### Same Side Interior Angles

**Same side interior angles** are two angles that are on the same side of the transversal and on the interior of (between) the two lines.

**Same Side Interior Angles Theorem:** If two parallel lines are cut by a transversal, then the same side interior angles are supplementary.

If \begin{align*}l || m\end{align*}, then \begin{align*}m\angle 1 + m\angle 2 = 180^\circ\end{align*}.

**Converse of the Same Side Interior Angles Theorem:** If two lines are cut by a transversal and the same side interior angles are supplementary, 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 same side of a transversal and between the two parallel lines crossed by the transversal? How would you describe these angles and what could you conclude about their measures?

### Examples

#### Example 1

Is \begin{align*}l || m\end{align*}? How do you know?

These angles are Same Side Interior Angles. So, if they add up to \begin{align*}180^\circ\end{align*}, then \begin{align*}l || m\end{align*}.

\begin{align*}130^\circ + 67^\circ = 197^\circ\end{align*}, therefore the lines are not parallel.

#### Example 2

Give two examples of same side interior angles in the diagram:

There are MANY examples of same side interior angles in the diagram. Two are \begin{align*}\angle 6\end{align*} and \begin{align*} \angle 10\end{align*}, and \begin{align*}\angle 8\end{align*} and \begin{align*}\angle 12\end{align*}.

#### Example 3

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

The given angles are same side interior angles. Because the lines are parallel, the angles add up to \begin{align*}180^\circ\end{align*}.

\begin{align*}(2x+43)^\circ + (2x-3)^\circ & = 180^\circ\\ (4x+40)^\circ & = 180^\circ\\ 4x & = 140\\ x & =35\end{align*}

#### Example 4

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

\begin{align*}y\end{align*} is a same side interior angle with the marked right angle. This means that \begin{align*}90^\circ +y=180\end{align*} so \begin{align*}y=90\end{align*}.

#### Example 5

Find the value of \begin{align*}x\end{align*} if \begin{align*}m\angle 3 = (3x +12)^\circ\end{align*} and \begin{align*}\ m\angle 5 = (5x + 8)^\circ\end{align*}.

These are same side interior angles, so set up an equation and solve for \begin{align*}x\end{align*}. Remember that same side interior angles add up to \begin{align*}180^\circ\end{align*}.

\begin{align*}(3x+12)^\circ+(5x+8)^\circ&=180^\circ\\ (8x+20)^\circ&=180^\circ\\ 8x&=160\\x&=20\end{align*}

### Review

For questions 1-2, use the diagram to determine if each angle pair is congruent, supplementary or neither.

- \begin{align*}\angle 5\end{align*} and \begin{align*}\angle 8\end{align*}
- \begin{align*}\angle 2\end{align*} and \begin{align*}\angle 3\end{align*}
- Are the lines parallel? Justify your answer.

In 4-5, use the given information to determine which lines are parallel. If there are none, write none. Consider each question individually.

- \begin{align*}\angle AFD\end{align*}
*and*\begin{align*}\angle BDF\end{align*} are supplementary - \begin{align*}\angle DIJ\end{align*}
*and*\begin{align*}\angle FJI\end{align*} are supplementary

For 6-8, what does the value of \begin{align*}x\end{align*} have to be to make the lines parallel?

- \begin{align*}m\angle 3 = (3x+25)^\circ\end{align*} and \begin{align*}m\angle 5 = (4x-55)^\circ\end{align*}
- \begin{align*}m\angle 4 = (2x+15)^\circ\end{align*} and \begin{align*}m\angle 6 = (3x-5)^\circ\end{align*}
- \begin{align*}m\angle 3 = (x+17)^\circ\end{align*} and \begin{align*}m\angle 5 = (3x-5)^\circ\end{align*}

For 9-10, determine whether the statement is true or false.

- Same side interior angles are on the same side of the transversal.
- Same side interior angles are congruent when lines are parallel.

### Review (Answers)

To see the Review answers, open this PDF file and look for section 3.6.