Chapter 2

Segments and Angles

Section 6Properties of Equality and Congruence

The photos to the left illustrate the Reflexive, Symmetric, and Transitive Properties of Equality. You can use these properties in geometry with statements about equality and congruence.

Example 1: Name Properties of Equality and Congruence

Name the property that the statement illustrates.

a. If GH ≅ JK then JK ≅ GH.

symmetric

b. DE = DE

reflexive

c. If <P ≅ <Q and <Q ≅ <R, then <P ≅ <R.

transitive

Checkpoint: Name Properties of Equality and Congruence

Name the property that the statement illustrates.

1. If DF = FG and FG = GH, then DF = GH.

transitive

1. <P ≅ <P

reflexive

1. If m<S = m<T, then m<T = m<S.

symmetric

Logical Reasoning In geometry, you are often asked to explain why statements are true. Reasons can include definitions, theorems, postulates or properties.

Example 2: Use Properties of Equality

In the diagram, N is the midpoint of MP, and P is the midpoint of NQ. Show that MN = PQ.

MN = NP ___Definition of midpoint____________

NP = PQ ___Definition of midpoint____________

MN = PQ ___Transitive Property_______________

Checkpoint: Use Properties of Equality and Congruence

<1 and <2 are vertical angles, and <2 ≅ <3. Show that <1 ≅ <3.

<1 ≅ <2 ___Vertical Angles___ Theorem

<2 ≅ <3 Given

<1 ≅ <3 __Transitive__ Property of Congruence

Example 3: Justify the Congruent Supplements Theorem

<1 and <2 are both supplementary to <3. Show that <1 ≅ <2.

1)

2)

3)

4)

5)

Checkpoint: Use Properties of Equality and Congruence

In the diagram, M is the midpoint of AB. Show that AB = 2 × AM.

MB = AM Definition of ___midpoint____

AB = AM + MB ____Segment Addition____ Postulate

AB = AM + AM __Substitution__ Property of Equality

AB = 2 × AM Distributive Property (simplify)