Question

what must be true for the sas criterion to apply?
○ a. two angles must be congruent
○ b. the ratios of all sides must be equal
○ c. all pairs of sides and angles must be congruent
○ d. two pairs of corresponding sides must be proportional, and the included angle must be congruent

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

a building casts a shadow of 20 meters, while a 5 - meter pole casts a shadow of 2 meters at the same time. what is the height of the building?
○ a. 25 meters
○ b. 50 meters
○ c. 40 meters
○ d. 30 meters

Explanation:

First Question (SAS Criterion)

The SAS (Side - Angle - Side) similarity criterion states that for two triangles to be similar (in the case of similarity) or congruent (in the case of congruence), two pairs of corresponding sides must be proportional (for similarity) or equal (for congruence) and the included angle must be congruent.

• Option a: SAS does not require two angles to be congruent. It focuses on sides and the included angle. Eliminate a.
• Option b: The ratio of all sides being equal is the SSS (Side - Side - Side) similarity criterion, not SAS. Eliminate b.
• Option c: All pairs of sides and angles being congruent is the definition of congruent triangles, not a requirement for SAS. Eliminate c.
• Option d: Matches the definition of the SAS (for similarity, proportional sides; for congruence, equal sides) criterion as it requires two pairs of corresponding sides to be proportional (or equal) and the included angle to be congruent.

To determine if two triangles are similar, we can use the SSS (Side - Side - Side) similarity criterion which states that if the ratios of the corresponding sides of two triangles are equal, then the triangles are similar.
For the first triangle with sides \(6\) m, \(9\) m, and \(12\) m and the second triangle with sides \(3\) m, \(4.5\) m, and \(6\) m:

• Calculate the ratio of the first pair of sides: \(\frac{6}{3}=2\)
• Calculate the ratio of the second pair of sides: \(\frac{9}{4.5} = 2\)
• Calculate the ratio of the third pair of sides: \(\frac{12}{6}=2\)

Since all the ratios of the corresponding sides are equal (\(2\) in this case), by the SSS similarity criterion, the two triangles are similar.

• Option a: The sides are proportional (ratios are all \(2\)), so a is incorrect.
• Option b: Since the ratios of all corresponding sides are equal, the triangles are similar by the SSS Criterion. This is correct.
• Option c: We are using the SSS criterion here, not SAS, so c is incorrect.
• Option d: The angles of similar triangles are equal, so d is incorrect.

Step 1: Set up the proportion

Let \(h\) be the height of the building. At the same time, the ratio of the height of an object to the length of its shadow is constant. So we can set up the proportion: \(\frac{h}{20}=\frac{5}{2}\)

Step 2: Solve for \(h\)

Cross - multiply: \(2h=20\times5\)
\(2h = 100\)
Divide both sides by \(2\): \(h=\frac{100}{2}=50\)

Answer:

d. Two pairs of corresponding sides must be proportional, and the included angle must be congruent

Second Question (Similar Triangles - SSS Criterion)