2-6 Prove Statements About Segments and Angles

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Presentation transcript:

2-6 Prove Statements About Segments and Angles Hubarth Geometry

A proof is a logical argument that shows a statement is true A proof is a logical argument that shows a statement is true. We will use a two-column proof. A two column-proof has numbered statements and corresponding reasons for each step of the argument in logical order. Ex 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

Theorems Congruence of Segments Segment congruence is reflexive, symmetric and transitive. Reflexive Symmetric Transitive Congruence of Angles Angle congruence is reflexive, symmetric and transitive Reflexive Symmetric Transitive

Ex 2 Name the Property Shown Name the property illustrated by the statement. a. Transitive Property of Angle Congruence b. Symmetric Property of Segment Congruence

Ex 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 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. Combine Like Terms AM = AB 2 1 7. b. 7. Division Property of Equality

Writing A Two-Column Proof Proof of the Symmetric Property of Angle Congruence Given: Prove: 1 2 Statements Reasons 1. 1. Given 2. 2. Def. of Congruent Angles 3. 3. Symmetric Property of Equality 4. 4. Definition of Congruent Angles

Practice 1. Four steps of a proof are shown. Give the reasons for the last two steps. GIVEN : AC = AB + AB 1. AC = AB + AB 2. AB + BC = AC 3. AB + AB = AB + BC 4. AB = BC Given Segment Addition Postulate Transitive Property of Equality Subtraction Property of Equality STATEMENT REASONS PROVE : AB = BC 2. Name the property illustrated by the statement. a. CD CD Reflexive Property of Congruence Symmetric Property of Congruence