Question

multiple choice - identify the choice that best completes the statement or answers the question. 1. two triangles are shown. which is a true statement about the two triangles? a. the triangles are not similar. b. △abc ~ △edf by aa similarity postulate c. △abc ~ △fde by sas similarity postulate d. △abc ~ △fde by aa similarity postulate 2. state if the triangles in each pair are similar. if so, state how they are similar. a. aa similarity b. sas similarity c. sss similarity d. not similar 3. the two triangles are similar. what is the length of the base of the larger triangle? a. 11 units b. 14 units c. 15 units d. 21 units

Explanation:

Step1: Analyze angles in first pair of triangles

In $\triangle ABC$, $\angle B = 90^{\circ}$ and $\angle A=37^{\circ}$, so $\angle C=180^{\circ}-90^{\circ} - 37^{\circ}=53^{\circ}$. In $\triangle EDF$, $\angle D = 90^{\circ}$ and $\angle E = 53^{\circ}$. The corresponding angles $\angle B=\angle D = 90^{\circ}$ and $\angle C=\angle E=53^{\circ}$. By AA (Angle - Angle) similarity postulate, $\triangle ABC\sim\triangle EDF$.

Step2: Analyze second pair of triangles

Only two angles are marked as congruent in the second - pair of triangles. Since we have two pairs of congruent angles, by AA similarity postulate, the triangles are similar.

Step3: Find base length in third pair of similar triangles

For similar triangles, the ratios of corresponding sides are equal. Let the base of the larger triangle be $x$. The ratio of the corresponding sides of the two similar triangles is $\frac{6}{18}=\frac{1}{3}$. Then $\frac{7}{x}=\frac{1}{3}$, so $x = 21$ units.

Answer:

1. b. $\triangle ABC\sim\triangle EDF$ by AA Similarity Postulate
2. a. AA similarity
3. d. 21 units