VERTICAL ANGLES THEOREM WORKSHEET

Problem 1 :

In the diagram shown below, WX and YZ are two straight lines intersecting. If m∠1 = 140°, find the measures of the other three angles. 

Problem 2 :

Look at the picture shown below and answer the following questions. 

(i)  Are 1 and 3 vertical angles ?

(i)  Are 2 and 4 vertical angles ?

Problem 3 :

In the diagram shown below, using Linear Pair Postulate, solve for x and y. Then, find the angle measures and analyze your results with Vertical Angles Theorem. 

Problem 4 :

In the stair railing shown at the right, if ∠6 has a measure of 130°, find the measures of the other three angles.

Problem 5 :

Problem 6 :

Problem 7 :

Problem 8 :

Problem 9 :

Use the diagram below.

a. Use the Vertical Angles Theorem to write an equation.

b. Solve your equation to find the value of x.

c. Find the measures of the acute angles formed by the lines.

d. Find the measures of the obtuse angles formed by the lines.

1. Answer :

∠1 and ∠2 form a linear pair, they are supplementary. 

m∠1 + m∠2  =  180°

Substitute m∠1 = 140°.

140° + m∠2  =  180°

Subtract 140° from each side.

m∠2  =  40°

∠1 and ∠3 are vertical angles, they are equal. 

m∠3  =  m∠1

m∠3  =  140°

∠2 and ∠4 are vertical angles, they are equal. 

m∠4  =  m∠2

m∠4  =  40°

Therefore,

m∠2  =  40°

m∠3  =  140°

m∠4  =  40°

2. Answer :

(i)  Are ∠1 and ∠3 vertical angles ?

(iI)  Are ∠2 and ∠4 vertical angles ?

Solution (i) : 

No. The sides of the angles do not form two pairs of opposite rays.

Solution (ii) : 

No. The sides of the angles do not form two pairs of opposite rays.

3. Answer :

Use the fact that the sum of the measures of angles that form a linear pair is 180°. 

Solving for x :

∠AED and ∠DEB form a linear pair.

m∠AED + m∠DEB  =  180°

Substitute m∠AED = (3x + 5)° and m∠DEB = (x + 15)°.

(3x + 5)° + (x + 15)°  =  180°

Simplify.

4x + 20  =  180

Subtract 20 from each side.  

4x  =  160

Divide each side by 4.

x  =  40

Solving for y :

∠AEC and ∠CEB form a linear pair. 

m∠AEC + m∠CEB  =  180°

Substitute m∠AEC = (y + 20)° and m∠CEB = (4y - 15)°.

(y + 20)° + (4y - 15)°  =  180°

Simplify.

5y + 5  =  180

Subtract 5 from each side.  

5y  =  175

Divide each side by 5.

y  =  35

Use substitution to find the angle measures :

m∠AED  =  (3x + 5)°  =  (3 • 40 + 5)°  =  125°

m∠DEB  =  (x + 15)°  =  (40 + 15)°  =  55°

m∠AEC  =  ( y + 20)°  =  (35 + 20)°  =  55°

m∠CEB  =  (4y - 15)°  =  (4 • 35 - 15)°  =  125°

So, the angle measures are 125°, 55°, 55°, and 125°. Because the vertical angles are congruent, the result is reasonable.

4. Answer :

∠5 and ∠6 form a linear pair.

m∠5 + m∠6  =  180°

Substitute m∠6  =  130°

m∠5 + 130°  =   180°

Subtract 130° from both sides.

m∠5  =   50°

∠5 and ∠7 are vertical angles, they are equal. 

m∠7  =  m∠5  =  50°

∠6 and ∠8 are vertical angles, they are equal. 

m∠8  =  m∠6  =  130°

Therefore, 

m∠5  =  50°

m∠7  =  50°

m∠8  =  130°

5. Answer :

Vertical angles are equal,

2r + 3 = 89

2r = 89 - 3

2r = 86

r = 86/2

r = 43

6. Answer :

Vertical angles are equal,

3x = 2x + 16

3x - 2x = 16

x = 16

7. Answer :

Sum of adjacent angles = 180

15t + 20t + 5 = 180

35t + 5 = 180

35t = 180 - 5

35t = 175

t = 175/35

t = 5

So, the value of t is 5.

8. Answer :

Find the values of x and y in the diagram below.

Vertical angles are equal.

7x - 2 = 11x - 34

7x - 11x = -34 + 2

-4x = -32

x = 32/4

x = 8

Sum of adjacent angles = 180

7x - 2 + 18y = 180

Applying the value of x, we get

7(8) - 2 + 18y = 180

56 - 2 + 18y = 180

54 + 18y = 180

18y = 180 - 54

18y = 126

y = 126/18

y = 7

9. Answer :

a) 6x = 4x + 8

b) Solving for x,

6x - 4x = 8

2x = 8

x = 8/2

x = 4

c) Finding acute angle :

Applying the value of x, we get

= 6(4)

= 24

So, the required acute angle is 24 degree.

d) Finding obtuse angle :

= 180 - 24

= 156

So, the required obtuse angle is 156 degree.

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