Question

a surveyor measures a triangle on a map where two sides are 50 m and 75 m, with an included angle of 60°. another triangle has side lengths 25 m and 37.5 m with an included angle of 60°. are the triangles similar? a. yes, by the aa criterion b. yes, by the sas criterion c. no, the angles do not match d. no, the sides are not proportional a flagpole has sides that form a triangle with the ground. the height of the flagpole is 8 meters, and the distance to the tip of its shadow is 10 meters, forming a 50° angle with the ground. a nearby pole has a height of 5 meters, a shadow length of 7 meters, and forms the same angle with the ground. are the triangles formed by the flagpole and the pole similar? a. no, the sides are not proportional b. no, the angles are not congruent c. yes, by the sas criterion d. yes, by the sss criterion

Explanation:

Step1: Check side - length ratios for the first pair of triangles

For the first pair of triangles, the ratios of corresponding sides are $\frac{25}{50}=\frac{1}{2}$ and $\frac{37.5}{75}=\frac{1}{2}$, and the included angles are equal ($60^{\circ}$). According to the Side - Angle - Side (SAS) similarity criterion, if the ratios of two pairs of corresponding sides of two triangles are equal and the included angles are congruent, the triangles are similar.

Step2: Check side - length ratios for the second pair of triangles

For the second pair of triangles, the height of the flag - pole is 8 meters and the distance to the tip of its shadow is 10 meters. For the nearby pole, the height is 5 meters and the shadow length is 7 meters. The ratio of the heights is $\frac{5}{8}$ and the ratio of the hypotenuses (distance to the tip of the shadow) is $\frac{7}{10}$. Since $\frac{5}{8}
eq\frac{7}{10}$, the sides are not proportional.

Answer:

1. b. Yes, by the SAS Criterion
2. a. No, the sides are not proportional