ALTERNATE INTERIOR ANGLES THEOREM

Alternate Interior Angles :

When two parallel lines are cut by a transversal, the pair of angles formed inside the parallel lines but on the opposite sides of the transversal are called alternate interior angles. 

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

Alternate Interior Angles Theorem :

If two parallel lines are cut by a transversal, then Alternate Interior Angles are congruent. 

Given : m||n, p is transversal. 

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

Proof :

Since m || n, by the Corresponding Angles Postulate, 

∠1 ≅ ∠5

By the definition of congruent angles,

m∠1  =  m∠5

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

m∠5 + m∠6  =  180°

Substitute m∠1 for m∠5.

m∠1 + m∠6  =  180°

Subtract m∠1 from each side. 

m∠6  =  180° - m∠1

m∠6  =  m∠4

Therefore, 

m∠4 ≅ m∠6

We can prove that m∠3 ≅ m∠5 using the same method. 

Alternate Interior Angles Theorem – Converse

When two lines are cut by a transversal, if alternate interior angles have equal measure, then the two lines are parallel. 

In the figure above, lines m and n are parallel. Because, a pair of alternate interior angles have equal measure. 

Example 1 :

In the figure shown below, m∠8 = 75°. Find m∠3.

Solution :

In the figure above, lines m and n are parallel, ∠5 and ∠8 form a linear pair. 

m∠5 + m∠8  =  180°

Substitute m∠8 = 75°. 

m∠5 + 75°  =  180°

Subtract 75° from each side. 

m∠5  =  105°

∠3 and ∠5 are alternate interior angles.

∠3 ≅ ∠5

m∠3  =  m∠5

Substitute m∠5 = 105°.

m∠3  =  105°

Example 2 :

In the figure shown below, m∠3 = 102°. Find the measures of ∠5, ∠11 and ∠13.

Solution :

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

∠3 and ∠5 are alternate interior angles. 

∠3 ≅ ∠5

m∠3  =  m∠5

Substitute m∠3 = 102°.

102°  =  m∠5

∠5 and ∠13 are corresponding angles. 

∠5 ≅ ∠13

m∠5  =  m∠13

Substitute m∠5 = 102°. 

102°  =  m∠13

∠11 and ∠13 are alternate interior angles. 

∠11 ≅ ∠13

m∠11  =  m∠13

Substitute m∠13 = 102°. 

m∠11  =  102°

Therefore, 

m∠5  =  102°

m∠11  =  102°

m∠13  =  102°

Example 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 m and n are parallel and p is transversal. Angles 5x° and (3x + 28)° are alternate interior angles and they are congruent. 

By the definition of congruent angles, 

5x°  =  (3x + 28)°

5x  =  3x + 28

Subtract 3x from each side. 

2x  =  28

Divide each side by 2.

x  =  14

Example 4 :

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

Solution :

In the figure above, a° and 118° are alternate interior angles and they are equal.

a°  =  118°

b° and 144° are alternate interior angles and they are equal.

b°  =  144°

In the figure above, 

x  =  a + b

=  118 + 144

=  262

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