Question

david drew △ pqr and △ stu so that ∠ p ≅ ∠ s, pr = 12, su = 3, pq = 20, and st = 5. are △ pqr and △ stu similar? if so, identify the similarity postulate or theorem that applies.
a. cannot be determined
b. similar - aa
c. similar - sas
d. similar - sss

Explanation:

Step1: Check Proportional Sides

We have two triangles, \(\triangle PQR\) and \(\triangle STU\), with \(\angle P\cong\angle S\). We need to check the ratios of the sides around the congruent angles.
For \(\triangle PQR\), the sides around \(\angle P\) are \(PQ = 20\) and \(PR=12\).
For \(\triangle STU\), the sides around \(\angle S\) are \(ST = 5\) and \(SU = 3\).
Calculate the ratios: \(\frac{PQ}{ST}=\frac{20}{5} = 4\) and \(\frac{PR}{SU}=\frac{12}{3}=4\). So the sides around the congruent angles are proportional.

Step2: Apply SAS Similarity

The SAS (Side - Angle - Side) similarity postulate states that if two sides of one triangle are proportional to two sides of another triangle and the included angles are congruent, then the triangles are similar. Here, \(\angle P\cong\angle S\) (included angle) and \(\frac{PQ}{ST}=\frac{PR}{SU} = 4\), so by SAS similarity, \(\triangle PQR\sim\triangle STU\).

Answer:

C. Similar - SAS