In previous grades, you learned that two figures are similar if all of the corresponding angles are congruent and all of the corresponding side lengths are proportional.

In this section, you will refine what you know about similar triangles in order to make arguments and solve problems about similar triangles when you only know the measures of two pairs of corresponding angles. This condition of similar triangles is called the angle-angle criterion.

Click on the image below to access the interactive. Use the interactive to investigate two triangles, Δ ABC and Δ HGI. Use the sliders to adjust the angle values. Click and drag on points A and B to adjust the size of Δ ABC, and click and drag on points G and I to adjust the size of Δ HGI.

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Use the interactive to answer the questions below.

• If you know the measures of two interior angles in a triangle, how can you calculate the measure of the third angle?

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  Check Your Answer

  Use the triangle sum to find the third angle.

• Suppose that in two triangles, the measures of two pairs of corresponding interior angles are equal.
  

  What does this tell you about the measures of the third interior angle in each triangle? Explain your reasoning.

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  Check Your Answer

  The measures of the third interior angle in each triangle must also be equal. In the example, m∠I = 180° − 62° − 49° = 69°. Likewise, m∠C = 180° − 62° − 49° = 69°. Both corresponding angles have a measure of 69°.
• In the interactive, check the box to show side lengths. What do you notice about the lengths of the corresponding sides of Δ ABC and Δ HGI?

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  Check Your Answer

  The corresponding side lengths are proportional.

• In the interactive, check the box to show side lengths. What do you notice about the lengths of the corresponding sides of Δ ABC and Δ HGI?

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  Check Your Answer

  The corresponding side lengths are proportional.

• Are triangles ABC and HGI similar? Explain how you know.

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  Check Your Answer

  Yes. Since all pairs of corresponding angles have equal measures, they are congruent. Also, all pairs of corresponding sides are proportional.

Pause and Reflect

For the angle-angle criterion for similarity, you know that two pairs of corresponding angles are congruent. Why do we not need to state that the third pair of corresponding angles is also congruent?

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Check Your Answer

The sum of the measures of the interior angles of any triangle is 180°. If you know the measures of two of these angles, then the third angle must be a particular number. If two triangles have two pairs of corresponding angles with the same measures, then the third angle in each triangle will also have the same measures.

Practice

1. In the diagram below, AB ∥ DC. AD and BC are transversals that intersect at point E. Use angle relationships between parallel lines to explain how you know that triangle ABE is similar to triangle DCE.
   
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   Need a hint?

   Look for pairs of alternate interior angles. Since AB ∥ DC, alternate interior angles will be congruent.
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   Check Your Answer

   • Since AB ∥ DC, ∠BAE is congruent to ∠CDE because they are alternate interior angles.
   • Since AB ∥ DC, ∠ABE is congruent to ∠DCE because they are alternate interior angles.
   • Since there are two pairs of corresponding angles that are congruent, the angle-angle criterion tells you that Δ ABE is similar to Δ DCE.

   

Use the diagram below to answer questions 2 − 4. Assume that BC || DE.


2. Write two equations that you can use to solve for x and y given that Δ ABC is similar to Δ ADE.

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   Need a hint?

   Identify two pairs of corresponding angles, and apply the angle-angle criterion.
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   Check Your Answer

   3x − 20 = 1.5x + 10
   80 = 8y
3. Determine the values of x and y.

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   Solve each equation from question 1 using properties of real numbers and inverse operations.
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   Check Your Answer

   x = 20, y = 10
4. Determine the measure of ∠ABC.

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   Substitute the value of x into the expression for m∠ABC.
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   Check Your Answer

   m∠ABC = 40°