2 5 Proving Statements about Segments Geometry StandardsObjectives

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 • Theorem 2. 1 – 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: Reasons: 1. 2. PQ ≅ XY PQ = XY 3. XY = PQ 4. XY ≅ PQ 1. Given 2. Definition of congruent segments 3. Symmetric Property of Equality 4. Definition of congruent segments

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 ≅ JL • PROVE: LK ≅ JL

Statements: 1. 2. 3. 4. 5. 6. _______________ LK = JK LK ≅ JK JK ≅ JL ________ Reasons: 1. 2. 3. 4. 5. 6. Given Transitive Property ________ Given Transitive Property

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

Statements: 1. 2. 3. 4. 5. 6. 7. Q is the midpoint of PR. PQ = QR PQ + QR = PR PQ + PQ = PR 2 ∙ PQ = PR PQ = ½ PR QR = ½ PR Reasons: 1. 2. Given Definition of a midpoint 3. Segment Addition Postulate 4. 5. 6. 7. 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 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

Ex. Writing a proof: Given: 2 AB = AC Copy or draw diagrams and label given info to help develop proofs A B C Prove: AB = BC Statements Reasons 1. 2 AB = AC 1. Given 2. AC = AB + BC 2. Segment addition postulate 3. 2 AB = AB + BC 3. Transitive 4. AB = BC 4. Subtraction Prop.

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

PROVE: a. AB = 2 AM EXAMPLE 3 1 AB b. AM = 2 STATEMENT REASONS 1. M is the midpoint of AB. 1. Given 2. AM 2. Definition of midpoint MB 3. AM = MB 4. AM + MB = AB 5. AM + AM = AB a. 6. 2 AM = AB b. 7. AM = 1 AB 2 3. Definition of congruent segments 4. Segment Addition Postulate 5. Substitution Property of Equality 6. Addition Property 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 REASONS STATEMENT 1. m∠ 1 = m∠ 3 1. Given 2. m∠ EBA = m∠ 3 + m∠ 22. Angle Addition Postulate 3. m∠ EBA = m∠ 1 + m∠ 23. 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