Question

1. if two triangles are similar, their corresponding angles are:

a. proportional
b. different
c. equal
d. supplementary

1. a triangular billboard frame has side lengths 6 meters, 9 meters, and 12 meters. another triangular frame has side lengths 3 meters, 4.5 meters, and 6 meters. are the frames similar?

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

1. two fences are intended to be parallel. one fence divides the triangle’s side into 9 m and 15 m segments, and the other divides the opposite side into 12 m and x segments. what is x to confirm the fences are parallel?

a. 24 m
b. 17 m
c. 20 m
d. 16 m

Explanation:

Question 13

Recall the definition of similar triangles: Similar triangles have corresponding angles equal and corresponding sides proportional. So we analyze each option:

• Option a: Corresponding sides are proportional, not angles. Eliminate.
• Option b: Corresponding angles of similar triangles are not different. Eliminate.
• Option c: By definition of similar triangles, corresponding angles are equal. Correct.
• Option d: Supplementary angles sum to 180°, which is not the case for corresponding angles of similar triangles. Eliminate.

To check similarity, we use the SSS (Side - Side - Side) similarity criterion, which states that if the ratios of the corresponding sides of two triangles are equal, the triangles are similar.
First, find the ratios of the corresponding sides:
For the first triangle with sides 6, 9, 12 and the second with sides 3, 4.5, 6:
Ratio of first sides: $\frac{3}{6}=\frac{1}{2}$
Ratio of second sides: $\frac{4.5}{9}=\frac{1}{2}$
Ratio of third sides: $\frac{6}{12}=\frac{1}{2}$
Since all three ratios are equal, by SSS criterion, the triangles are similar.

• Option a: Angles do match (since sides are proportional, angles are equal). Eliminate.
• Option b: SAS criterion requires two sides and included angle, here we have all three sides. Eliminate.
• Option c: Sides are proportional (ratios are equal). Eliminate.
• Option d: Correct as per SSS similarity.

Step 1: Apply the Basic Proportionality Theorem (Thales' theorem)

The Basic Proportionality Theorem states that if a line is drawn parallel to one side of a triangle, intersecting the other two sides, then it divides those sides proportionally. Here, the two fences are parallel, so we can apply the theorem. So we have the proportion $\frac{9}{15}=\frac{12}{x}$.

Step 2: Solve for \(x\)

Cross - multiply the proportion: \(9\times x = 15\times12\)
\(9x=180\)
Divide both sides by 9: \(x=\frac{180}{9} = 20\)

Answer:

c. Equal

Question 14