Question

△xyz ~ △pqr in each pair. find the unknown measures.
14.
15.

Explanation:

Step1: Find the measure of angle \( b \)

Since \(\triangle XYZ\sim\triangle PQR\), corresponding angles are equal. In \(\triangle XYZ\), using the angle - sum property of a triangle (\(A + B + C=180^{\circ}\)), if \(\angle Y = 40^{\circ}\) and \(\angle Z=89^{\circ}\), then \(\angle X=180-(40 + 89)=51^{\circ}\). \(\angle R\) corresponds to \(\angle X\), so \(b = 51^{\circ}\)

Step2: Find the length of side \(a\)

Since \(\triangle XYZ\sim\triangle PQR\), the ratios of corresponding sides are equal. \(\frac{XY}{PQ}=\frac{YZ}{QR}\). We know \(XY = 12\mathrm{cm}\), \(PQ = 20\mathrm{cm}\), \(YZ = 9\mathrm{cm}\), and \(QR=a\). Using the proportion \(\frac{12}{20}=\frac{9}{a}\). Cross - multiply: \(12a=20\times9\). Then \(12a = 180\), and \(a=\frac{180}{12}=15\mathrm{cm}\)

Step3: For problem 15, find the measure of angle \(s\)

Since \(\triangle XYZ\sim\triangle PQR\), corresponding angles are equal. \(\angle Y\) corresponds to \(\angle Q\). So \(s = 58^{\circ}\)

Step4: Find the length of side \(y\)

Since \(\triangle XYZ\sim\triangle PQR\), \(\frac{XY}{PQ}=\frac{XZ}{PR}\). We know \(XY = 48\mathrm{m}\), \(PQ = 30\mathrm{m}\), \(PR = 35\mathrm{m}\), and \(XZ=y\). Using the proportion \(\frac{48}{30}=\frac{y}{35}\). Cross - multiply: \(30y=48\times35\). Then \(30y = 1680\), and \(y=\frac{1680}{30}=56\mathrm{m}\)

Answer:

For problem 14: \(a = 15\mathrm{cm}\), \(b = 51^{\circ}\)
For problem 15: \(s = 58^{\circ}\), \(y = 56\mathrm{m}\)