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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:
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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
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 Definition of congruent segments Example 1: Symmetric Property of Segment Congruence
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 Given Transitive Property _______________ Given Transitive Property Statements: Reasons:
Example 3: Using Segment Relationships • GIVEN: Q is the midpoint of PR. • PROVE: PQ = ½ PR and QR = ½ PR.
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 Statements: Reasons:
1. Four steps of a proof are shown. Give the reasons for the last two steps. REASONS STATEMENT 1. 1. AC = AB + AB Given 2. 2. AB + BC = AC Segment Addition Postulate ? 3. 3. AB + AB = AB + BC ? 4. 4. AB = BC for Example 1 GUIDED PRACTICE GIVEN :AC = AB + AB PROVE :AB = BC
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
A B C Copy or draw diagrams and label given info to help develop proofs Ex. Writing a proof: Given: 2AB = AC Prove: AB = BC Statements Reasons • 2AB = AC • AC = AB + BC • 2AB = AB + BC • AB = BC • Given • Segment addition postulate • Transitive • Subtraction Prop.
GIVEN: Mis the midpoint of AB. a. AB= 2AM PROVE: AM=AB b. 1 2 EXAMPLE 3 Use properties of equality
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 a. AB= 2AM PROVE: 5. AM + AM = AB 5. Substitution Property of Equality AM=AB b. 1 1 a. 6. 2AM = AB 6. Addition Property 2 2 b. 7. AM=AB 7. Division Property of Equality EXAMPLE 3
Write a two-column proof Write a two-column proof for this situation 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 4. m∠ 1 +m∠ 2 = m∠ DBC
