Suppose you know that a circle measures 360 degrees and you want to find what kind of angle one-quarter of a circle is. After completing this Concept, you'll be able to apply the basic properties of equality and congruence to solve geometry problems like this one.
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CK-12 Properties of Equality and Congruence
James Sousa: Introduction to Proof Using Properties of Equality
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James Sousa: Introduction to Proof Using Properties of Congruence
Guidance
The basic properties of equality were introduced to you in Algebra I. Here they are again:
- Reflexive Property of Equality:
- Symmetric Property of Equality: If , then
- Transitive Property of Equality: If and , then
- Substitution Property of Equality: If and , then
- Addition Property of Equality: If , then or
- Subtraction Property of Equality: If , then or
- Multiplication Property of Equality: If , then or
- Division Property of Equality: If , then or
- Distributive Property:
Just like the properties of equality, there are properties of congruence. These properties hold for figures and shapes.
- Reflexive Property of Congruence: or
- Symmetric Property of Congruence: If , then . Or, if , then
- Transitive Property of Congruence: If and , then . Or, if and , then
When you solve equations in algebra you use properties of equality. You might not write out the property for each step, but you should know that there is an equality property that justifies that step. We will abbreviate “Property of Equality” “” and “Property of Congruence” “” when we use these properties in proofs.
Example A
Solve and write the property for each step (also called “to justify each step”).
Example B
, and . Are points , and collinear?
Set up an equation using the Segment Addition Postulate.
Because the two sides of the equation are not equal, and are not collinear.
Example C
If and , prove that is an acute angle.
We will use a 2-column format, with statements in one column and their reasons next to it, just like Example A.
CK-12 Properties of Equality and Congruence
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 9-11, 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 ______________.