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3.7: Same Side Interior Angles

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

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.

f-d_a8d9b91df19f74f34fa164377fd63a7dbc7c055fc5ff57ecbe9634bf+IMAGE_TINY+IMAGE_TINY.png
Figure 3.7.1

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

f-d_91ce9eaa5c4a427538d80d88915519d0a21cfbabb9534f1f08cb0850+IMAGE_TINY+IMAGE_TINY.png
Figure 3.7.2

If lm, then m1+m2=180.

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

f-d_67d340685341a2f8c3028d57936483cd844c7c3451729fd0f1cfc6b0+IMAGE_TINY+IMAGE_TINY.png
Figure 3.7.3

then lm

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?

Example 3.7.1

Is lm? How do you know?

Solution

These angles are Same Side Interior Angles. So, if they add up to 180, then lm.

130+67=197, therefore the lines are not parallel.

f-d_74e7e9a69fe84ce8ccc252f406f9133f3718d58847748e2a3b59e5a1+IMAGE_TINY+IMAGE_TINY.png
Figure 3.7.4

Example 3.7.2

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

f-d_acf685f70adc87b4075812b7ecc116dbceabac3970f8ed695303c88b+IMAGE_TINY+IMAGE_TINY.png
Figure 3.7.5

Solution

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

Example 3.7.3

Find the value of x.

f-d_44014a86cf753c8bea839356ae4c8b578df175e0f1e2aab4cdf96ccc+IMAGE_TINY+IMAGE_TINY.png
Figure 3.7.6

Solution

The given angles are same side interior angles. Because the lines are parallel, the angles add up to 180.

(2x+43)+(2x3)=180(4x+40)=1804x=140x=35

Example 3.7.4

Find the value of y.

f-d_ef9c9e5e1682147552c76009265b7367a902b7a14102006cb31b7352+IMAGE_TINY+IMAGE_TINY.pngFigure 3.7.7

Solution

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

Example 3.7.5

Find the value of x if m3=(3x+12) and m5=(5x+8).

f-d_7d690a17ce6b2e3bc2ec2070038ceed705eb9a6272064c8c84195102+IMAGE_TINY+IMAGE_TINY.png
Figure 3.7.8

Solution

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.

(3x+12)+(5x+8)=180(8x+20)=1808x=160x=20

Review

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

f-d_c433bbb430c5a21eef1034baabdc9ea175e53f1b202b64c654c172d5+IMAGE_TINY+IMAGE_TINY.png
Figure 3.7.9
  1. 5 and 8
  2. 2 and 3
  3. Are the lines parallel? Justify your answer.
    f-d_9ea9b140c5da22ccd6c69206ecf7887faa5b7828b1f90c387659b054+IMAGE_TINY+IMAGE_TINY.png
    Figure 3.7.10

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

f-d_a9669d1041f370e9e54907e878108a83838d6cf9377d8d1f8aac5a10+IMAGE_TINY+IMAGE_TINY.png
Figure 3.7.11
  1. AFD and BDF are supplementary
  2. DIJ and FJI are supplementary

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

f-d_963dbdfc7b826d8117ff8f1171f6c6bf93e5c8d514fd8352bb797b67+IMAGE_TINY+IMAGE_TINY.png
Figure 3.7.12
  1. m3=(3x+25) and m5=(4x55)
  2. m4=(2x+15) and m6=(3x5)
  3. m3=(x+17) and m5=(3x5)

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.

Review (Answers)

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

Vocabulary

Term Definition
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.
transversal A line that intersects two other lines.

Additional Resource

Interactive Element

Video: Same Side Interior Angles Principles - Basic

Activities: Same Side Interior Angles Discussion Questions

Study Aids: Angles and Transversals Study Guide

Practice: Same Side Interior Angles

Real World: Alternate Exterior Angles


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