Question

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quadrilateral abcd and quadrilateral pqrs are similar. the lengths of (overline{ab}) and (overline{cd}) are 15 units each, and the lengths of (overline{ad}) and (overline{bc}) are 10 units each.
use the given information to complete the following sentences.
if the length of (overline{pq}) is 6 units, then the length of (overline{ps}) is (\boldsymbol{\text{units}}). if (mangle adc) is (62^{circ}) and (mangle bcd) is (118^{circ}), then (mangle qrs) is (\boldsymbol{\text{°}}).
note: picture not drawn to scale.

Explanation:

Step1: Identify Similar Quadrilaterals

Since \(ABCD \sim PQRS\), corresponding sides are proportional. \(AB = CD = 15\), \(AD = BC = 10\), and \(PQ = 6\). Let's find the scale factor. The ratio of \(PQ\) to \(AB\) is \(\frac{PQ}{AB}=\frac{6}{15}=\frac{2}{5}\).

Step2: Find Length of \(PS\)

\(PS\) corresponds to \(AD\). So, \(PS = AD\times\frac{2}{5}\). Substituting \(AD = 10\), we get \(PS = 10\times\frac{2}{5}=4\) units.

Step3: Find \(m\angle QRS\)

In a parallelogram (since \(AB = CD\) and \(AD = BC\), \(ABCD\) is a parallelogram), consecutive angles are supplementary. \(m\angle ADC + m\angle BCD=62^\circ + 118^\circ = 180^\circ\), so \(ABCD\) is a parallelogram. For similar figures, corresponding angles are equal. \(\angle QRS\) corresponds to \(\angle BCD\), so \(m\angle QRS = 118^\circ\).

Answer:

The length of \(\overline{PS}\) is 4 units. The measure of \(\angle QRS\) is \(118^\circ\).