Question

what do you notice about the angle measures of the similar triangles above?
the angle - angle criterion described below allows us to determine if two triangles are similar by observing just their angle measures.
angle - angle criterion

• if two pairs of corresponding angles in a pair of triangles are ______, then the two triangles are ______.
• rewrite the angle - angle criterion in your own words:

in 1 - 3, use the angle - angle criterion to determine if the triangles are similar. explain your choice.
1.
2.

1. jace constructed triangle abc, where (mangle a = 53^{circ}), and (mangle b = 112^{circ}). ruthie constructed triangle rst, where (mangle s = 112^{circ}), and (mangle t = 15^{circ}). are the two triangles similar? explain.

Explanation:

Step1: Fill AA criterion blanks

If two pairs of corresponding angles in a pair of triangles are congruent (equal), then the two triangles are similar.

Step2: Rewrite AA criterion in own words

If two triangles share two pairs of matching angles that have the same measure, the triangles are similar.

Step3: Solve problem 1: Find missing angles

First triangle: $180^\circ - 87^\circ - 58^\circ = 35^\circ$
Second triangle: $180^\circ - 87^\circ - 36^\circ = 57^\circ$
Only one pair of equal angles ($87^\circ$), so no two matching pairs.

Step4: Solve problem 2: Find missing angles

First triangle: $180^\circ - 80^\circ - 60^\circ = 40^\circ$
Second triangle: $180^\circ - 80^\circ - 54^\circ = 46^\circ$
Only one pair of equal angles ($80^\circ$), so no two matching pairs.

Step5: Solve problem 3: Find missing angles

Triangle ABC: $180^\circ - 53^\circ - 112^\circ = 15^\circ$
Triangle RST: $180^\circ - 112^\circ - 15^\circ = 53^\circ$
Two pairs of equal angles: $\angle A = \angle R = 53^\circ$, $\angle B = \angle S = 112^\circ$ (or $\angle C = \angle T = 15^\circ$).

Answer:

1. Filled Angle-Angle Criterion:

• If two pairs of corresponding angles in a pair of triangles are congruent (equal), then the two triangles are similar.

1. Rewritten criterion: If two triangles have two sets of corresponding angles with identical measures, the triangles are similar.
2. Problem 1: The triangles are not similar. They only share one pair of equal angles ($87^\circ$), so they do not meet the angle-angle criterion.
3. Problem 2: The triangles are not similar. They only share one pair of equal angles ($80^\circ$), so they do not meet the angle-angle criterion.
4. Problem 3: The triangles are similar. Triangle ABC has angles $53^\circ, 112^\circ, 15^\circ$, and triangle RST has angles $53^\circ, 112^\circ, 15^\circ$. This gives two pairs of corresponding equal angles, satisfying the angle-angle criterion.