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October 12, 2015 October 12, 2015 October 12, 2015 October 12, 2015 October 12, 2015 (2.8) Proving Angle Relationships Objective: Write proofs involving angles (supplementary, complementary, congruent, and right). Why: Construct viable arguments. October 12, 2015 Obj: Write proofs involving angles (supplmentary, complementary, congruent, and right). Protractor Postulate: Given any angle, the measure can be put into one-to-one correspondence with real numbers between 0 and 180. 80 70 60 110 50 40 100 90 90 80 C 110 120 70 120 130 60 130 140 50 140 30 100 40 0° 150 150 30 160 20 160 10 A 0 20 10 170 180 0 B Angle Addition Postulate: If D is in the interior of then m ABD + m ABC, DBC = m 170 180 A D ABC B C October 12, 2015 (2.8) Obj: EX. Find m ABD if m A (8x+1)o D (5x-2)o B C Write proofs involving angles (supplmentary, complementary, congruent, and right). ABC = 90o. Justify each step. October 12, 2015 Obj: Write proofs involving angles (supplmentary, complementary, congruent, and right). Supplement Theorem: If 2 s form a linear pair, then they are supplementary. Complement Theorem: If the noncommon sides of two adjacent angles form a right angle, then the angles are complementary angles. October 12, 2015 Obj: Ex. 6 and Write proofs involving angles (supplmentary, complementary, congruent, and right). 7 form a linear pair. If m m 7 = 5x + 12, find x, m 6, and m 6 = 3x + 32 and 7. Justify each step. October 12, 2015 October 12, 2015 Obj: Write proofs involving angles (supplmentary, complementary, congruent, and right). Congruent Supplements Theorem: s suppl. to same 2 or = 1 s are = . 3 Congruent Complements Theorem: s compl. to same or = s are= . October 12, 2015 Obj: Write proofs involving angles (supplmentary, complementary, congruent, and right). (Proof): One Case of the Congruent Complements Theorem If 2 angles are complementary to the same angle, then they are congruent to each other. Given: Prove: Statements Reasons October 12, 2015 October 12, 2015 Vertical Angles Theorem: Vert. Obj: Write proofs involving angles (supplmentary, complementary, congruent, and right). s are = . Proof: D A Given: diagram as shown Prove: ABC = Statement B C DBE Reason E October 12, 2015 Obj: Ex. B A Given: DB bis. ADC Prove: 3 2 = Statement 1 2 D Reason Write proofs involving angles (supplmentary, complementary, congruent, and right). 3 C October 12, 2015 Right Angle Theorems: Obj: Write proofs involving angles (supplmentary, complementary, congruent, and right). 1. Perpendicular lines interest to form 4 right angles. 2. All right angles are congruent. 3. Perpendicular lines form congruent adjacent angles. 4. If two angles are congruent and supplementary, then each angle is a right angle. 5. If two congruent angles form a linear pair, then they are right angles. October 12, 2015 Obj: Given: ABC is a right angle. Prove: ABD and Write proofs involving angles (supplmentary, complementary, congruent, and right). A CBD are complementary D B Statement Reason C October 12, 2015 Obj: Write proofs involving angles (supplmentary, complementary, congruent, and right). HW: (HR) (2.8) Pg. 156: 10, 13, 15, 16, 21, 25