2.6 Prove Statements About Segments and Angles

Objectives

Write proofs involving segment and angle addition

Write proofs involving segment and angle congruence

Two-Column Proof

Recall what a two-column proof was… Two-Column Proof – A proof format

used in geometry in which an argument is presented with two columns, statements and reasons, to prove conjectures and theorems are true. Also referred to as a formal proof.

Statements ReasonsProof:

Two-Column Proof

Given: PR = QS

Prove the following.

Prove: PQ = RS

1. GivenPR = QS1.

2. Subtraction PropertyPR – QR = QS – QR2.

3. Segment Addition Postulate

PR – QR = PQ; QS – QR = RS

3.

4. SubstitutionPQ = RS4.

Proof:Statements Reasons

Example 1:

Prove the following.

Prove:

Given:

Your Turn:

Proof:

Statements Reasons

1. Given

2. Transitive Property

3. Given

4. Addition Property

AC = AB, AB = BX

AC = BX

CY = XD

AC + CY = BX + XD

5. Segment Addition Property

AC + CY = AY; BX + XD = BDAY = BD 6. Substitution

1.

2.

3.

4.

5.

6.

Your Turn:

Congruence of Segments

Theorem 2.1 (Congruence of Segments)Congruence of segments is reflexive, symmetric, and transitive.

Reflexive Property: AB ABSymmetric Property: If AB CD, then CD AB.Transitive Property: If AB CD and CD EF, then AB EF.

Congruence of Angles

Theorem 2.2 (Congruence of Angles)Congruence of angles is reflexive, symmetric, and transitive.

Reflexive Property: A A Symmetric Property: If A B , then B A Transitive Property: If A B and B C , then A C .

Prove the following.

Prove:

Given:

Example 2:

1. Given1.

2. Definition of congruent segments2.

3. Given3.

4. Transitive Property4.

5. Transitive Property5.

Proof:

Statements Reasons

Example 2:

Prove the following.

Prove:

Given:

Your Turn:

Proof:

Statements Reasons

1. Given

2. Transitive Property

3. Given

4. Transitive Property

5. Symmetric Property

1.

2.

3.

4.

5.

Your Turn:

Assignment

Geometry: Pg. 116 – 119

#3 – 13, 16, 21, 22