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.
- Symmetric: If , then _________.
- Transitive: If and , then _________.
- Substitution: If and , then _________.
- Distributive: If , then_____.
- Given points , and and , and . Determine if and are collinear.
- Given points and and and . Are the three points collinear? Is the midpoint?
- If and , explain how must be an obtuse angle.