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Alternate Interior Angles

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Alternate Interior Angles

What if you were presented with two angles that are on the interior 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 interior angle theorems to find the measure of unknown angles.

Watch This

CK-12 Alternate Interior Angles

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

James Sousa: Angles and Transversals

Then watch this video.

James Sousa: Proof that Alternate Interior Angles Are Congruent

Finally, watch this video.

James Sousa: Proof of Alternate Interior Angles Converse

Guidance

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

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

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

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

If then l || m.

Example A

Find the value of x.

The two given angles are alternate interior angles and equal.

(4x-10)^\circ & = 58^\circ\\4x & = 68\\x & = 17

Example B

True or false: alternate interior angles are always congruent.

This statement is false, but is a common misconception. Remember that alternate interior angles are only congruent when the lines are parallel.

Example C

What does x have to be to make a || b?

The angles are alternate interior angles, and must be equal for a || b. Set the expressions equal to each other and solve.

3x+16^\circ & = 5x-54^\circ\\70 = 2x\\35 = x

To make a || b, \ x = 35.

CK-12 Alternate Interior Angles

Guided Practice

Use the given information to determine which lines are parallel. If there are none, write none. Consider each question individually.

  1. \angle EAF \cong \angle FJI
  2. \angle EFJ \cong \angle FJK
  3. \angle DIE \cong \angle EAF

Answers:

1. None

2. \overleftrightarrow{CG} || \overleftrightarrow{HK}

3. \overleftrightarrow{BI} || \overleftrightarrow{AM}

Practice

  1. Is the angle pair \angle 6 and \angle 3 congruent, supplementary or neither?
  2. Give two examples of alternate interior angles in the diagram:

For 3-4, find the values of x.

For question 5, use the picture below. Find the value of x.

  1. m\angle 4 = (5x - 33)^\circ, \ m\angle 5 = (2x + 60)^\circ
  1. Are lines l and m parallel? If yes, how do you know?

For 7-10, what does the value of x have to be to make the lines parallel?

  1. m\angle 4 = (3x-7)^\circ and m\angle 5 = (5x-21)^\circ
  2. m\angle 3 = (2x-1)^\circ and m\angle 6 = (4x-11)^\circ
  3. m\angle 3 = (5x-2)^\circ and m\angle 6 = (3x)^\circ
  4. m\angle 4 = (x-7)^\circ and m\angle 5 = (5x-31)^\circ

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