PROBLEMS ON COINTERIOR ANGLES

Find the values of the variables in the following :

Problem 1 :

Solution :

y and 127 are cointerior angles are consecutive interior angles.

y + 127 = 180

Subtracting 127 on both sides.

y = 180 - 127

y = 53

Problem 2 :

Solution :

z and 70 are cointerior angles are consecutive interior angles.

z + 70 = 180

Subtracting 70 on both sides.

y = 180 - 70

y = 110

Problem 3 :

Solution :

g and 57 are cointerior angles are consecutive interior angles.

g + 57 = 180

Subtracting 57 on both sides.

g = 180 - 57

g = 123

Problem 4 :

Solution :

Since the opposite sides are parallel.

Problem 5 :

Solution :

19x - 2 + 42x - 1 = 180

61x - 3 = 180

Adding 3.

61x = 180 + 3

61x = 183

Dividing by 61 on both sides.

x = 183/61

x = 3

Problem 6 :

Solution :

4 + 16x + 80 = 180

16x + 84 = 180

Subtracting 84.

16x = 180 - 84

16x = 96

Dividing by 16 on both sides.

x = 96/16

x = 6

Problem 7 :

Solution :

97 + 15x + 8 = 180

15x + 105 = 180

Subtracting 105 on both sides.

15x = 180 - 105

15x = 75

Dividing by 15 on both sides.

x = 75/15

x = 5

Problem 8 :

a) Name a pair of complementary angles, a pair of supplementary angles, and a pair of adjacent angles.

b) Are ∠KGH and ∠LKG adjacent angles? Are ∠FGK and ∠FGH adjacent angles? Explain.

c) ∠1 is a complement of ∠2, and m∠2 = 5°. Find m∠1.

d) ∠3 is a supplement of ∠4, and m∠3 = 148°. Find m∠4.

Solution :

a)

• ∠LKG and  ∠FGK are complementary angles, because they add upto 90 degree.
• ∠GKL and  ∠KGH are supplementary angles, because they add upto 180 degree.
• ∠FGK and ∠KGH are adjacent angles.

b) 

• ∠KGH and ∠LKG are not adjacent angles, because they are not next to each other.
• ∠FGK and ∠FGH are not adjacent angles, because they are not next to each other.

c) ∠1 is a complement of ∠2

m∠2 = 5°

m∠1 = 90 - m∠2

= 90 - 5

= 85

d) ∠3 is a supplement of ∠4

m∠3 = 148°

m∠4 = 180 - m∠3

= 180 - 148

= 32

Problem 9 :

a. ∠1 is a complement of ∠2, and m∠1 = 62°. Find m∠2.

b. ∠3 is a supplement of ∠4, and m∠4 = 47°. Find m∠3.

Solution :

a) ∠1 is a complement of ∠2

m∠1 = 62°

m∠2 = 90 - m∠1

= 90 - 62

= 28

d) ∠3 is a supplement of ∠4

m∠4 = 47°

m∠3 = 180 - m∠4

= 180 - 47

= 133

Problem 10 :

What is the value of x?

Solution :

Oak trail and Pine trail are parallel and Ash trail is transversal. Supplementary angle of 125 degree is 180 - 125 that is 55.

Corresponding angles are equal, then x = 55.

Problem 11 :

Find the values of x and y.

Solution :

m∠ADC + m∠DCB = 180

m∠ADC + 65 = 180

m∠ADC = 180 - 65

m∠ADC = 115

m∠ADC + m∠DAB = 180

115 + y = 180

y = 65

Since it is parallelogram, opposite sides will be equal.

x + 4 = 12

x = 12 - 4

x = 8

Problem 12 :

Find FG and m∠G.

Solution :

Since it is parallelogram, the opposite sides are equal and opposite angles are equal.

m∠G = 60

FG = HE

8 = HE

So, the measure of FG is 8

Problem 13 :

Find the values of x and y.

Solution :

Problem 14 :

For what values of x and y is quadrilateral ABCD a parallelogram? Explain your reasoning.

Solution :

∠A and ∠D are co-interior angles

3x - 32 + y + 87 = 180

3x + y + 55 = 180

3x + y = 180 - 55

3x + y = 125 ---- (1)

4y + 2x = 180

x + 2y = 90 -----(2)

(1) - 3(2) ==> 3x + y - (3x + 6y) = 125 - 270

y - 6y = -145

-5y = -145

y = 145/5

y = 29

Applying the value of y in (1), we get

3x + 29 = 125

3x = 125 - 29

3x = 96

x = 96/3

x = 32

Problem 15 :

You shoot a pool ball, and it rolls back to where it started, as shown in the diagram. The ball bounces off each wall at the same angle at which it hits the wall.

a. The ball hits the first wall at an angle of 63°. So m∠AEF = m∠BEH = 63°. What is m∠AFE? Explain your reasoning.

b. Explain why m∠FGD = 63°.

c. What is m∠GHC? m∠EHB?

d. Is quadrilateral EFGH a parallelogram? Explain your reasoning.

Solution :

a) m∠AEF = m∠BEH = 63°

m∠EAF = 90

m∠AFE + m∠AEF + m∠EAF = 180

m∠AFE + 63 + 90 = 180

m∠AFE + 153 = 180

m∠AFE = 180 - 153

m∠AFE = 27

b.  m∠FGD = 63° because the ball bounces at the same angle

c.  m∠GHC = 27  and m∠EHB = 27

d. Since opposite angles are equal, then it is parallelogram.