### Alternate Interior Angles

**Alternate interior angles** are two angles that are on the interior of \begin{align*}l\end{align*} and \begin{align*}m\end{align*}, 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.

If \begin{align*}l || m\end{align*}, then \begin{align*}\angle 1 \cong \angle 2\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*}.

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?

### Examples

For Examples 1 and 2, use the given information to determine which lines are parallel. If there are none, write none. Consider each question individually.

#### Example 1

\begin{align*}\angle EAF \cong \angle FJI\end{align*}

None

#### Example 2

\begin{align*}\angle EFJ \cong \angle FJK\end{align*}

\begin{align*}\overleftrightarrow{CG} || \overleftrightarrow{HK}\end{align*}

#### Example 3

Find the value of \begin{align*}x\end{align*}.

The two given angles are alternate interior angles and equal.

\begin{align*}(4x-10)^\circ & = 58^\circ\\ 4x & = 68\\ x & = 17\end{align*}

#### Example 4

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 5

What does \begin{align*}x\end{align*} have to be to make \begin{align*}a || b\end{align*}?

The angles are alternate interior angles, and must be equal for \begin{align*}a || b\end{align*}. Set the expressions equal to each other and solve.

\begin{align*}3x+16^\circ & = 5x-54^\circ\\ 70 = 2x\\ 35 = x\end{align*}

To make \begin{align*}a || b, \ x = 35\end{align*}.

### Review

- Is the angle pair \begin{align*}\angle 6\end{align*} and \begin{align*}\angle 3\end{align*} congruent, supplementary or neither?
- Give two examples of alternate interior angles in the diagram:

For 3-4, 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*}

- Are lines \begin{align*}l\end{align*} and \begin{align*}m\end{align*} parallel? If yes, how do you know?

For 7-10, what does the value of \begin{align*}x\end{align*} have to be to make the lines parallel?

- \begin{align*}m\angle 4 = (3x-7)^\circ\end{align*} and \begin{align*}m\angle 5 = (5x-21)^\circ\end{align*}
- \begin{align*}m\angle 3 = (2x-1)^\circ\end{align*} and \begin{align*}m\angle 6 = (4x-11)^\circ\end{align*}
- \begin{align*}m\angle 3 = (5x-2)^\circ\end{align*} and \begin{align*}m\angle 6 = (3x)^\circ\end{align*}
- \begin{align*}m\angle 4 = (x-7)^\circ\end{align*} and \begin{align*}m\angle 5 = (5x-31)^\circ\end{align*}

### Review (Answers)

To see the Review answers, open this PDF file and look for section 3.4.