What if you were presented with two angles that are on opposite sides of a transversal, but inside the lines? How would you describe these angles and what could you conclude about their measures? After completing this Concept, you'll be able to answer these questions using your knowledge of alternate interior angles.
Watch This
CK12 Foundation: Chapter3AlternateInteriorAnglesA
Watch the portions of this video dealing with alternate interior angles.
James Sousa: Angles and Transversals
James Sousa: Proof that Alternate Interior Angles Are Congruent
James Sousa: Proof of Alternate Interior Angles Converse
Guidance
Alternate Interior Angles are two angles that are on the interior of
Alternate Interior Angles Theorem: If two parallel lines are cut by a transversal, then the alternate interior angles are congruent.
Proof of Alternate Interior Angles Theorem:
Given:
Prove:
Statement  Reason 

1. 
Given 
2. 
Corresponding Angles Postulate 
3. 
Vertical Angles Theorem 
4. 
Transitive PoC 
There are several ways we could have done this proof. For example, Step 2 could have been
Converse of Alternate Interior Angles Theorem: If two lines are cut by a transversal and alternate interior angles are congruent, then the lines are parallel.
Example A
Find
Example B
Find the measure of the angle and
The two given angles are alternate interior angles so, they are equal. Set the two expressions equal to each other and solve for
Example C
Prove the Converse of the Alternate Interior Angles Theorem.
Given:
Prove:
Statement  Reason 

1. 
Given 
2. 
Vertical Angles Theorem 
3. 
Transitive PoC 
4. 
Converse of the Corresponding Angles Postulate 
Watch this video for help with the Examples above.
CK12 Foundation: Chapter3AlternateInteriorAnglesB
Vocabulary
Alternate Interior Angles are two angles that are on the interior of
Guided Practice
1. Is
2. What does
3. List the pairs of alternate interior angles:
Answers:
1. First, find
2. Because these are alternate interior angles, they must be equal for
3. Alternate Interior Angles:
Interactive Practice
Practice
 Is the angle pair
∠6 and∠3 congruent, supplementary or neither?  Give two examples of alternate interior angles in the diagram:
For 34, find the values of
For question 5, use the picture below. Find the value of

m∠4=(5x−33)∘, m∠5=(2x+60)∘
 Are lines
l andm parallel? If yes, how do you know?
For 712, what does the value of

m∠4=(3x−7)∘ andm∠5=(5x−21)∘ 
m∠3=(2x−1)∘ andm∠6=(4x−11)∘ 
m∠3=(5x−2)∘ andm∠6=(3x)∘ 
m∠4=(x−7)∘ andm∠5=(5x−31)∘ 
m∠3=(8x−12)∘ andm∠6=(7x)∘ 
m∠4=(4x−17)∘ andm∠5=(5x−29)∘
For questions 1315, use the picture below.
 What is the alternate interior angle to
∠4 ?  What is the alternate interior angle to
∠5 ?  Are the two lines parallel? Explain.