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2-7 Proving Segment Relationships Ruler Postulate (2.8): The points on any line or line segment can be paired with real numbers so that, given any two points A and B on a line, A corresponds to zero and B corresponds to a positive real number. (This postulate establishes a number line on any line) Segment Addition Postulate (2.9): Ex 1: Proof with Segment Addition Given: PR = QS P Q R Prove: PQ = RS Statement Reason Statement Reason OR S Segment Congruence Reflexive Property: Symmetric Property: Transitive Property: Proof of Transitive Property of Congruence (with segments) ̅̅̅̅ , 𝑃𝑄 ̅̅̅̅ ≅ 𝑅𝑆 ̅̅̅̅ ̅̅̅̅̅ ≅ 𝑃𝑄 Given: 𝑀𝑁 M ̅̅̅̅ ̅̅̅̅̅ ≅ 𝑅𝑆 Prove: 𝑀𝑁 N Statement Ex 2: Proof with Segment Congruence ̅̅̅̅ ≅ 𝑋𝑍, ̅̅̅̅̅ 𝑋𝑍 ̅̅̅̅ ≅ 𝑊𝑋 ̅̅̅̅̅ Given: 𝑊𝑌 = 𝑌𝑍, 𝑌𝑍 ̅̅̅̅̅ ≅ 𝑊𝑌 ̅̅̅̅̅̅ Prove: 𝑊𝑋 Statement P Reason Y 3cm W Reason Z X Q R S 2-8 Proving Angle Relationships ⃗⃗⃗⃗⃗ and a number r between 0 and 180, there is exactly one ray Protractor Postulate (2.10): Given 𝐴𝐵 with endpoint A, extending on either side of ⃗⃗⃗⃗⃗ 𝐴𝐵 , such that the measure of angle formed is r. Angle Addition Postulate (2.11): Ex 1: Angle Addition: The time is 4 o’ clock and ten seconds. What are the measures of the angles between the minutes and second hands and between the second and hour hands? 2.3 Supplement Theorem: If two angles from a linear pair, then they are __________________ angles. 2.4 Complement Theorem: If the noncommon sides of two adjacents angles from a right angle, then the angles are ________________________ angles. Ex 2: Supplementary Angles If ∠1 and ∠2 form a linear pair and 𝑚∠2 = 67, find 𝑚∠1. Statement Given Theorem 2.5: Congruence of angles is reflexive, symmetric, and transitive. Reflexive Property: Symmetric Property: Transitive Property: Proof: Symmetric Property of Congruence Statement Reason Theorems 2.6 Congruent Supplement Theorem: Angles supplementary to the same angle or to congruent angles are congruent. 2.7 Congruent Complement Theorem: Angles complementary to the same angle or to congruent angles are congruent Ex 3: Use Supplementary Angles Given: 1 and 2 form a linear pair, 2 and 3 form a linear pair Prove: 1 3 Statement Reason Vertical Angles Theorem: If two angles are vertical angles, then they are congruent. Ex 4: If 1 and 2 are vertical angles and the m1 x and m2 228 3x , find m1 and m2 . Right Angle Theorems 2.9.1 Perpendicular lines intersect to form four right angles 2.10 All right angles are congruent. 2.11 Perpendicular lines form congruent adjacent angles. 2.12 If two angles are congruent and supplementary, then each angle is a right angle. 2.13 If two congruent angles form a linear pair, then they are right angles.