VERTICAL ANGLES AND LINEAR PAIRS

About "Vertical angles and linear pairs"

Vertical Angles and Linear Pairs :

Vertical Angles :

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

m∠1 and m∠3 are vertical angles

m∠2 and m∠4 are vertical angles

Linear Pair :

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

m∠5 and m∠6 are a linear pair. 

Vertical angles and linear pairs - Examples

Example 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.

Example 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 º 15)°  =  125°

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

Example 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° 

After having gone through the stuff given above, we hope that the students would have understood "Vertical angles and linear pairs". 

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