4.5 Segment and Angle Proofs

Basic geometry symbols you need to know Word(s) Symbol Definition Point A Line AB Line Segment AB Ray Angle ABC Measure of angle ABC Congruent

Substitution properties Reflexive properties Transitive properties Supplementary Angles Complementary Angles Congruent Angles Substitution properties If you prove 2 parts are congruent, they can substitute Reflexive properties The same part is equal to the same part. AB=AB Transitive properties If part 1 = part 2, and part 2 = part 3, then 1 = 3 Symmetric properties If A = B then B = A.

Vocabulary Proof – a logical argument that shows a statement is true Two – column proof – numbered statements in one column, corresponding reason in other Statement Reasons

Segment Addition Postulate -if B is between A and C, then AB + BC = AC Postulate – a rule that is accepted without proof. Write forwards and backwards – reverse it and it‘s still true Segment Addition Postulate -if B is between A and C, then AB + BC = AC converse….. Reverse it…. - if AB + BC = AC, then B is between A and C Draw it….

Angle Addition Postulate – - If P is the interior (inside) of <RST then the measure of < RST is equal to the sum of the measures of <RSP and <PST. Draw it -

Theorem – a statement that can be proven Theorem 4.1 – Congruence of Segments Reflexive – Symmetric – Transitive – Theorem 4.2 – Congruence of Angles Reflexive Symmetric Transitive

Write a two-column proof EXAMPLE 1 Write a two-column proof Write a two-column proof for the situation in Example 4 from Lesson 2.5. GIVEN: m∠ 1 = m∠ 3 PROVE: m∠ EBA = m∠ DBC STATEMENT REASONS 1. m∠ 1 = m∠ 3 1. Given 2. m∠ EBA = m∠ 3 + m∠ 2 2. Angle Addition Postulate 3. m∠ EBA = m∠ 1 + m∠ 2 3. Substitution Property of Equality 4. m∠ 1 + m∠ 2 = m∠ DBC 4. Angle Addition Postulate 5. m∠ EBA = m∠ DBC 5. Transitive Property of Equality

GUIDED PRACTICE for Example 1 GIVEN : AC = AB + AB PROVE : AB = BC ANSWER STATEMENT REASONS 1. AC = AB + AB 1. Given 2. AB + BC = AC 2. Segment Addition Postulate 3. AB + AB = AB + BC 3. Transitive Property of Equality 4. AB = BC 4. Subtraction Property of Equality

EXAMPLE 2 Name the property shown Name the property illustrated by the statement. a. If R T and T P, then R P. b. If NK BD , then BD NK . SOLUTION Transitive Property of Angle Congruence a. b. Symmetric Property of Segment Congruence

GUIDED PRACTICE for Example 2 Name the property illustrated by the statement. 2. CD CD Reflexive Property of Congruence ANSWER 3. If Q V, then V Q. Symmetric Property of Congruence ANSWER

Solving for x. Based on the properties learned, if you know 2 “parts” are congruent, you set them equal to each other and solve. m<A=2x+15, m<B=4x-3 2x+15=4x-3 15=2x-3 18=2x 9=x 2(9)+15=18+15=33 B A

EXAMPLE 3 Use properties of equality Prove this property of midpoints: If you know that M is the midpoint of AB ,prove that AB is two times AM and AM is one half of AB. GIVEN: M is the midpoint of AB . PROVE: a. AB = 2 AM b. AM = AB 2 1

Use properties of equality EXAMPLE 3 Use properties of equality STATEMENT REASONS 1. M is the midpoint of AB. 1. Given 2. AM MB 2. Definition of midpoint 3. AM = MB 3. Definition of congruent segments 4. AM + MB = AB 4. Segment Addition Postulate 5. AM + AM = AB 5. Substitution Property of Equality 6. 2AM = AB a. 6. Distributive Property AM = AB 2 1 7. b. 7. Division Property of Equality

Solve a multi-step problem EXAMPLE 4 Solve a multi-step problem GIVEN: B is the midpoint of AC . C is the midpoint of BD . PROVE: AB = CD STATEMENT REASONS 1. B is the midpoint of AC . C is the midpoint of BD . 1. Given 2. AB BC 2. Definition of midpoint 3. BC CD 3. Definition of midpoint 4. AB CD 4. Transitive Property of Congruence 5. AB = CD 5. Definition of congruent segments