2.5 Proving Statements about Segments Geometry

Standards/Objectives: Students will learn and apply geometric concepts. Objectives: Justify statements about congruent segments. Write reasons for steps in a proof.

Definitions Theorem: A true statement that follows as a result of other true statements. Two-column proof: Most commonly used. Has numbered statements and reasons that show the logical order of an argument.

NOTE: Put in the Definitions/Properties/ Postulates/Theorems/Formulas portion of your notebook Segment congruence is reflexive, symmetric, and transitive. Examples: Reflexive: For any segment AB, AB ≅ AB Symmetric: If AB ≅ CD, then CD ≅ AB Transitive: If AB ≅ CD, and CD ≅ EF, then AB ≅ EF

Example 1: Symmetric Property of Segment Congruence Given: PQ ≅ XY Prove XY ≅ PQ Statements: PQ ≅ XY PQ = XY XY = PQ XY ≅ PQ Reasons: Given Definition of congruent segments Symmetric Property of Equality

Example 2: Using Congruence Use the diagram and the given information to complete the missing steps and reasons in the proof. GIVEN: LK = 5, JK = 5, JK ≅ JL PROVE: LK ≅ JL

_______________ LK = JK LK ≅ JK JK ≅ JL ________________ Given Statements: Reasons: _______________ LK = JK LK ≅ JK JK ≅ JL ________________ Given Transitive Property _______________

Example 3: Using Segment Relationships GIVEN: Q is the midpoint of PR. PROVE: PQ = ½ PR and QR = ½ PR.

Definition of a midpoint Substitution Property Distributive property Statements: Reasons: Q is the midpoint of PR. PQ = QR PQ + QR = PR PQ + PQ = PR 2 ∙ PQ = PR PQ = ½ PR QR = ½ PR Given Definition of a midpoint Segment Addition Postulate Substitution Property Distributive property Division property Substitution

GUIDED PRACTICE for Example 1 1. Four steps of a proof are shown. Give the reasons for the last two steps. GIVEN : AC = AB + AB PROVE : AB = BC STATEMENT REASONS 1. AC = AB + AB 1. Given 2. AB + BC = AC 2. Segment Addition Postulate 3. AB + AB = AB + BC 3. ? 4. AB = BC 4. ?

GUIDED PRACTICE for Example 1 ANSWER GIVEN : AC = AB + AB PROVE : AB = BC ANSWER 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

Ex. Writing a proof: Given: 2AB = AC Prove: AB = BC A B C 2AB = AC Copy or draw diagrams and label given info to help develop proofs Ex. Writing a proof: Given: 2AB = AC Prove: AB = BC A B C Statements Reasons 2AB = AC AC = AB + BC 2AB = AB + BC AB = BC Given Segment addition postulate Transitive Subtraction Prop.

EXAMPLE 3 Use properties of equality GIVEN: M is the midpoint of AB . PROVE: a. AB = 2 AM b. AM = AB 2 1

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

Write a two-column proof EXAMPLE 1 Write a two-column proof for this situation 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