Question

1. triangle pqr is similar to triangle wxy. which proportion describes the relationship between corresponding sides of the triangles? a. $\frac{qr}{xy}=\frac{6}{3}$ b. $\frac{pq}{wx}=\frac{2}{4}$ c. $\frac{qr}{wx}=\frac{3}{4}$ d. $\frac{pq}{xy}=\frac{2}{6}$ 19. find the $mangle q$. a. $48^{circ}$ b. $91^{circ}$ c. $135^{circ}$ d. $50^{circ}$

Explanation:

Step1: Recall similarity - side ratio

For similar triangles, the ratios of corresponding sides are equal. In \(\triangle PQR\) and \(\triangle WXY\), \(PQ = 2\) in, \(WX=4\) in, \(QR = 3\) in, \(XY = 6\) in. The ratio of corresponding sides is \(\frac{PQ}{WX}=\frac{2}{4}\) and \(\frac{QR}{XY}=\frac{3}{6}=\frac{1}{2}\).

Step2: Analyze options

• Option a: \(\frac{QR}{XY}=\frac{3}{6}

eq\frac{6}{3}\), so a is incorrect.

• Option b: \(\frac{PQ}{WX}=\frac{2}{4}\), this is correct.
• Option c: \(\frac{QR}{WX}=\frac{3}{4}\), but \(QR\) and \(WX\) are not corresponding sides, so c is incorrect.
• Option d: \(PQ\) and \(XY\) are not corresponding sides, \(\frac{PQ}{XY}=\frac{2}{6}\) is incorrect.

Step3: Solve for \(x\) in the second - part

In a polygon, if \(RQ\) is a straight - line segment, then \(12x+3x + 15=180\) (since the sum of angles on a straight line is \(180^{\circ}\)).
Combining like terms: \(15x+15 = 180\).
Subtract 15 from both sides: \(15x=180 - 15=165\).
Divide both sides by 15: \(x = 11\).
Then \(m\angle Q=3x + 15=3\times11+15=33 + 15=48^{\circ}\).

Answer:

1. B. \(\frac{PQ}{WX}=\frac{2}{4}\)
2. A. \(48^{\circ}\)