different types of proofs, but the main one is called the two-column proof.0587

And a two-column proof is just a way of organizing your reasoning, and it is deductive reasoning.0594

When you use two-column proofs, you use them to show how to come up with some kind of conclusion.0605

Remember: with deductive reasoning, we talked about having some true statements,0617

and using facts and different definitions and so on to come up with a conclusion.0623

And a two-column proof is just a way of organizing those things.0632

For a two-column proof, you are going to have a given statement, and the given statement is just whatever is given to you,0638

the information that is given; and it can be maybe the values of angles, or the values of the side measures--whatever.0652

Whatever they give you, whatever is given to you, is going to go right here, as given.0662

That is going to be given, and then they are going to give you a "prove" statement, what to prove.0668

Given this information, your conclusion--how will you get to this right here?0677

They are going to give you both statements; and then, on this side, you are going to have a diagram or some kind of drawing,0682

some kind of picture of this proof--some kind of drawing, maybe a diagram; that is going to go right here.0693

And then, right below it, you are going to have something that looks like this.0709

And it is a two-column proof, so you are going to have two columns.0719

On this column, you are going to have statements; on this column, you are going to have reasons.0722

You are going to state something--you are going to state your facts, your different things.0733

And then, on this side, you are going to have reasons for that statement.0739

You can't just say something--you have to have a reason; you have to back it up with something.0744

Why is that statement true? You are going to do numbers 1, 2, 3, 4...and it is going to go on.0748

And then, your reasons: 1, 2, 3, 4...you have to have a reason for every statement you write down.0756

Now, any time you do a two-column proof, the first statement is always going to be your given statement.0763

Whatever is written here, you are also going to write here.0772

Now, you don't always have to have 4 or 5; it is usually going to be around 4, 5, or 6, but you can have less; you can have more.0784

It depends on the proof; but your last statement is going to be this statement.0793

Whatever is written here is going to be your last statement.0801

And then, for number 5, you are going to have a reason for that statement.0804

This is what a two-column proof looks like; now, if you are so confused by what a two-column proof is, think of directions.0808

From your house to, let's say, school, or from your house to a friend's house, you have a starting point.0821

You are starting at some place, and you are going to head over to school, or your friend's house, wherever it is.0833

You have directions; if you are to give someone directions to school from your house--or anywhere--0843

the starting point...you have point A to point B; you have steps to get from point A to point B.0850

If you are at home, how are you going to get to point B?0860

You make a right here, make a left here, or whatever it may be; you have directions.0864

There are steps to get there; this is exactly the same thing.0869

The given statement...this is where you start; that is point A--that is your starting point; that is like your house.0874

This statement right here, the "prove" statement, is point B--that is where you have to end up at; that is your destination.0882

You have to go from point A to point B; but again, you can't just snap your fingers and get there.0892

You have steps; you have directions to get there.0901

For each (maybe "make a right turn"; "make a left turn here"), you can't skip any steps, because it has to lead from point A,0905

and then through all of these steps, you are going to end up at point B.0916

And that is what a two-column proof is; they are just saying how you get from here to here.0924

And all you have to do is list out your statements: the starting point, point A, is going to be on line 1, statement 1.0928

Your last statement is going to be right here, your "prove" statement.0937

And you are just going to have reasons for that: why is this statement true? Why is this statement true?0941

Now, when you write your given statement for step 1, your reason is always going to be "Given."0946

That is the reason; this statement is true because it is given--that is given to you.0956

So, step 1 is this part right here, the given statement; and the reason for that is "Given."0964

So, here is an example of a proof: now, here, the statements are just listed, and the reasons are just listed.0975

It is a two-column proof; you can draw a line out like this and draw a line down like this.0983

Or you can just do it like this; as long as you have two columns, a column for statements and a column for reasons, you still have a two-column proof.0992

Again, here is your given statement; you have a few things that they give you; and prove this.1002

So, look at the statements: now, for step 1 (they are not all listed out, so let me write them out here),1011

AC is 21; that is this right here; now, you have to write out all of them.1026

So then, see how only this is written out; so I am going to write in the other ones.1031

