Question

determining similarity are the triangles similar? if so, state the similarity and the postulate or theorem that justifies your answer. 13. triangle rpq with sides rp=30, pq=24, qr=18 and angle at p is 37°. triangle efd with sides ef=15, fd=25, de=20 and angle at d is 37°.

Explanation:

Step1: Check Proportional Sides

Calculate ratios of corresponding sides:
$\frac{RP}{FD} = \frac{30}{25} = \frac{6}{5}$,
$\frac{PQ}{DE} = \frac{24}{20} = \frac{6}{5}$,
$\frac{QR}{EF} = \frac{18}{15} = \frac{6}{5}$.
All ratios are equal, so sides are proportional.

Step2: Check Included Angle

Both triangles have a $37^\circ$ angle. For $\triangle RPQ$ and $\triangle FDE$, the $37^\circ$ angle is included between the proportional sides.

Step3: Apply SAS Similarity

By SAS (Side - Angle - Side) Similarity Theorem, if two sides of one triangle are proportional to two sides of another triangle and the included angles are equal, the triangles are similar.

Answer:

Yes, the triangles are similar by the SAS (Side - Angle - Side) Similarity Theorem. $\triangle RPQ \sim \triangle FDE$ (or $\triangle PQR \sim \triangle DEF$ depending on correspondence)