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Pa i r s o f A n g l e s 1. Adjacent Angles Two angles that share a common side and a common vertex, but do not overlap. Angle ABC is adjacent to angle CBD 1. they have a common side: (line segment CB) 2. they have a common vertex: (point B) In the figure shown, a and b are adjacent angles. They have a common vertex (.O) a common side (Ray OA.) Which of the following are adjacent angles? Why? They Adjacent Angles they share a vertex and a side They are NOT Adjacent Angles they only share a vertex, not a side They are NOT Adjacent Angles they only share a side, not a vertex They are NOT Adjacent Angles angle PSQ and angle PSR overlap Solved Examples on Adjacent Angles Find and , if are adjacent angles. Choices: A. 90 B. 26 C. 80 D. 16 Solution: Step1: Step 2: = 58 - 32 = 26 Correct Answer : B and B C A D . Exercises: Given: <DOC = 45; <BOA= 35; <COB = 42 C D B Find: 1. m<DOB m<COA m<DOA m<DOC + m<BOA m<DOB – m<BOA O A Answers: 1. 2. 3. 4. 5. m<DOB = 87 m<COA =77 m<DOA = 122 m<DOC + m<BOA = 80 m<DOB – m<BOA = 52 2. Complementary angles Two angles are complementary if their sum is 90° . These two angles (40° and 50°) are Complementary Angles, because they add up to 90°. Notice that together they make a right angle. These two are complementary because 27° + 63° = 90° These two angles are complementary. Sample Problems for Complementary Angles 1. If one angle is 60 degree find the measure of the other angle if the two angles are complementary to each other. Sol :- Since the two angles are complementary angles they must add up to 90 degree. So, Angle 1 + Angle 2 = 90 degree 60 + angle 2 = 90 degree. Angle 2 = 90 – 60 Angle 2 = 30 degrees 2. If one angle is 39 degree find the measure of angle 2 if the two angles are complementary to each other. Solution: 90 – 39 = 51 Angle 2 = 51 degrees. 3. Angle 1 measures 25 degrees and angle 2 measures 65 degrees. Are the angles complementary to each other? Ans : Yes, They are complementary to each other 4. Find the complement angle of 40 degrees? Solution: 40 + x = 90 x = 90 – 40 x = 50. The complementary angle is 50. 5. Find the measures of the complementary angle of the following: a. 22 degrees b. 63 degrees c. 45 degrees d. 75 degrees Solution: 1) 2) 3) 4) 90-22=68 90-3o=27 90-45=45 90-75=15 ASSIGNMENT: 1. Find whether the angles 72 degree and 18 degree are complementary angles. 2. Two complementary angles are the ratio 2 : 3, find these angles 3. The two complementary angles are (4x + 8)° and (4x + 10)°. Find the value of x from the given data. 4. (10 – 3x)˚ and (90 – 2x)˚ are complimentary angles. Solve for x. 5. Two complementary angles are A (x + 4)0 and B(2x – 7)0. Find the value of x and the measure of each angle. 3. Linear pair Two angles are a linear pair if they have a common side and their other sides are opposite rays. If angle 1 and angle 2 are a linear pair, then angle1+ angle 2 = 180° . 2. Supplementary angles Two angles are supplementary if their sum is 180° . 3. Linear pair Two angles are a linear pair if they have a common side and their A linear pair is a pair of adjacent angles formed when two lines intersect. Linear pairs are: Angles 1 and 2 angles 3 and 4, angles 2 and 4, angles 1 and 3. 4. Vertical angles Two angles are vertical angles iff the sides of one angle are opposite rays to the sides of the other. Vertical angles are the "opposite angles" that are formed by two intersecting lines. Note: If Angle 1 and angle 2 are vertical angles, then measure of angle 1 is equal to the measure of angle 2 .