Deductive Reasoning

“The proof is in the pudding.” “Indubitably.” Le pompt de pompt le solve de crime!" Je solve le crime. Pompt de pompt pompt." Deductive Reasoning 2-4 Special Pairs of Angles WE

2-4 Written Exercises Determine the measures of the complements and supplement of each angle. measure sums up to 900. Supplementary Complementary 180 – 10 = 170 90 – 20 = 70 1 180 – 72.5 = 107.5 90 – 72.5 = 17.5 2 180 – x 90 – x 3 90 – 2y 180 – 2y 4

2 complementary angles are congruent. Find their measures. 5 x + x = 90 450 and 450 2x = 90 x = 45 2 supplementary angles are congruent. Find their measures. 6 x + x = 180 900 and 900 2x = 180 x = 90

Name the angles. In the diagram, is a right angle. 7 Name another right angle.

Name the angles. 8 Two complementary angles.

Name the angles. 9 Two congruent supplementary angles.

Name the angles. 10 Two noncongruent supplementary angles.

Name the angles. 11 Two acute vertical angles.

Name the angles. 12 Two obtuse vertical angles.

In the diagram, bisects and U T 35 V 60 S 25 60 O 60 25 W 60 Z X 35 120 Y Label completely ! Vertical Angle Th. Vertical Angle Th. Vertical Angle Th. Now you answer the questions. Vertical Angle Th. 60+60+35+x = 180 x = 25

In the diagram, bisects and U T 35 V 60 S 25 60 O 25 60 W 60 Z X 35 Y 35 13 14 155

19 (3x-5) 3x -5 = 70 70 3x = 75 Vertical Angles divide by 3 x = 25

20 Vertical Angles (3x+8) (6x-22) 3x + 8 = 6x - 22 3x + 30 = 6x 30 = 3x divide by 3 10 = x

21 64 36 Vertical Angles 4x 4x = 64 + 36 4x = 100 Divide by 4 X = 25

22 2 1 27 4 27 3 are supplements are supplements a] If , find . 180 – 27 = 153

22 2 1 x 4 x 3 are supplements are supplements b] If , find . 180 – x

22 y 2 1 x y 4 x 3 are supplements are supplements c] If 2 angles are congruent, must their supplements be congruent? YES !

23 Given: Prove: g g 2 3 Label completely first. 1 4 ? ? Reasons Statements Vert. Angles are congruent Given Vert. Angles are congruent Transitive Prop. Of Equality Note the flow is better without the given first.

24 If and are supplementary, Then find the values of x, and . Start with a labeled diagram. 2x + x – 15 = 180 2x x - 15 A B 3x – 15 = 180 3x = 195 A = 2(65) B = 65 - 15 Divide by 3 B = 50 A = 130 x = 65

25 If and are supplementary, Then find the values of x, and . Start with a labeled diagram. X + 16 +2x– 16 = 180 3x = 180 X + 16 2x - 16 A B Divide by 3 A = 60 + 16 x = 60 B = 2(60) - 16 B = 120 - 16 A = 76 B = 104

26 If and are complementary, Then find the values of y, and . Start with a labeled diagram. 3y + 5 + 2y = 90 3y+5 C 5y + 5 = 90 2y D 5y = 85 D = 2(17) C = 3(17) + 5 divide by 5 C = 51 + 5 D = 34 y = 17 C = 56

27 If and are complementary, Then find the values of y, and . Start with a labeled diagram. y – 8 + 3y + 2 = 90 y - 8 C 4y - 6 = 90 3y + 2 D 4y = 96 D = 3(24) + 2 C = 24 - 8 divide by 4 D = 72 + 2 y = 24 C = 16 D = 74

Use the information to find an equation and solve. 28 Find the measure of an angle that is twice as large as its supplement. x = 2( ) 180 – x x = 180 – 2x 3x = 180 180 – 60 = 120 x = 60

Use the information to find an equation and solve. 29 Find the measure of an angle that is half as large as its complement. 2 2 x = 90 - x Multipy by 2 to get rid of fractions 2x = 90 - x 90 – 30 = 60 3x = 90 x = 30

Use the information to find an equation and solve. 30 The measure of a supplement of an angle is 12 more than twice the measure of the angle. 180 – x = 2x 12 + 180 = 12 + 3x 180 – 56 = 124 168 = 3x 56 = x

Use the information to find an equation and solve. A supplement of an angle is six times as large as the complement of the angle. 31 180 – x = 6( ) 90 - x Supplement 180 – 72 = 108 180 – x = 540 – 6x 180 + 5x = 540 Complement 5x = 360 90 – 72 = 18 x = 72

32 Find the values of x and y. 47 x (3x – 8) 47 + 2y – 17 = 180 (2y – 17) 2y + 30 = 180 x + 3x – 8 = 180 2y = 150 4x – 8 = 180 y = 75 4x = 188 x = 47

33 Find the values of x and y. 33 = 3(33) - y 50 3x - y x 33 = 99 - y 33 2x – 16 = 50 - 66 = - y 2x - 16 2x = 66 66 = y x = 33

C’est fini. Good day and good luck.

Deductive Reasoning