PROBLEMS ON VERTICALLY OPPOSITE ANGLES

Find the unknown angles in the following figures.

Problem 1 :

Solution :

Since a and 133 are vertically opposite, they will be equal.

So, ∠a = 133.

Problem 2 :

Solution :

Since c and 52 are vertically opposite, they will be equal.

So, ∠c = 52

Problem 3 :

Solution :

Since b and 73 are vertically opposite, they will be equal.

So, ∠b = 73

Problem 4 :

Solution :

81 = 65 + d

Subtracting 65 on both sides, we get

81 - 65 = d

d = 16

Problem 5 :

Find the value of m.

Solution :

m + 20 = 100

Subtracting 20 on both sides.

m = 100 - 20

m = 80

Problem 6 :

Solution :

3t + 12 = 66

Subtracting 12 on both sides.

3t = 66 - 12

3t = 54

Dividing by 3, we get

t = 54/3

t = 18

Problem 7 :

Find the value of p.

Solution :

2p + 30 = 108

Subtracting 30 on both sides.

2p = 108 - 30

2p = 78

Dividing by 2, we get

p = 78/2

p = 39

Problem 8 :

Find the value of z.

Solution :

58 and 2z - 10 are vertically opposite angles.

58 = 2z - 10

Add 10 on both sides.

68 = 2z

dividing by 2

z = 68/2

z = 34

Problem 9 :

Find the value of y.

Solution :

102 - 2y and 78 are vertically opposite angles.

78 = 102 - 2y

Add 2y on both sides.

2y + 78 = 102

Subtracting 78, we get

2y = 102 - 78

2y = 24

Dividing by 2 on both sides.

y = 24/2

y = 12

Problem 10 :

Find the value of r.

Solution :

126 and 180 - 3r are vertically opposite angles.

126 = 180 - 3r

Add 3r on both sides.

126 + 3r = 180

Subtracting 126 on both sides.

3r = 180 - 126

3r = 54

r = 54/3

r = 18

Problem 11 :

Tell whether the angles are adjacent or vertical. Then find the value of x.

Solution :

The shown angles x and 35 are next to each other. Then it must be adjacent angles.

x + 35 = 90

x = 90 - 35

x = 55

Problem 12 :

The measures of two adjacent angles have a ratio of 3 : 5. The sum of the measures of the two adjacent angles is 120°. What is the measure of the larger angle

Solution :

The sum of two adjacent angles = 120

The given angles be 3x and 5x.

3x + 5x = 120

8x = 120

x = 120/8

x = 15

3x ==> 3(15) ==> 45

5x ==> 5(15) ==> 75

So, the given angle measures are 45 and 75.

Problem 13 :

The iron cross is a skiing trick in which the tips of the skis are crossed while the skier is airborne. Find the value of x in the iron cross shown.

Solution :

127 and 2x + 41 are vertical angles, they must be equal.

127 = 2x + 41

2x = 127 - 41

2x = 86

x = 86/2

x = 43

So, the value of x is 43.

Problem 14 :

Determine whether the statement is always, sometimes, or never true.

a) When the measure of ∠1 is 70°, the measure of ∠3 is 110°. When the measure of ∠4 is 120°, the measure of ∠1 is 60°.

b) ∠2 and ∠3 are congruent.

c)  The measure of ∠1 plus the measure of ∠2 equals the measure of ∠3 plus the measure of ∠4.

Solution :

a) ∠1 = ∠3 (Vertically opposite angles)

It is never true.

b) ∠2 and ∠3 may be congruent. So, it is sometimes.

c)  Given that,

∠1 + ∠2 = ∠3 + ∠4

∠1 and ∠2 are supplementary angles and ∠3 and ∠4 are supplementary. Then they must be equal.

Problem 15 :

For safety reasons, a ladder should make a 15° angle with a wall. Is the ladder shown leaning at a safe angle? Explain.

Solution :

a) The adjacent angle of 120 degree is 60 degree. The sum of interior angles of a triangle is 180 degree.

Let x be the angle with a wall.

60 + 90 + x = 180

150 + x = 180

x = 180 - 150

x = 30

So, the ladder shown leaning at a safe angle.

Problem 16 :

What are the measures of the other three angles formed by the intersection?

Solution :

<1 + 132 = 180

<1 = 180 - 132

<1 = 48

<1 and <3 are vertically opposite angles.

<2 = 132 (vertically opposite angles)

Problem 17 :

Describe and correct the error in naming a pair of vertical angles.

Solution :

<ACB and <BCD are next to each other, then they are called as adjacent angles.