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Aim: What are the theorems related to angles? (Day 2) Do Now: B C Q D A If AQD is a straight angle, mAQB 6 x 6, mBQC 10 x, and mCQD 4 x 6, find the measure and type of each angle: a) AQB 70º, acute b) BQC 90º, right c ) BQD 120º, obtuse Geometry Lesson: Angle Theorems (Day 2) 1 Theorem #4: If two angles are straight angles, then they are congruent. A O B G P H Given that AOB and GPH are straight angles, what can we say? AOB GPH Geometry Lesson: Angle Theorems (Day 2) 2 Def: Supplementary Angles are two angles whose measures add up to 180 degrees. A 1 O 2 C If AOC is a straight angle, what can we say about the sum m1 m2 ? m1 m2 =180 1 is "supplementary" to 2. 2 is "supplementary" to 1. 1 is "the supplement" of 2. 2 is "the supplement" of 1. Geometry Lesson: Angle Theorems (Day 2) 3 Ex: Supplementary Angles In each case, A and B are supplementary. Determine mB for each: 3) 1) 130º A A 50º B B mA x, mB ______ 180 x 2) B 96 º 84º A 4) A B mA 48 2 x 180 (48 2 x ) mB _________ mB 132 2 x Geometry Lesson: Angle Theorems (Day 2) 4 Def: A Linear Pair of angles is two adjacent angles whose union is a straight angle. If we join these two supplementary angles together along ray OD, what is formed by the outer rays OA and OC ? Ans : Straight angle AOC D D D or line AOC. A A O O O C C Angles AOD and DOC "form a linear pair". Theorem #5: A linear pair of angles is supplementary. Geometry Lesson: Angle Theorems (Day 2) 5 Ex: Supplementary Angles Proof Given: 1 is supplementary to 2 3 is supplementary to 2 Prove: m1 m3 Statement 1) 1 suppl. 2 2) 3 suppl. 2 3) m1 m2 180 4) m3 m2 180 5) m1 m2 m3 m2 6) m2 m2 7) m1 m3 1 2 3 Reason 1) 2) 3) 4) 5) 6) 7) Given Given Def. suppl. angles Def. suppl. angles Substitution Postulate Reflexive Postulate Subtraction Postulate Geometry Lesson: Angle Theorems (Day 2) 6 Theorems Theorem #6:#6:Supplements of the same angle, or congruent angles, are congruent. Given: 1 3 1 3 2 1 suppl. 2 4 3 suppl. 4 What can we say about angles 2 and 4 ? 2 4 Geometry Lesson: Angle Theorems (Day 2) 7 A Ex: Suppl. Angles Proof Given: AB and CD instersect at X C Prove: 1 3 1 Statement 1) AB and CD 2 X D 3 B Reason 1) Given instersect at X 2) 1 and 2 form a linear pair. 2) Def. Of linear pair 3 and 2 form a linear pair. 3) 1 suppl. 2 3 suppl. 2 4) 1 3 3) A linear pair of angles is supplementary. 4) Supplements of the same, or congruent angles, are Geometry Lesson: Angle Theorems 8 congruent. (Day 2) Def: A Vertical Pair of angles is two angles in which the sides of one angle are opposite rays to the sides of the other angle. A 1 C 2 4 3 When two lines intersect, two pairs of vertical angles are D created. 1 and 3 are vertical angles. B 2 and 4 are vertical angles. Theorem #7: Vertical angles are congruent. Geometry Lesson: Angle Theorems (Day 2) 9 Ex 1,2,3: Compl./Suppl./Vert. Angles: 1) Given: ABCD , 5 6 Prove: 4 7 4 5 A B 2) Given: EC bisects ADB F 6 7 C D C D BF intersects EC at D Prove: ADE FDC A 3) Given: BE intersects AD at C BAC compl. ACB B EDC compl. DCE A Prove: BAC EDC B E Geometry Lesson: Angle Theorems (Day 2) C D E 10 Ex 4, 5: Suppl. Angles 4) Two supplementary angles have measures in the ratio 2:7. Determine the measure of both angles. 5) The measure of an angle is 20 less than 4 times its supplement. Find the measure of the angle and its supplement. Geometry Lesson: Angle Theorems (Day 2) 11