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In this lesson, you investigated and then used informal arguments to show angle relationships among parallel lines with a transversal, interior, and exterior angle relationships with a triangle, and the angle-angle criterion for similar triangles.

You also used these relationships to write and solve equations.

Angle Relationships in Parallel Lines

Interactive exercise. Assistance may be required. Review angle relationships by identifying the alternate interior, alternate exterior, and corresponding angles. Click on the image to access the interactive.

After observing all the angles, answer the following:

List all the alternate interior angles.
Interactive popup. Assistance may be required.
Need a hint?
Click on the Alternate Interior tab to view the angles.

Interactive popup. Assistance may be required.
Check Your Answer
∠HEF and ∠GFL, ∠OEF and ∠IFG
List all the alternate exterior angles.
Interactive popup. Assistance may be required.
Need a hint?
Click on the Alternate Exterior tab to view the angles.

Interactive popup. Assistance may be required.
Check Your Answer
∠HEG and ∠NFL, ∠OEG and ∠IFN
List all the corresponding angles.
Interactive popup. Assistance may be required.
Need a hint?
Click on the Corresponding Angles tab to view the angles.

Interactive popup. Assistance may be required.
Check Your Answer
∠HEG and ∠IFG, ∠OEG and ∠LFG, ∠OEF and ∠LFN, ∠HEF and ∠IFN
Consecutive interior angles are Interactive button. Assistance may be required. _______________. supplementary
Consecutive exterior angles are Interactive button. Assistance may be required. _______________. supplementary

Below is a diagram reviewing the above information.

interior and exterior angles of triangle ABC

Angle Relationships in Triangles

First, review the interior angles of a triangle.

Interactive exercise. Assistance may be required. Click on the image to access the interactive. Move one of the points, and note the sum of the interior angles of the triangle.

Now, note the relationships between the measure of the exterior angle and the interior angles.

Interactive exercise. Assistance may be required. Click on the image to access the interactive. Move point A while observing the angle measures.

Fill in the blanks.

The sum of the measures of the interior angles of a triangle is Interactive button. Assistance may be required. _______________. 180°
The measure of an exterior angle is equal to the Interactive button. Assistance may be required. _______________ sum of the measures of the two remote (opposite) interior angles.

Below is a diagram reviewing the above information.

Angle-Angle Criterion for Similar Triangles

Interactive exercise. Assistance may be required. Click on the image to access the interactive. Move the points around while observing the angles and ask yourself: do any of the interior angle measures change?

NOTE: Angles with the same measures are marked with the same marks.

Angles R and N are congruent and corresponding angles. At any time, were the measures of these two angles not equal? Angles Q and M are congruent and corresponding angles. At any time, were the measures of these two angles not equal?
Interactive popup. Assistance may be required.
Check Your Answer
No
If two pairs of corresponding angles in two triangles are congruent, then the two triangles are Interactive button. Assistance may be required. _______________ similar.

Below is a diagram reviewing the above information.