What if you were presented with two angles that are on the interior of two parallel lines cut by a transversal but on opposite sides of the transversal? 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 and apply alternate interior angle theorems to find the measure of unknown angles.
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CK12 Alternate Interior Angles
Watch the portions of this video dealing with alternate interior angles.
James Sousa: Angles and Transversals
Then watch this video.
James Sousa: Proof that Alternate Interior Angles Are Congruent
Finally, watch this video.
James Sousa: Proof of Alternate Interior Angles Converse
Guidance
Alternate interior angles are two angles that are on the interior of \begin{align*}l\end{align*}
Alternate Interior Angles Theorem: If two parallel lines are cut by a transversal, then the alternate interior angles are congruent.
If \begin{align*}l  m\end{align*}
Converse of Alternate Interior Angles Theorem: If two lines are cut by a transversal and the alternate interior angles are congruent, then the lines are parallel.
If then \begin{align*}l  m\end{align*}
Example A
Find the value of \begin{align*}x\end{align*}
The two given angles are alternate interior angles and equal.
\begin{align*}(4x10)^\circ & = 58^\circ\\ 4x & = 68\\ x & = 17\end{align*}
Example B
True or false: alternate interior angles are always congruent.
This statement is false, but is a common misconception. Remember that alternate interior angles are only congruent when the lines are parallel.
Example C
What does \begin{align*}x\end{align*}
The angles are alternate interior angles, and must be equal for \begin{align*}a  b\end{align*}
\begin{align*}3x+16^\circ & = 5x54^\circ\\ 70 = 2x\\ 35 = x\end{align*}
To make \begin{align*}a  b, \ x = 35\end{align*}
CK12 Alternate Interior Angles
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Guided Practice
Use the given information to determine which lines are parallel. If there are none, write none. Consider each question individually.

\begin{align*}\angle EAF \cong \angle FJI\end{align*}
∠EAF≅∠FJI 
\begin{align*}\angle EFJ \cong \angle FJK\end{align*}
∠EFJ≅∠FJK 
\begin{align*}\angle DIE \cong \angle EAF\end{align*}
∠DIE≅∠EAF
Answers:
 None

\begin{align*}\overleftrightarrow{CG}  \overleftrightarrow{HK}\end{align*}
CG←→HK←→− 
\begin{align*}\overleftrightarrow{BI}  \overleftrightarrow{AM}\end{align*}
BI←→AM←→−
Explore More
 Is the angle pair \begin{align*}\angle 6\end{align*}
∠6 and \begin{align*}\angle 3\end{align*}∠3 congruent, supplementary or neither?  Give two examples of alternate interior angles in the diagram:
For 34, find the values of \begin{align*}x\end{align*}
For question 5, use the picture below. Find the value of \begin{align*}x\end{align*}

\begin{align*}m\angle 4 = (5x  33)^\circ, \ m\angle 5 = (2x + 60)^\circ\end{align*}
m∠4=(5x−33)∘, m∠5=(2x+60)∘
 Are lines \begin{align*}l\end{align*} and \begin{align*}m\end{align*} parallel? If yes, how do you know?
For 710, what does the value of \begin{align*}x\end{align*} have to be to make the lines parallel?
 \begin{align*}m\angle 4 = (3x7)^\circ\end{align*} and \begin{align*}m\angle 5 = (5x21)^\circ\end{align*}
 \begin{align*}m\angle 3 = (2x1)^\circ\end{align*} and \begin{align*}m\angle 6 = (4x11)^\circ\end{align*}
 \begin{align*}m\angle 3 = (5x2)^\circ\end{align*} and \begin{align*}m\angle 6 = (3x)^\circ\end{align*}
 \begin{align*}m\angle 4 = (x7)^\circ\end{align*} and \begin{align*}m\angle 5 = (5x31)^\circ\end{align*}