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 units. if \\( m \angle adc \\) is \\( 62 ^ { \circ } \\) and \\( m \angle bcd \\) is \\( 118 ^ { \circ } \\), then \\( m \angle qrs \\) is

Explanation:

Step1: Find the scale factor

Since quadrilateral \(ABCD\) and \(PQRS\) are similar, and \(AB = 15\) units, \(PQ=6\) units. The scale factor \(k=\frac{PQ}{AB}=\frac{6}{15}=\frac{2}{5}\).

Step2: Calculate the length of \(PS\)

Given \(AD = 10\) units. Using the scale - factor, \(PS=AD\times k\). Substitute \(AD = 10\) and \(k=\frac{2}{5}\), we get \(PS = 10\times\frac{2}{5}=4\) units.

Step3: Use the property of similar polygons for angles

Similar polygons have corresponding angles equal. In parallelogram \(ABCD\), \(\angle BCD\) and \(\angle QRS\) are corresponding angles.

Answer:

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