QUESTIONS ON EXTERIOR ANGLES OF A POLYGON

Find, giving reasons, the value of x:

Problem 1 :

Solution:

The sum of exterior angles of a polygon is 360°.

x° + 116° + 122° = 360°

x° + 238° = 360°

x° = 360 - 238

x° = 122

Problem 2 :

Solution:

2x + x = 108°

3x = 108°

x = 36°

Problem 3 :

Solution:

∠CDE = 90°

∠CBE + 100° + 90° + 110° = 360°

∠CBE + 300° = 360°

∠CBE = 60°

∠ABC + ∠CBE = 180°

x + 60° = 180°

x = 120°

Problem 4 :

Solution:

83° + x = 117°

x = 117° - 83°

x = 34°

Find the values of the variables in the following diagrams:

Problem 5 :

Solution:

a + 33° + 62° = 180°

a + 95° = 180°

a = 180° - 95°

a = 85°

2b = a + 33°

2b = 85° + 33°

2b = 118°

b = 59°

Problem 6 :

Solution:

x + 140° = 180°

x = 180° - 140°

x = 40°

z + y = 140°

y = 140° - z

y = x

y = 40°

140° - z = 40°

z = 100°

180° = t + 100° + 50°

180° = t + 150°

t = 30°

Problem 7 :

Solution:

Problem 8 :

Solution:

a = 65° + 65°

a = 130°

Problem 9 :

Solution:

70° + 60° = a

a = 130°

Problem 10 :

Solution:

t + 105° = 180°

t = 180° - 105°

t = 75°

t = 3x

75° = 3x

x = 25°

3x + y = 105°

3(25) + y = 105°

75 + y = 105°

y = 105° - 75°

y = 30°

Problem 11 :

Solution:

a + 130° = 180°

a = 180° - 130°

a = 50°

c + 40° + 40° = 180°

c + 80° = 180°

c = 180° - 80°

c = 100°

Vertically opposite angles will be equal.

The triangle which is at below

a + c + d = 180

50 + 100 + d = 180

d + 150 = 180

d = 180 - 150

d = 30

Problem 12 :

In FGH, m∠F = 42 and an exterior angle at vertex H has a measure of 104. What is m∠G?

1) 34    2) 62      3) 76     4) 146

Solution :

Exterior angle = 104

Remote interior angles are m∠F and m∠G

m∠F + m∠G = 104

42 + m∠G = 104

m∠G = 104 - 42

m∠G = 62

Problem 13 :

In the diagram below of ABC, side BC is extended to point D, m∠A = x, m∠B = 2x + 15, and m∠ACD = 5x + 5.

What is m∠B?

1) 5       2) 20        3) 25      4) 55

Solution :

m∠A = x, m∠B = 2x + 15, and m∠ACD = 5x + 5.

Exterior angle m∠ACD = 5x + 5.

Remote interior angles are m∠A = x and m∠B = 2x + 15

m∠A + m∠B = m∠ACD

x + 2x + 15 = 5x + 5

3x + 15 = 5x + 5

3x - 5x = 5 - 15

-2x = -10

x = 5

∠B = 2x + 15 ==> 2(5) + 15

= 10 + 15

∠B = 25

So, option 3) is correct.

Problem 14 :

In ABC shown below, side AC is extended to point D with m∠DAB = (180 − 3x)°, m∠B = (6x − 40)°, and m∠C = (x + 20)°.

What is m∠BAC?

1) 20º      2) 40º     3) 60º      4) 80º

Solution :

180 - 3x = 6x - 40 + x + 20

180 - 3x = 7x - 20

180 + 20 = 7x + 3x

10x = 200

x = 200/10

x = 20

Problem 15 :

In the diagram below of ABC, BC is extended to D.

If m∠A = x² − 6x, m∠B = 2x − 3, and m∠ACD = 9x + 27, what is the value of x?

1) 10     2) 2     3) 3     4) 15

Solution :

9x + 27 = x² − 6x + 2x - 3

9x + 27 = x² − 4x - 3

x² − 4x - 3 - 9x - 27 = 0

x² − 13x - 30 = 0

(x - 10)(x - 3) = 0

x = 10 and x = 3

So, the value of x is 3.