VERTICAL ANGLES AND LINEAR PAIRS

Vertical Angles :

Two angles are vertical angles, if their sides form two pairs of opposite rays.

∠1 and ∠3 are vertical angles

∠2 and ∠4 are vertical angles

Linear Pair :

Two adjacent angles are a linear pair, if their non-common sides are opposite rays.

∠5 and ∠6 are a linear pair.

Solved Problems

Problem 1 :

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

(i)  Are ∠2 and ∠3 a linear pair ?

(ii)  Are ∠3 and ∠4 a linear pair ?

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

(iv)  Are ∠2 and ∠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 :

∠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.

Problem 3 :

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

Solution :

∠5 and ∠6 form a linear pair, they are supplementary.

m∠5 + m∠6  =  180°

Substitute m∠6  =  130°

m∠5 + 130°  =   180°

Subtract 130° from both sides.

m∠5  =   50°

∠6 and ∠7 also form a linear pair. So, it follows that 

m∠7  =  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°

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