In the last section, you investigated relationships between pairs of angles created when a transversal crosses two parallel lines. In this section, you will investigate and use informal arguments to describe two important relationships among the angles of a triangle:

• the sum of the measures of the interior angles and
• the measure of an exterior angle as compared to certain interior angles.

Sum of the Measures of Interior Angles of a Triangle

Click on the image below to access the interactive. Use the interactive to investigate the relationship among the measures of the three interior angles of a triangle. Click on the arrow in the bottom left corner or slowly drag the slider at the bottom. Notice the congruent angles.

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Use the interactive to answer the questions below.

• In the interactive, m ∥ n. How is ∠BCA(∠Z) related to ∠CBE(∠Z)?

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  Check Your Answer

  They are alternate interior angles, and they are congruent.

• In the interactive, m ∥ n. How is ∠BAC(∠X) related to ∠DBA(∠X)?

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  Check Your Answer

  They are alternate interior angles, and they are congruent.

• Examine the three angles with a common vertex at point B. What is special about these three angles? What does that tell you about the angle measures?

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  Check Your Answer

  These three angles form a straight line, and a straight line has an angle measure of 180°. The sum of the measures of these three angles must be equal to 180°.

Exterior Angles of a Triangle

A triangle has several exterior angles, or angles formed outside the triangle by extending one of the three sides. There can be up to six exterior angles depending on which sides are extended.

In the diagram shown, triangle ABC has three exterior angles: ∠MAC, ∠ABN, and ∠PCB.

Click on the image below to access the interactive. Use the interactive below to investigate the relationship between an exterior angle of a triangle and the two opposite, or remote, interior angles. Click and drag the red vertex of the triangle to change the shape of the triangle. Use the sliders to move the interior angles to the same vertex as the exterior angle so that you can compare the measures of the three angles.

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Need additional directions?

Use the interactive to answer the questions below.

• How do the purple and green interior angles compare to the exterior angle?

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  The purple and green interior angles line up along the exterior angle.

• How does the sum of the measures of the two interior angles (purple and green) compare to the measure of the exterior angle?

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  The sum of the measures of the two interior angles is the same as the measure of the exterior angle.

• If the measure of the purple interior angle is 75° and the measure of the green interior angle is 20°, what is the measure of the exterior angle? How do you know?

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  Check Your Answer

  95°, because the measure of the exterior angle is equal to the sum of the measures of the two remote (opposite) interior angles; i.e., 75° + 20° = 95°.

Pause and Reflect

For both the sum of the interior angles of a triangle and the measure of an exterior angle of a triangle, straight lines were important. Thinking about the number of degrees there are in an angle created by one straight line, why do you think that is the case?

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Check Your Answer

An angle that is actually a straight line (called a straight angle) has a measure of 180°. A straight line helps you see why the sum of the measures of the interior angles of any triangle is 180°. A straight line is also useful for measuring the exterior angle of a triangle.

Practice

1. In the diagram below, DE ∥ AC. Use the angle relationships created by parallel lines to explain why
   a° + b° + c° = 180°.
   
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   What is the relationship between ∠DBA and ∠BAC for parallel lines DE and AC? What is the relationship between ∠EBC and ∠BCA?
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   Check Your Answer

   ∠DBA and ∠BAC are alternate interior angles between two parallel lines, so m∠DBA = a°. Likewise, ∠EBC and ∠BCA are alternate interior angles between two parallel lines, so
   m∠EBC = c°. Since ∠DBA, ∠ABC, and ∠EBC form a straight line, the sum of their measures is 180°. Therefore, a° + b° + c° = 180°.
2. Write an equation that you can use to solve for x, then determine the value of x.
   
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   What is the sum of the measures of the interior angles of a triangle?
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   1 over 2 ( 1 2 x)° + (2x)° + 58° = 180°
   x = 484 over 5 4 5 °
3. Write an equation that you can use to solve for x, then determine the value of x.
   
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   What is the relationship between the measure of an exterior angle and the measures of the two remote (opposite) interior angles?
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   Check Your Answer

   3 over 4 ( 3 4 x)° + 31° = 82°
   x = 68