AB = 2y, and BC = 3y - 9; those are all of your statements.1037

Now, here, AB is 2y; so I can write that in; so use this diagram to help you get from here, point A, to point B.1050

Write it in: AB is 2y; BC is 3y - 9; and then, AC, the whole thing, is 21.1065

And, given this information, they want you to prove that y equals 6.1080

Step 1: All that I did was to copy down all of the given statements right there.1088

And the reason for that is "Given."1095

Now, the next step: AB + BC = AC...well, that is because, since I have AB, and I have BC, and I have AC,1102

AC, the whole thing, is 21; but since I need to solve for y, I need to look at where my destination is.1119

Where am I trying to get to?--to what y is--my value for y.1127

Well, y, I see here, is from AB, and from BC, not from AC.1131

So, how do I mention these parts, these segments, in relation to the whole thing?1139

This is part of the segment; AB is part, and BC is another part, of this whole segment, AC.1149

If you remember, from chapter 1, we talked about segments, and then their parts.1160

I can say that AB + BC = AC; this part, plus this part, equals the whole thing: AB + BC = AC.1170

And the reason for this step, this statement, AB + BC = AC: if you remember, that is called the Segment Addition Postulate.1186

And you can just write it like this for short: the Segment Addition Postulate.1206

Now, why did I write this down--why is this step here?1214

It is because I know that, in order for me to find the value of y, I have to look at these parts, AB and BC.1218

I can't just look at the whole thing; so when I have to look at the parts, compared to the whole thing,1227

then I have to use the Segment Addition Postulate.1233

And then, what happened here? The next step: 2y + 3y - 9 = 21; so how did I get from this step to this step?1237

What happened here? Well, I know that AB is what?--AB is 2y; BC is this; so, guess what happened right here.1252

You see that...and 21; AC is 21; so, since AB is 2y, just replace AB for 2y, and then replace this for this, and replace this for this.1272

Whenever you do replacing, whenever you replace something for something else, in an equation or expression, that is the substitution property of equality.1292

Now, remember: be careful not to write "sub." because that can mean subtraction; "substitution" is the shortest you can write it.1313

Or you can write the whole thing out.1323

The next step: from here, 5y - 9 = 21--well, how did you get from this step to this step?1327

You did this plus this; you just simplified it, and more specifically, you added; so this would be the addition property (and this is for the "equality").1340

Now, here, number 5: you did 5y, and then you added 9 to both sides; this is the addition property, because you added.1364

And then, from here, how did you get from this step to this step?1385

You divided by 5 on both sides; so this is the division property of equality.1391

And then, since we have this statement, which is the same as this statement right here, we have arrived to our destination, to point B.1406

And once you do that, then you are done; so as long as you start here and you end up here, then you are done.1416

The next example: the measure of angle CDE (there is angle CDE) and the measure of angle EDF are supplementary.1431

Prove that x = 40.1442

So, here I am going to write in...now, for this one, this is x, and this is 3x + 20.1446

Sometimes, they give you the information on the diagram; they might not always give it to you in the given.1466

The given is very important, but you have to look at the diagram, too, because they might label something--1472

an angle, or give you some measure or length, and they might just write it in the diagram.1478

So, that is very, very important to have; if there is no diagram, then draw one in, because that is going to help you.1485

Especially if you are very visual--if you are a visual learner--then you should draw it in and write in whatever is given to you.1491

Number 1: Angle CDE and angle EDF are supplementary.1506

Now, to review over supplementary: supplementary means that two angles add up to 180 degrees.1514

These two angles right here, angle CDE and angle EDF, form a linear pair, meaning that,1529

when you put them together, they form a line; see how there is a line right there--so they are a linear pair.1545

And linear pairs are always supplementary, because a line measures 180 degrees.1555

So, if you have two angles that form a line, then they are supplementary.1563

If you look at supplementary angles, supplementary angles are just any two angles that add up to 180.1571

So, supplementary angles don't always form a linear pair; sometimes they do; sometimes they don't.1576

If you just have two angles that are separated, then they don't form a linear pair, but they can still be supplementary.1582

