Question

(a) complete the congruence statements below.
$\angle p\cong$ (choose one)
$\angle q\cong$ (choose one)
$\angle r\cong$ (choose one)
(b) find the following ratios.
write your answers as fractions in simplest form.
$\frac{ac}{pq}=\square$
$\frac{cb}{qr}=\square$
$\frac{ab}{pr}=\square$
(c) select the true statement.
$\circ \triangle pqr$ and $\triangle acb$ are similar.

Explanation:

Step1: Match sides for similarity

First, order the sides of each triangle by length:

• $\triangle PQR$: $PQ=12$, $QR=20$, $PR=24$
• $\triangle ACB$: $AC=21$, $CB$ (unknown, but we use proportionality), $AB=42$

Find the scale factor: $\frac{AC}{PQ}=\frac{21}{12}=\frac{7}{4}$, $\frac{AB}{PR}=\frac{42}{24}=\frac{7}{4}$. So the corresponding angles are:
$\angle P \cong \angle C$, $\angle Q \cong \angle A$, $\angle R \cong \angle B$

Step2: Calculate side ratios

Ratio 1: $\frac{AC}{PQ}$

$\frac{AC}{PQ}=\frac{21}{12}=\frac{7}{4}$

Ratio 2: $\frac{CB}{QR}$

First find $CB$: since scale factor is $\frac{7}{4}$, $CB=20\times\frac{7}{4}=35$. Then $\frac{CB}{QR}=\frac{35}{20}=\frac{7}{4}$

Ratio 3: $\frac{AB}{PR}$

$\frac{AB}{PR}=\frac{42}{24}=\frac{7}{4}$

Step3: Verify similarity

All corresponding side ratios are equal ($\frac{7}{4}$), so the triangles are similar.

Answer:

(a)
$\angle P \cong \angle C$
$\angle Q \cong \angle A$
$\angle R \cong \angle B$

(b)
$\frac{AC}{PQ}=\frac{7}{4}$
$\frac{CB}{QR}=\frac{7}{4}$
$\frac{AB}{PR}=\frac{7}{4}$

(c)
$\triangle PQR$ and $\triangle ACB$ are similar.