CONSECUTIVE INTERIOR ANGLES THEOREM

Consecutive Interior Angles :

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

In the figure above, ∠4 and ∠5 are consecutive interior angles, and also ∠3 and ∠6 are consecutive angles.

Consecutive Interior Angles Theorem :

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

Given : m||n, p is transversal. 

Prove : ∠4 and ∠5 are supplementary and ∠3 and ∠6 are supplementary. 

Solved Problems

Problem 1 :

In the figure shown below, m∠3 = 105°. Find the measure of ∠6.

Solution :

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

By Theorem, ∠3 and ∠6 are supplementary. 

m∠3 + m∠6  =  180°

Substitute m∠3 = 105°. 

105° + m∠6  =  180°

Subtract 105° from each side. 

m∠6  =  75°

Problem 2 :

In the figure shown below, m∠3 = 102°. Find the measures ∠6, ∠12 and ∠13.

Solution :

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

By Theorem, ∠3 and ∠6 are supplementary. 

m∠3 + m∠6  =  180°

Substitute m∠3 = 102°. 

102° + m∠6  =  180°

Subtract 102° from each side. 

m∠6  =  78°

By Theorem, ∠3 and ∠12 are supplementary. 

m∠3 + m∠12  =  180°

Substitute m∠3 = 102°. 

102° + m∠12  =  180°

Subtract 102° from each side. 

m∠12  =  78°

By Theorem, ∠12 and ∠13 are supplementary. 

m∠12 + m∠13  =  180°

Substitute m∠12 = 78°. 

78° + m∠13  =  180°

Subtract 78° from each side. 

m∠13  =  102°

Therefore, 

m∠6  =  78°

m∠12  =  78°

m∠13  =  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 above, lines m and n are parallel and p is transversal.

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

5x° + (3x + 28)°  =  180°

5x + 3x + 28  =  180

8x + 28  =  180

Subtract 28 from each side. 

8x  =  152

Divide each side by 8.

x  =  19

Problem 4 :

Using a 3^rd parallel Line – Auxiliary Line, find the value of x. 

Solution :

In the figure above, a° and 50° are corresponding angles and they are equal.

a°  =  50°

By Theorem, b° and 100° are supplementary. 

b° + 100°  =  180°

Subtract 100° from each side. 

b°  =  80°

In the figure above, 

x  =  a + b

=  50 + 80

=  130

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