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Aim: What are the theorems related to angles? (Day 1) A B C Do Now: Given: EOD BOA OC bisects DOB Prove: OC bisects AOE Statement D E O Reason 1) EOD BOA 1) Given 2) OC bisects DOB 2) Given 3) COD BOC 3) Def. Angle Bisector 4) COD EOD BOC BOA 4) Addition Post. (1,3) 5)COE COD EOD, COA BOC BOA 5) Partition Post. 6) COE COA 6) Substitution Post. 1 Def. Angle Bisector 7) OC bisects AOE Geometry Lesson: Angle Theorems 7) (Day 1) Theorem #1: All right angles are congruent. 1 2 Given that 1 and 2 are right angles, what statements can we make? m1 90, m2 90 m1 m2 1 2 Def. Right Angles Transitive Post. Def. congruent angles Geometry Lesson: Angle Theorems (Day 1) 2 Def: Def:Complementary Angles are two angles whose measures add up to 90 degrees. A 1 O If AOC is a right angle, what can we say about the sum m1 m2 ? m1 m2 =90 2 C 1 is "complementary" to 2. 2 is "complementary" to 1. 1 is "the complement" of 2. 2 is "the complement" of 1. Geometry Lesson: Angle Theorems (Day 1) 3 Ex: Complementary Angles In each case, A and B are complementary. Determine mB for each: 1) B 20º 3) 70º 90-x B x A A 2) B 50º 40º A 4) mA 3x 8 mB 90 _________ (3x 8) mB 98 3x Geometry Lesson: Angle Theorems (Day 1) 4 Ex: Complementary Angles Proof A Given: 1 is complementary to 2 3 is complementary to 2 Prove: m1 m3 Statement 1) 1 compl. 2 2) 3 compl. 2 3) m1 m2 90 4) m3 m2 90 5) m1 m2 m3 m2 6) m2 m2 7) m1 m3 1) 2) 3) 4) 5) 6) 7) C D 1 2 3 B P Reason Given Given Def. comp. angles Def. comp. angles Substitution Postulate Reflexive Postulate Subtraction Postulate Geometry Lesson: Angle Theorems (Day 1) 5 Theorems#2: #2,Complements 3: of the same angle, or congruent Theorem angles, are congruent. 2 is compl. 1 1 3 is compl. 1 What can we say about angles 2 and 3 ? 2 3 2 3 Theorem #3: If the union of two adjacent angles is a right angle, the angles are complementary. If ABC is a right angle, and D A 1 2 B 1 2 ABC , what can we say about 1 and 2 ? 1 compl. 2 C Geometry Lesson: Angle Theorems (Day 1) 6 Ex 1: Compl. Angles Given: PA PD PC PB Prove: 1 3 Statement C A 12 D 3 P Reason B 1) PA PD PC PB 1) Given 2) APD, CPB are rt. ' s 2) Def. perpendicular lines 3) 1 2 APD 3) Partition Postulate 2 3 CPB 4) 1 compl. 2 3 compl. 2 5) 1 3 4) If the union of two adjacent angles is a right angle, the angles are compl. 5)Complements of the same angle, or congruent angles, are congruent. Geometry Lesson: Angle Theorems (Day 1) 7 Ex 2,3,4: Compl. Angles: 2) Given: PB AD, QC AD m1 m3 Prove: m2 m4 P• •Q 4 2 3 1 • A B C 3) Given: PB AD, 2 and 3 are complementary Prove: m1 m3 •D 4) If A is the complement of B. If mA 5 x 6, and mB 3x 4, what is: a) x b) mA c) mB Geometry Lesson: Angle Theorems (Day 1) 8