Question

1. which of the following is not a criterion for triangle similarity?

a. aa
b. sss
c. ssa
d. sas

1. a surveyor measures two triangular plots of land. in the first plot, the two sides are 40 meters and 60 meters, with an included angle of 70°. in the second plot, the two sides are 20 meters and 30 meters, with the same included angle. are the plots similar?

a. yes, by the sas criterion
b. no, the sides are not proportional
c. yes, by the aa criterion
d. no, the angles do not match

1. a surveyor measures a triangular plot of land and finds two sides are 45 meters and 60 meters. the hypotenuse of the triangle is 75 meters. which theorem justifies this relationship?

a. triangle proportionality theorem
b. aa similarity criterion
c. sas similarity criterion
d. pythagorean theorem

Explanation:

Question 4

To determine which is not a triangle similarity criterion, we recall the criteria: AA (Angle - Angle), SSS (Side - Side - Side) similarity, and SAS (Side - Angle - Side) similarity are valid for triangle similarity. SSA (Side - Side - Angle) is not a valid similarity (or congruence in some cases) criterion for triangles as it does not guarantee similarity.

First, check the ratios of the sides. For the first plot, sides are 40 m and 60 m. For the second plot, sides are 20 m and 30 m. The ratio of 40/20 = 2 and 60/30 = 2, so the sides are proportional. The included angle is the same (70°) for both. The SAS (Side - Angle - Side) similarity criterion states that if two sides of one triangle are proportional to two sides of another triangle and the included angles are equal, the triangles are similar.

We have a triangle with two sides 45 m, 60 m and hypotenuse 75 m. The Pythagorean Theorem states that for a right - triangle, $a^{2}+b^{2}=c^{2}$, where $c$ is the hypotenuse and $a,b$ are the other two sides. Let's check: $45^{2}+60^{2}=2025 + 3600=5625$ and $75^{2}=5625$. So the Pythagorean Theorem justifies this relationship.

Answer:

c. SSA

Question 5