Question

which property of similar triangles is always true? a. only the included angle must be congruent b. only one pair of corresponding angles is congruent c. corresponding angles are congruent, and corresponding sides are proportional d. all sides are congruent if ∠p≅∠s, pq = 18m, pr = 24m, and st = 12m, find the missing side su to prove similarity between △pqr and △stu using sas. a. 18m b. 14m c. 16m d. 20m which of the following situations does not use the sas similarity criterion? a. determining the similarity of two bird nests based on size ratios b. proving that two bridges have proportional supports c. verifying similarity between two mountain slopes d. proving congruence of two triangles

Explanation:

Step1: Recall property of similar triangles

Similar - triangles have corresponding - angles congruent and corresponding - sides proportional.

Step2: Analyze each option for the first question

Option a is incorrect as it's not just the included angle. Option b is incorrect as all corresponding angles are congruent in similar triangles. Option d is for congruent triangles, not similar ones. So, for the first question, the correct property is that corresponding angles are congruent and corresponding sides are proportional.

Step3: Use SAS similarity for the second question

For $\triangle PQR$ and $\triangle STU$ with $\angle P\cong\angle S$, by SAS (Side - Angle - Side) similarity, $\frac{PQ}{ST}=\frac{PR}{SU}$. Substitute $PQ = 18m$, $PR = 24m$, and $ST = 12m$ into $\frac{18}{12}=\frac{24}{SU}$. Cross - multiply: $18\times SU=24\times12$. Then $SU=\frac{24\times12}{18}=16m$.

Step4: Understand SAS similarity criterion for the third question

SAS similarity criterion is used for showing similarity of triangles by comparing two pairs of sides and the included angle. Proving congruence of two triangles uses SAS congruence criterion which is different from SAS similarity. Options a, b, and c can potentially use similarity concepts including SAS similarity for non - triangular shapes approximated as triangular in some cases, but proving congruence of two triangles is not an application of SAS similarity criterion.

Answer:

1. c. Corresponding angles are congruent, and corresponding sides are proportional
2. c. 16m
3. d. Proving congruence of two triangles