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Geometry 2.4 Special Pairs of Angles In this lesson we will learn about: •Complimentary angles •Supplementary angles •Vertical angles 7 Complementary angles (Comp ‘s) Two angles whose measures have the sum 90. Each angle is called a complement of the other. Y . . W . Z 7 7 X XYW is a complement of WYZ If an angle has a measure of x o, What is the measure of its complement? o 90 - x o 7 Complementary angles (Comp R T are 7 complementary o 30 S 7 60 R and o T ‘s) 7 Supplementary angles (Supp ‘s) Two angles whose measures have the sum 180. Each angle is called a supplement of the other. D 7 DEG is a supplement of GEF E G F If an angle has a measure of x o, What is the measure of its supplement? o o 180 - x 7 50 o 130 o B A 7 A and B are supplementary 7 Supplementary angles (Supp ‘s) 7 Example A supplement of an angle is three times as large as a complement of the angle. Find the measure of the angle. Let x = measure of the angle Then 180 – x = the measure of its supplement And 90 – x = the measure of its complement 180 – x 180 – x 2x x = 3 (90 – x) = 270 – 3x = 90 = 45 The angle is 45o and its complement is 90 – 45 = 45o and its supplement is 180 – 45 = 135o. 7 Vertical angles (Vert ‘s) Two angles such that the sides of one angle are opposite rays to the sides of the other angle. When two lines intersect, they form two pairs of vertical angles. Basically, fold an angle over its vertex to find its vertical angle. “V”ertical starts with a V, so fold the “V” down to find its “V”ertical angle. 3 are vert 7 2 and 7 4 are vert 7 7 3 4 7 2 1 and 7 1 ‘s ‘s In a proof, these will both work. You will never use Vertical Angles Defn. Vertical Angles Thm. Vertical angles are congruent. 7 3 1 2 ‘s 2 Think: What do you know about the sum of the measure of supplementary ‘s 1 and 3 ? The sum = 180 7 7 1 2 are vertical 7 7 7 Prove: 1 and 7 Given: ‘s 2 and 3 ? The sum = 180 7 1 and 7 7 1 7 2 are vertical 7 Given: Prove: 2 Statements 7 m = 180 = 180 =m 2 + m = m =m 2 3 3 Angle Add. Post 2. Substitution Reflexive Subtr. POE 4. 3 1 1. 3. 7 4. 3 3 3 3 7 7 7 3. 1 + m 2 + m 1 + m m 1 Reasons 7 7 7 2. m m m 7 7 7 7 1. ‘s 2 Example 4 m 7 7 In this diagram, m 5 Name two other angles congruent to 5. 9 7 8 7 4 5 6 7 8 because vertical angles are congruent(vert angles thm.) 7 7 by the Transitive Property ( m<7=m<4 and m<4=m<5 ) Homework P.53 Let’s do #33 Together P. 52 (1-33 Odd) P. 57 (1-11 Odd)