On to our proof: the reason for #1 is "Given."1591

And then, #2: I want to find x, so if I know that these two angles are supplementary, meaning that they add up to 180,1600

and then I know that these two angles together add up to 180, then I can find x that way.1614

But then, there are steps that I need to take to get there.1621

The next step is going to be that the measure of angle CDE, plus the measure of angle EDF, equals 180.1625

Now, we know that, since it says "supplementary," I can just say, "Well, since they are supplementary, then I add them together, and they equal 180 degrees."1640

And that is because of the definition of supplementary angles.1651

The definition of supplementary angles says that, if two angles are supplementary, then they add up to 180.1662

Any time you go from something supplementary to then making them add up to 180, then that would just be the definition of supplementary angles.1674

Any time you do this step, the reason will be "definition of supplementary angles."1683

The next step: now that I gave these two angles, adding them up to 180, now I have to use x somewhere, because I need to prove that x equals 40.1692

So, this angle right here became x, and then the measure of angle EDF is 3x + 20.1709

So, what happened here? Instead of writing this one, you wrote x; and instead of writing this one, you wrote 3x + 20.1722

Step #3: Since you replaced something, that is the substitution property of equality.1731

Now, #4: This right here, in the last proof--this could be the addition property, because you are adding it.1746

But it could also be the substitution property, because you are just substituting in these two for this value.1759

Let's just write "substitution property of equality."1766

And then, #5: To get from here to here, you subtracted 20 from both sides, so #5 is going to be the subtraction property.1773

And just so that you don't get confused, you can write it out, or you can just write "subtraction property," or "subtract. of equality."1798

In the next step, you divide it by 4 to get x = 40, and that is the division property, because you divided.1808

And then, we know that this is the final step, because that is what that is.1821

OK, another example: The measure of angle AXC and the measure of angle DYF...1831

oh, this is supposed to be written as "equal"; so then they are equal.1848

The measure of angle AXC and this angle are the same; and the measure of angle 1, this one, is equal to the measure of angle 3.1858

And by doing this, this is showing that they are the same.1872

So, if I do this one time, and I do this one time, that means that they are the same.1876

And I have to prove that the measure of angle 2 is equal to the measure of angle 4.1883

For this one, I don't have any statements, so we are going to have to do the statements on our own, and then come up with the reasons as we go along.1892

#1: I am going to write that the measure of angle AXC equals the measure of angle DYF,1899

and that the measure of angle 1 equals the measure of angle 3.1913

OK, and my reason for that is "Given."1925

Now, my next step: since I know that I am trying to prove this and this, that these two are equal,1932

I need to break down this big angle into its parts.1948

I know that angle AXC equals the measure of angle 1 plus the measure of angle 2.1959

So, let me write that out: the measure of angle 1, plus the measure of angle 2, equals the measure of angle AXC.1968

And the reason why I do that is because I need to somehow get that angle 2 in there somewhere.1983

And I know that these equal each other, and I know that the measure of angle 1 and the measure of angle 3 equal each other.1992

How am I going to come up with angle 2?1999

I can say that this one, plus this one, equals this big thing.2006

I am getting it in there somehow: the measure of angle 1, plus the measure of angle 2, equals the measure of angle AXC.2013

Now, I am going to do the same thing for this one, in the same step: the measure of angle 3 plus the measure of angle 4 equals the measure of angle DYF.2018

And the reason for that, if you remember from Chapter 1: this is the Angle Addition Postulate.2040

And my third step: Since all of this equals this, and all of this equals that, look at my first step right here.2060

I know that they equal each other; well, if these equal each other, doesn't that mean that all of its parts equal each other?2077

If this big angle and this big angle equal each other, then angles 1 and 2 together equal angles 3 and 4 together.2085

So, my next step is going to be: The measure of angle 1, plus the measure of angle 2,2095

equals the measure of angle 3, plus the measure of angle 4, because all of this right here equals AXC,2103

and all of this right here equals the measure of angle DYF.2114

And they equal each other; that means that all of this and all of this equal each other; so that is the only step right there.2118

Step 3: I basically just used this right here, and I substituted in the parts for that.2126

