# Same Side Interior Angles

## Angles on the same side of a transversal and inside the lines it intersects.

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Same Side Interior Angles

What if you were presented with two angles that are on the same side of a transversal, but inside the lines? How would you describe these angles and what could you conclude about their measures?

### 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. \begin{align*}\angle 3\end{align*} and \begin{align*}\angle 5\end{align*} are same side interior angles.

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

So, if \begin{align*}l \ || \ m\end{align*} and both are cut by \begin{align*}t\end{align*}, then \begin{align*}m \angle 3 + m \angle 5 = 180^\circ\end{align*} and \begin{align*}m \angle 4 + m \angle 6 = 180^\circ\end{align*}.

Converse of the Same Side Interior Angles Theorem: If two lines are cut by a transversal and the consecutive interior angles are supplementary, then the lines are parallel.

#### Recognizing Same Side Interior Angles

Using the picture above, list all the pairs of same side interior angles.

Same Side Interior Angles: \begin{align*}\angle 4\end{align*} and \begin{align*}\angle 6\end{align*}, \begin{align*}\angle 5\end{align*} and \begin{align*}\angle 3\end{align*}.

#### Measuring Angles

1. Find \begin{align*}m \angle 2\end{align*}.

Here, \begin{align*}m \angle 1 = 66^\circ\end{align*} because they are alternate interior angles. \begin{align*}\angle 1\end{align*} and \begin{align*}\angle 2\end{align*} are a linear pair, so they are supplementary.

\begin{align*}m\angle 1 + m\angle 2 & = 180^\circ\\ 66^\circ + m\angle 2 & = 180^\circ\\ m\angle 2 & = 114^\circ\end{align*}

This example shows why if two parallel lines are cut by a transversal, the same side interior angles are supplementary.

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

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

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

### Examples

#### Example 1

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

These 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*}113^\circ + 67^\circ = 180^\circ\end{align*}, therefore \begin{align*}l \ || \ m\end{align*}.

#### Example 2

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 3

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.

1. \begin{align*}\angle 5\end{align*} and \begin{align*}\angle 8\end{align*}
2. \begin{align*}\angle 2\end{align*} and \begin{align*}\angle 3\end{align*}

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

1. \begin{align*}\angle AFD\end{align*} and \begin{align*}\angle BDF\end{align*} are supplementary
2. \begin{align*}\angle DIJ\end{align*} and \begin{align*}\angle FJI\end{align*} are supplementary

For 6-11, what does the value of \begin{align*}x\end{align*} have to be to make the lines parallel?

1. \begin{align*}m\angle 3 = (3x+25)^\circ\end{align*} and \begin{align*}m\angle 5 = (4x-55)^\circ\end{align*}
2. \begin{align*}m\angle 4 = (2x+15)^\circ\end{align*} and \begin{align*}m\angle 6 = (3x-5)^\circ\end{align*}
3. \begin{align*}m\angle 3 = (x+17)^\circ\end{align*} and \begin{align*}m\angle 5 = (3x-5)^\circ\end{align*}
4. \begin{align*}m\angle 4 = (3x+12)^\circ\end{align*} and \begin{align*}m\angle 6 = (4x-1)^\circ\end{align*}
5. \begin{align*}m\angle 3 = (2x+14)^\circ\end{align*} and \begin{align*}m\angle 5 = (3x-2)^\circ\end{align*}
6. \begin{align*}m\angle 4 = (5x+16)^\circ\end{align*} and \begin{align*}m\angle 6 = (7x-4)^\circ\end{align*}

For 12-13, determine whether the statement is true or false.

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

For questions 14-15, use the image.

1. What is the same side interior angle with \begin{align*}\angle 3\end{align*}?
2. Are the lines parallel? Explain.

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