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Properties of Equality and Congruence

Logical rules that allow equations to be manipulated and solved.

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Properties of Equality and Congruence

What if you wanted to solve an equation and justify each step? What mathematical properties could you use in your justification? After completing this Concept, you be able to see how the properties of equality from Algebra I relate to geometric properties of congruence.

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CK-12 Foundation: Chapter2PropertiesofEqualityandCongruenceA

James Sousa: Introduction to Proof Using Properties of Equality

Guidance

The basic properties of equality were introduced to you in Algebra I. Here they are again:

For all real numbers , and :

Examples
Reflexive Property of Equality
Symmetric Property of Equality and or
Transitive Property of Equality and , then and , then
Substitution Property of Equality If , then can be used in place of and vise versa. If and , then
Addition Property of Equality If , then . If , then
Subtraction Property of Equality If , then . If , then
Multiplication Property of Equality If , then . If , then
Division Property of Equality If , then . If , then
Distributive Property

Recall that if and only if . and represent segments, while and are lengths of those segments, which means that and are numbers. The properties of equality apply to and .

This also holds true for angles and their measures. if and only if . Therefore, the properties of equality apply to and .

Just like the properties of equality, there are properties of congruence. These properties hold for figures and shapes.

For Line Segments For Angles
Reflexive Property of Congruence
Symmetric Property of Congruence If , then If , then
Transitive Property of Congruence If and , then If and , then

When you solve equations in algebra you use properties of equality. You might not write out the logical justification for each step in your solution, but you should know that there is an equality property that justifies that step. We will abbreviate “Property of Equality” “PoE” and “Property of Congruence” “PoC.”

Example A

Solve and justify each step.

Example B

Given points , and , with , and . Are , and collinear?

Set up an equation using the Segment Addition Postulate.

Because the two sides are not equal, and are not collinear.

Example C

If and , prove that is an acute angle.

We will use a two-column format, with statements in one column and their corresponding reasons in the next. This is formally called a two-column proof.

Statement Reason
1. and Given (always the reason for using facts that are told to us in the problem)
2. Substitution PoE
3. Subtraction PoE
4. is an acute angle Definition of an acute angle,

Watch this video for help with the Examples above.

CK-12 Foundation: Chapter2PropertiesofEqualityandCongruenceB

Vocabulary

The properties of equality and properties of congruence are the logical rules that allow equations to be manipulated and solved.

Guided Practice

Use the given property or properties of equality to fill in the blank. , and are real numbers.

1. Symmetric: If , then ______________.

2. Distributive: If , then ______________.

3. Transitive: If and , then ______________.

Answers:

1.

2.

3.

Practice

For questions 1-8, solve each equation and justify each step.

For questions 9-12, use the given property or properties of equality to fill in the blank. , and are real numbers.

  1. Symmetric: If , then _________.
  2. Transitive: If and , then _________.
  3. Substitution: If and , then _________.
  4. Distributive: If , then_____.
  5. Given points , and and , and . Determine if and are collinear.
  6. Given points and and and . Are the three points collinear? Is the midpoint?
  7. If and , explain how must be an obtuse angle.

Vocabulary

Real Number

Real Number

A real number is a number that can be plotted on a number line. Real numbers include all rational and irrational numbers.
Right Angle Theorem

Right Angle Theorem

The Right Angle Theorem states that if two angles are right angles, then the angles are congruent.
Same Angle Supplements Theorem

Same Angle Supplements Theorem

The Same Angle Supplements Theorem states that if two angles are supplementary to the same angle then the two angles are congruent.
Vertical Angles Theorem

Vertical Angles Theorem

The Vertical Angles Theorem states that if two angles are vertical, then they are congruent.

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