Processing math: 100%
Skip to main content
Library homepage
 

Text Color

Text Size

 

Margin Size

 

Font Type

Enable Dyslexic Font
K12 LibreTexts

3.6: Alternate Exterior Angles

( \newcommand{\kernel}{\mathrm{null}\,}\)

Angles on opposite sides of a transversal, but outside the lines it intersects.

Alternate exterior angles are two angles that are on the exterior of l and m, but on opposite sides of the transversal.

f-d_e0e1b24cabd3907ddb79e7f5a22391da41b338ac18504818807dfee3+IMAGE_TINY+IMAGE_TINY.png
Figure 3.6.1

Alternate Exterior Angles Theorem: If two parallel lines are cut by a transversal, then the alternate exterior angles are congruent.

f-d_c0e20623c4579ab6d6cba701289c1c80107494240ba3825fcfa7adda+IMAGE_TINY+IMAGE_TINY.png
Figure 3.6.2

If lm, then 12.

Converse of the Alternate Exterior Angles Theorem: If two lines are cut by a transversal and the alternate exterior angles are congruent, then the lines are parallel.

If

f-d_9c7ec17fb02a7141e2063fd5ea712f254b1356eb99032e1f7c1b5905+IMAGE_TINY+IMAGE_TINY.png
Figure 3.6.3

then lm.

What if you were presented with two angles that are on the exterior of two parallel lines cut by a transversal but on opposite sides of the transversal? How would you describe these angles and what could you conclude about their measures?

For Examples 3.6.1 and 3.6.2, use the following diagram:

f-d_acf685f70adc87b4075812b7ecc116dbceabac3970f8ed695303c88b+IMAGE_TINY+IMAGE_TINY.png
Figure 3.6.4

Example 3.6.1

Give an example of a pair of alternate exterior angles.

Solution

1 and 14 (many other possibilities)

Example 3.6.2

Give another example of a pair of alternate exterior angles.

Solution

2 and 13 (many other possibilities, must be different than answer to Example 1)

Example 3.6.3

Find the measure of each angle and the value of \)y\).

f-d_b442f21414ee3d898b7ce8a5109c44ba344a3687b10599412089ed5b+IMAGE_TINY+IMAGE_TINY.png
Figure 3.6.5

Solution

The angles are alternate exterior angles. Because the lines are parallel, the angles are equal.

(3y+53)=(7y55)108=4y27=y

If y=27,theneachangleis\([3(27)+53]=134.

Example 3.6.4

The map below shows three roads in Julio’s town.

f-d_3b7db739f102b41478b199f5e4f610ccbd88018373e4d5ce6e85dc47+IMAGE_TINY+IMAGE_TINY.png
Figure 3.6.6

Solution

Julio used a surveying tool to measure two angles at the intersections in this picture he drew (NOT to scale). Julio wants to know if Franklin Way is parallel to Chavez Avenue.

The 130 angle and \angle a are alternate exterior angles. If ma=130, then the lines are parallel.

a+40=180bytheLinearPairPostulatea=140

140130, so Franklin Way and Chavez Avenue are not parallel streets.

Example 3.6.5

Which lines are parallel if AFGIJM?

f-d_a9669d1041f370e9e54907e878108a83838d6cf9377d8d1f8aac5a10+IMAGE_TINY+IMAGE_TINY.png
Figure 3.6.7

Solution

These two angles are alternate exterior angles so if they are congruent it means that CGHK.

Review

  1. Find the value of x if m1=(4x+35), m8=(7x40):
    f-d_7d690a17ce6b2e3bc2ec2070038ceed705eb9a6272064c8c84195102+IMAGE_TINY+IMAGE_TINY.png
    Figure 3.6.8
  2. Are lines 1 and 2 parallel? Why or why not?
    f-d_9b8cfcfc792afb149f56f2ed620d3111c7c34146302914c254a0c500+IMAGE_TINY+IMAGE_TINY.png
    Figure 3.6.9

For 3-6, what does the value of x have to be to make the lines parallel?

f-d_963dbdfc7b826d8117ff8f1171f6c6bf93e5c8d514fd8352bb797b67+IMAGE_TINY+IMAGE_TINY.png
Figure 3.6.10
  1. m2=(8x) and m7=(11x36)
  2. m1=(3x+5) and m8=(4x3)
  3. m2=(6x4) and m7=(5x+10)
  4. m1=(2x5) and m8=(x)

For 7-10, determine whether the statement is true or false.

  1. Alternate exterior angles are always congruent.
  2. If alternate exterior angles are congruent then lines are parallel.
  3. Alternate exterior angles are on the interior of two lines.
  4. Alternate exterior angles are on opposite sides of the transversal.

Review (Answers)

To see the Review answers, open this PDF file and look for section 3.5.

Resources

Vocabulary

Term Definition
alternate exterior angles Alternate exterior angles are two angles that are on the exterior of two different lines, but on the opposite sides of the transversal.

Additional Resource

Interactive Element

Video: Alternate Exterior Angles Principles - Basic

Activities: Alternate Exterior Angles Discussion Questions

Study Aids: Angles and Transversals Study Guide

Practice: Alternate Exterior Angles

Real World: Alternate Exterior Angles


This page titled 3.6: Alternate Exterior Angles is shared under a CK-12 license and was authored, remixed, and/or curated by CK-12 Foundation via source content that was edited to the style and standards of the LibreTexts platform.

CK-12 Foundation
LICENSED UNDER
CK-12 Foundation is licensed under CK-12 Curriculum Materials License

Support Center

How can we help?