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# 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

### 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 l||m\begin{align*}l || m\end{align*}, then m1+m2=180\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  l||m\begin{align*}l || m\end{align*}

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

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

130+67=197\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 6\begin{align*}\angle 6\end{align*} and 10\begin{align*} \angle 10\end{align*}, and 8\begin{align*}\angle 8\end{align*} and 12\begin{align*}\angle 12\end{align*}.

#### Example 3

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

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

(2x+43)+(2x3)(4x+40)4xx=180=180=140=35\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 y\begin{align*}y\end{align*}.

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

#### Example 5

Find the value of x\begin{align*}x\end{align*} if m3=(3x+12)\begin{align*}m\angle 3 = (3x +12)^\circ\end{align*} and  m5=(5x+8)\begin{align*}\ m\angle 5 = (5x + 8)^\circ\end{align*}.

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

(3x+12)+(5x+8)(8x+20)8xx=180=180=160=20\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. 5\begin{align*}\angle 5\end{align*} and 8\begin{align*}\angle 8\end{align*}
2. 2\begin{align*}\angle 2\end{align*} and 3\begin{align*}\angle 3\end{align*}

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

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

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

1. m3=(3x+25)\begin{align*}m\angle 3 = (3x+25)^\circ\end{align*} and m5=(4x55)\begin{align*}m\angle 5 = (4x-55)^\circ\end{align*}
2. m4=(2x+15)\begin{align*}m\angle 4 = (2x+15)^\circ\end{align*} and m6=(3x5)\begin{align*}m\angle 6 = (3x-5)^\circ\end{align*}
3. m3=(x+17)\begin{align*}m\angle 3 = (x+17)^\circ\end{align*} and m5=(3x5)\begin{align*}m\angle 5 = (3x-5)^\circ\end{align*}

For 9-10, 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.

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

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### Vocabulary Language: English Spanish

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 the two lines.

supplementary angles

Two angles that add up to $180^\circ$.

transversal

A line that intersects two other lines.