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Alternate Exterior Angles

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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? After completing this Concept, you'll be able to answer these questions and apply alternate exterior angle theorems to find the measure of unknown angles.

Watch This

CK-12 Alternate Exterior Angles

Watch the portions of this video dealing with alternate exterior angles.

James Sousa: Angles and Transversals

Then watch this video.

James Sousa: Proof of Alternate Exterior Angles Converse

Guidance

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

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

If l || m , then \angle 1 \cong \angle 2 .

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 then l || m .

Example A

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

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

(3y+53)^\circ & = (7y-55)^\circ\\108 = 4y\\27 = y

If y = 27 , then each angle is [3(27) + 53]^\circ = 134^\circ .

Example B

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

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^\circ angle and \angle a are alternate exterior angles. If m\angle a = 130^\circ , then the lines are parallel.

\angle a + 40^\circ & = 180^\circ && \text{by the Linear Pair Postulate}\\\angle a & = 140^\circ

140^\circ \neq 130^\circ , so Franklin Way and Chavez Avenue are not parallel streets.

Example C

Which lines are parallel if \angle AFG \cong \angle IJM ?

These two angles are alternate exterior angles so if they are congruent it means that \overleftrightarrow{CG} || \overleftrightarrow{HK} .

CK-12 Alternate Exterior Angles

Guided Practice

Give THREE examples of pairs of alternate exterior angles in the diagram below:

Answers:

There are many examples of alternate exterior angles in the diagram. Here are some possible answers:

1.  \angle 1 and  \angle 14

2.  \angle 2 and  \angle 13

3.  \angle 12 and  \angle 13

Practice

  1. Find the value of x if m\angle 1 = (4x + 35)^\circ, \ m\angle 8 = (7x - 40)^\circ :
  2. Are lines 1 and 2 parallel? Why or why not?

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

  1. m\angle 2 = (8x)^\circ and m\angle 7 = (11x-36)^\circ
  2. m\angle 1 = (3x+5)^\circ and m\angle 8 = (4x-3)^\circ
  3. m\angle 2 = (6x-4)^\circ and m\angle 7 = (5x+10)^\circ
  4. m\angle 1 = (2x-5)^\circ and m\angle 8 = (x)^\circ

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.

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