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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

What if you were presented with two angles that are on the same side of the transversal and on the interior of the two lines? 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 same side interior angle theorems to find the measure of unknown angles.

### Watch This

Watch the portions of this video dealing with same side interior angles.

Then watch this video.

Finally, watch this video.

### Guidance

Same side interior angles are two angles that are on the same side of the transversal and on the interior of 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 $l || m$ , then $m\angle 1 + m\angle 2 = 180^\circ$ .

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 $l || m$ .

#### Example A

Find $z$ .

$z+116^\circ=180^\circ$ so $z = 64^\circ$ by Same Side Interior Angles Theorem.

#### Example B

Is $l || m$ ? How do you know?

These angles are Same Side Interior Angles. So, if they add up to $180^\circ$ , then $l || m$ .

$130^\circ + 67^\circ = 197^\circ$ , therefore the lines are not parallel.

#### Example C

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

There are MANY examples of same side interior angles in the diagram. Two are $\angle 6$ and $\angle 10$ , and $\angle 8$ and $\angle 12$ .

### Guided Practice

1. Find the value of $x$ .

2. Find the value of $y$ .

3. Find the value of $x$ if $m\angle 3 = (3x +12)^\circ$ and $\ m\angle 5 = (5x + 8)^\circ$ .

1. The given angles are same side interior angles. Because the lines are parallel, the angles add up to $180^\circ$ .

$(2x+43)^\circ + (2x-3)^\circ & = 180^\circ\\(4x+40)^\circ & = 180^\circ\\4x & = 140\\x & =35$

2. $y$ is a same side interior angle with the marked right angle. This means that $90^\circ +y=180$ so $y=90$ .

3. These are same side interior angles so set up an equation and solve for $x$ . Remember that same side interior angles add up to $180^\circ$ .

$(3x+12)^\circ+(5x+8)^\circ&=180^\circ\\ (8x+20)^\circ&=180^\circ\\ 8x&=160\\x&=20$

### Practice

For questions 1-2, determine if each angle pair below is congruent, supplementary or neither.

1. $\angle 5$ and $\angle 8$
2. $\angle 2$ and $\angle 3$

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

1. $\angle AFD$ and $\angle BDF$ are supplementary
2. $\angle DIJ$ and $\angle FJI$ are supplementary

For 6-8, what does the value of $x$ have to be to make the lines parallel?

1. $m\angle 3 = (3x+25)^\circ$ and $m\angle 5 = (4x-55)^\circ$
2. $m\angle 4 = (2x+15)^\circ$ and $m\angle 6 = (3x-5)^\circ$
3. $m\angle 3 = (x+17)^\circ$ and $m\angle 5 = (3x-5)^\circ$

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.

### Vocabulary Language: English Spanish

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 the two lines.
supplementary angles

supplementary angles

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

transversal

A line that intersects two other lines.