Question

use the figure to answer the question. which statement gives enough additional information about the figure to prove △pro is similar to △sto? a. $overline{rp}paralleloverline{st}$ b. $ot = \frac{1}{2}or$ c. $overline{ps}congoverline{rt}$ d. $overline{st}congoverline{rp}$ e. $angle ropcongangle tos$

Explanation:

Step1: Recall similarity - criteria

If two - triangles have two pairs of corresponding angles equal, they are similar. Also, if a pair of corresponding angles are equal and the sides including these angles are in proportion, the triangles are similar. Another way is if the corresponding sides of two triangles are in proportion, the triangles are similar.

Step2: Analyze option A

If $\overline{RP}\parallel\overline{ST}$, then $\angle R\cong\angle T$ and $\angle P\cong\angle S$ (alternate - interior angles). By the AA (angle - angle) similarity criterion, $\triangle PRO\sim\triangle STO$.

Step3: Analyze option B

$OT = \frac{1}{2}OR$ only gives a ratio of two sides but no information about angles or other side - ratios to prove similarity.

Step4: Analyze option C

$\overline{PS}\cong\overline{RT}$ gives information about non - corresponding sides of the two triangles and does not help in proving similarity.

Step5: Analyze option D

$\overline{ST}\cong\overline{RP}$ gives information about non - corresponding sides of the two triangles and does not help in proving similarity.

Step6: Analyze option E

$\angle ROP\cong\angle TOS$ is a pair of vertical angles. But this alone is not enough to prove the similarity of $\triangle PRO$ and $\triangle STO$ without more information about other angles or side - ratios.

Answer:

A. $\overline{RP}\parallel\overline{ST}$