Question

what proportional segment lengths verify that $overline{xz}paralleloverline{pq}$? fill in the boxes to correctly complete the proportion. $\frac{16}{}=\frac{}{}$

Explanation:

Step1: Use similar - triangle property

If $\overline{XZ}\parallel\overline{PQ}$, then $\triangle XPZ\sim\triangle XQY$ (by AA similarity, as the corresponding angles are equal). For similar triangles, the ratios of corresponding sides are equal. That is, $\frac{XP}{XQ}=\frac{XZ}{PQ}$. We know that $XP = 5$, $XQ=5 + 16=21$, $PZ = 2.5$, and $ZY = 8$.

Step2: Set up the proportion

The proportion of the sides of similar triangles gives us $\frac{5}{21}=\frac{2.5}{2.5 + 8}=\frac{5}{21}=\frac{2.5}{10.5}$. The proportion we want is $\frac{XP}{XQ}=\frac{PZ}{ZY}$. Substituting the values, we get $\frac{5}{21}=\frac{2.5}{10.5}$. Cross - multiplying to check: $5\times10.5=52.5$ and $21\times2.5 = 52.5$. The proportion for the given similar - triangle situation is $\frac{5}{21}=\frac{2.5}{10.5}$.

Answer:

$\frac{5}{21}=\frac{2.5}{10.5}$