# VERTICAL ANGLES AND LINEAR PAIRS WORKSHEET

Problem 1 :

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

Problem 2 :

In the diagram shown below, Solve for "x" and "y". Then, find the angle measures.

Problem 3 :

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

Problem 1 :

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

(i)  Are m∠2 and m∠3 a linear pair ?

(ii)  Are m∠3 and m∠4 a linear pair ?

(iii)  Are m∠1 and m∠3 vertical angles ?

(iv)  Are m∠2 and m∠4 vertical angles ?

Solution :

Solution (i) :

No. The angles are adjacent but their non-common sides are not opposite rays.

Solution (ii) :

Yes. The angles are adjacent and their non-common sides are opposite rays.

Solution (iii) :

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

Solution (iv) :

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

Problem 2 :

In the diagram shown below, Solve for x and y. Then, find the angle measures.

Solution :

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

Solving for x :

m∠AED and m∠DEB are a linear pair. So, the sum of their measures is 180°.

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 both sides.

4x  =  160

Divide both sides by 4.

x  =  40

Solving for y :

m∠AEC and m∠CEB are a linear pair. So, the sum of their measures is 180°.

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 both sides.

5y  =  175

Divide both sides by 5.

y  =  35

Use substitution to find the angle measures :

mAED  =  (3x + 5)°  =  (3 • 40 + 5)°  =  125°

mDEB  =  (x + 15)°  =  (40 + 15)°  =  55°

mAEC  =  ( y + 20)°  =  (35 + 20)°  =  55°

mCEB  =  (4y º 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.

Problem 3 :

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

Solution :

m∠6 and m∠7 are a linear pair. So, the sum of their measures is 180°.

m∠6 + m∠7  =  180°

Substitute m∠6  =  130°

130° + m∠7  =   180°

Subtract 130° from both sides.

m∠7  =   5

m∠6 and m∠5 are also a linear pair. So, it follows that m∠7  =  50°.

m∠6 and m∠8 are vertical angles. So, they are congruent and they have same measure.

m∠8  =  m∠6  =  130°

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