4.5 Segment and Angle Proofs

1. 4.5 Segment and Angle Proofs

2. Basic geometry symbols you need to know

4. 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

5. 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….

6. 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 -

7. 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

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

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

10. a. IfRTandTP, then RP. b. IfNKBD, thenBDNK. a. Transitive Property of Angle Congruence b. Symmetric Property of Segment Congruence EXAMPLE 2 Name the property shown Name the property illustrated by the statement. SOLUTION

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

12. 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

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

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

15. Shopping Mall EXAMPLE 4 Solve a multi-step problem 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.

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

17. GIVEN: Bis the midpoint of AC. Cis the midpoint of BD. 4. Transitive Property of Congruence PROVE: AB = CD REASONS STATEMENT 1. 1. Bis the midpoint of AC. Given Cis the midpoint of BD. ABBC 2. 2. Definition of midpoint 3. 3. BCCD Definition of midpoint ABCD 4. 5. AB = CD 5. Definition of congruent segments EXAMPLE 4 Solve a multi-step problem