So, step 3 is going to be the substitution property; and I can put "equality."2135

Then, for #4: Now, always keep in mind what you have to prove.2146

I have to prove that this one equals this one; so I have to somehow get rid of this and this.2156

Now, look back at step 1; if you look back at step 1, see how the measure of angle 1 equals the measure of angle 3.2167

Well, here is the measure of angle 1, and here is the measure of angle 3.2177

So, since they are the same, I can use the substitution property to replace...2180

maybe measure of angle 3 for this, or measure of angle 1 for that, because they are the same.2186

I am going to just substitute in the measure of angle 1 in place of 3, since they are the same.2194

Do you see that? This is the measure of angle 1, in place of the measure of angle 3, since they are the same, since they equal each other.2209

And my reason for this one is, again, the substitution property; this property is actually used quite often.2218

And then, here, since these are the same, I can just subtract it out.2230

Then, these cancel out; I get that the measure of angle 2 equals the measure of angle 4.2238

My reason is the subtraction property of equality.2247

And I know that I am done, because this stuff is the same as that stuff.2257

Now, I know that this seems really long; but once you get used to it, and once you get more familiar with proofs,2262

they actually become kind of fun, and it is not so long; it is not so bad.2272

It is just that, since we are going over each step, and we are going over each reason, it just seems a lot longer than it is.2276

These next few examples...we are going to just go over the properties that we went over.2289

Name the property of equality that justifies each statement.2304

If 5 = 3x - 4, then 3x - 4 = 5: well, this one right here...remember when we had the property "if a = b, then b = a"?2309

This property is the same property as if I said that if ab = 10, then 10 = ab.2334

And this is the symmetric property of equality, meaning that it is the same on both sides; so you flip it, and it is the same.2347

The next one: If 3 times the difference of x and 3/5 equals 1, then 3x - 5 = 1.2360

So, what happened here--how did you get from this to this?2371

Well, it looks like this was distributed over to everything; this became 3 times x, which is 3x; and 3 times this; 3 times 5/3...2375

this is over 1; you can cross-cancel that out, and that becomes 5.2391

Then, that is how you got that 5; and minus 1; this was the distributive property of equality.2397

You can just also write "distributive...equality."2411

Name the property: Here, this is written: if 2 times the measure of angle 180...2417

no, if 2 times the measure of angle ABC equals 180 (that is how it is supposed to be written), then the measure of angle ABC equals 90.2439

OK, so if 2 times this angle is 180, then if you solve out for the measure of angle ABC, you get 90 degrees.2456

And so, how did you get from this step to that step over there?2467

Well, it looks like you divided the 2; and the measure of angle ABC equals 90; so this one was the division property of equality,2472

because you divided the 2 to get the answer--divided 2 into both sides.2487

Name the property of equality that justifies each statement.2497

For xy, xy = xy; well, this one right here--if something equals itself, this is different than the symmetric property.2501

The symmetric property is when you have something equaling something else.2511

And then, you can reverse it and say that the second thing equals the first thing.2516

In this one, there is no second thing; it is just one thing, and that one thing equals itself, so a = a; apple = apple; xy = xy.2523

Any time you have that, it is the reflexive property; "reflexive," or "reflexive property," and this is used quite often, too, in proofs.2537

If EF = GH, and GH = JK, then EF = JK.2554

Well, if 1 = 2 and 2 = 3, then 1 = 3; this is the transitive property of equality.2563

The next one: if AB + IJ = MX + IJ, then AB = MX.2585

What happened from here to get that? It looks like this happened: this is the subtraction property of equality.2595

The next one: if PQ = 5, and PQ + QR = 7, then 5 + QR = 7.2611

So, this was the equation; there is a value of PQ, and then 5 was replaced for PQ; this is the substitution property of equality.2621

Be careful when you are writing "subtraction" and "substitution."2640

It would probably be best to just write out the whole word.2644

But if you are going to write it like this, then make sure it is obvious what you are writing--subtraction property or substitution property.2648

And that is it for this lesson; we will work on some proofs for the next lesson.2660

So, we will see you next time--thank you for watching Educator.com.2666

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