3.4. Alternate Interior Angles http://www.ck12.org
Guidance
Alternate Interior Anglesare two angles that are on theinterior oflandm, but on opposite sides of the transversal.
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:l||m
Prove:^63 ∼=^66
TABLE3.2:
Statement Reason
1.l||m Given2.^63 ∼=^67 Corresponding Angles Postulate
3.^67 ∼=^66 Vertical Angles Theorem
4.^63 ∼=^66 Transitive PoC
There are several ways we could have done this proof. For example, Step 2 could have been^62 ∼=^6 6 for the same
reason, followed by^62 ∼=^6 3. We could have also proved that^64 ∼=^6 5.
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
Findm^6 1.
m^62 = 115 ◦because they are corresponding angles and the lines are parallel.^6 1 and^6 2 are vertical angles, so
m^61 = 115 ◦also.
(^6) 1 and the 115◦angle are alternate interior angles.
Example B
Find the measure of the angle andx.
The two given angles are alternate interior angles so, they are equal. Set the two expressions equal to each other and
solve forx.
( 4 x− 10 )◦= 58 ◦
4 x= 68 ◦
x= 17 ◦
Example C
Prove the Converse of the Alternate Interior Angles Theorem.
Given:landmand transversalt
6 3 ∼= (^66)