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 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. (duh) 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

EXAMPLE 4 Solve a multi-step problem Shopping Mall Walking down a hallway at the mall, you notice the music store is halfway between the food court and the shoe store. The shoe store is halfway between the music store and the bookstore. Prove that the distance between the entrances of the food court and music store is the same as the distance between the entrances of the shoe store and bookstore.

EXAMPLE 4 Solve a multi-step problem SOLUTION STEP 1 Draw and label a diagram. STEP 2 Draw separate diagrams to show mathematical relationships. STEP 3 State what is given and what is to be proved for the situation. Then write a proof.

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