CONSECUTIVE EXTERIOR ANGLES THEOREM

Consecutive Exterior Angles :

If two lines are cut by a transversal, the pair of angles on the same side of the transversal and outside the two lines are called consecutive exterior angles.

In the figure above, ∠1 and ∠8 are consecutive exterior angles, and also ∠2 and ∠7 are consecutive angles.

Consecutive Exterior Angles Theorem :

If two parallel lines are cut by a transversal, then the pairs of consecutive exterior angles are supplementary.

Given : m||n, p is transversal.

Prove : ∠1 and ∠8 are supplementary and ∠2 and ∠7 are supplementary.

Solved Problems

Problem 1 :

In the figure shown below, m∠1 = 105°. Find the measure of ∠8.

Solution :

In the figure above, lines m and n are parallel and p is transversal.

By Theorem,  ∠1 and ∠8 are supplementary.

m∠1 + m∠8  =  180°

Substitute m∠1 = 105°. 

105° + m∠8  =  180°

Subtract 105° from each side. 

m∠8  =  75°

Problem 2 :

In the figure shown below, m∠1 = 102°. Find the measures ∠8, ∠15 and ∠10.

Solution :

In the figure above, lines m and n are parallel, p and q are parallel.

By Theorem,  ∠1 and ∠8 are supplementary.

m∠1 + m∠8  =  180°

Substitute m∠1 = 102°. 

102° + m∠8  =  180°

Subtract 102° from each side. 

m∠8  =  78°

By Theorem,  ∠1 and ∠10 are supplementary.

m∠1 + m∠10  =  180°

Substitute m∠1 = 102°. 

102° + m∠10  =  180°

Subtract 102° from each side. 

m∠10  =  78°

By Theorem,  ∠10 and ∠15 are supplementary.

m∠10 + m∠15  =  180°

Substitute m∠10 = 78°. 

78° + m∠15  =  180°

Subtract 78° from each side. 

m∠15  =  102°

Therefore, 

m∠8  =  78°

m∠10  =  78°

m∠15  =  102°

Problem 3 :

In the figure shown below, lines m and n are parallel and p is transversal. Find the value of x. 

Solution :

In the figure shown above, lines m and n are parallel and p is transversal.

By Theorem, (3x + 28)° and 5x° are supplementary. 

(3x + 28)° + 5x°  =  180°

3x + 28 + 5x  =  180

8x + 28  =  180

Subtract 28 from each side. 

8x  =  152

Divide each side by 8.

x  =  19

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