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Worksheet – Section 2-8 Proving Angle Relationships Objectives: Understand the Angle Addition Postulate and use it to find unknown angle measures Understand Supplements and Compliments and use to find unknown angle measures Use algebra to find unknown angle measure Use angle relation theorems to prove relationships with 2 column proofs Angle Addition Postulate R is in the interior of ∠PQS if and only if m∠PQR + m∠RQS = m∠PQS. Example: Find the measure of angle 1 if the measure of angle 2 is 56 degrees and Practice: If and , find the measure of angle 3. Justify each step. Supplements and Complements If two angles forma linear pair, then they are supplementary angle If ∠1 and ∠2 form a linear pair, then m∠1 + m∠2 = 180. If the noncommon sides of two adjacent angles form a right angle, then the angles are complementary angles. If ⊥ , then m∠3 + m∠4 = 90. Example: Suppose and form a linear pair. If and . Find x, , and . Justify each step Practice: a. Find the measure of each numbered angle. m∠7 = 5x + 5, m∠8 = x – 5 b. Find the measure of each numbered angle. m∠5 = 5x, m∠6 = 4x + 6, m∠7 = 10x, m∠8 = 12x – 12 c. Find the measure of each numbered angle. m∠11 = 11x, m∠13 = 10x + 12 d. Find the measure of each numbered angle. and are complimentary and Proving Angle Relationships The following theorems hold true for angles and can be used in proofs dealing with angles Congruent Supplements Theorem Angles supplement to the same angle or congruent angles are congruent. Congruent Compliments Theorem Angles compliment to the same angle or to congruent angles are congruent. Vertical Angles Theorem If two angles are vertical angles, then they are congruent. Theorem (Definition of Perpendicular lines) Perpendicular lines intersect to form four right angles. Theorem (Definition of right angles) All right angles are congruent. Theorem (Definition of Perpendicular lines) Perpendicular lines form congruent adjacent angles. Example: Write a two-column proof. Given: ∠ ABC and ∠CBD are complementary. ∠DBE and ∠CBD form a right angle. Prove: ∠ ABC ≅ ∠DBE Statements Reasons Example: Complete each proof. 1. Given: ⊥ ; ∠1 and ∠3 are complementary. Prove: ∠2 ≅ ∠3 Proof: Statements Reasons a. ⊥, ∠1 and ∠3 are complementary a. ______________ b. _________________ b. Definition of ⊥ c. m∠ ABC = 90 c. Def. of right angle d. m∠ ABC =m∠1 + m∠2 d. ______________ e. 90 = m∠1 + m∠2 e. Substitution f. ∠1 and ∠2 are compliments f. _______________ g. ∠2 ≅ ∠3 g.________________ Practice: Given: ∠1 and ∠2 form a linear pair. m∠1 + m∠3 = 180 Prove: ∠2 ≅ ∠3 Proof: Statements a. ∠1 and ∠2 form a linear pair. m∠1 + m∠3 = 180 Reasons b. _________________ b. Def. of Linear Pair c. ∠1 is suppl. to ∠3. c. ________________ d. Congruent Supplements d. _________________ a. Given Homework Problems Find the measure of each numbered angle and name the theorems that justify your work. 1. m∠2 = 57 2. m∠5 = 22 3. m∠1 = 38 4. m∠13 = 4x + 11, m∠14 = 3x + 1 5. ∠9 and ∠10 are complementary. ∠7 ≅ ∠9, m∠8 = 41 6. m∠2 = 4x – 26, m∠3 = 3x + 4 7. Complete the following proof. Given: ∠QPS ≅ ∠TPR Prove: ∠QPR ≅ ∠TPS Proof: Statements Reasons 7. Complete the following proof. Given: bisects Prove: ∠2 ≅ ∠3 Proof: Statements Reasons