Question

what are the values of n and k? (triangles are not drawn to scale.)
show your work here
n=
k=

Explanation:

Step1: Determine the similarity of triangles

Since \(AB\parallel ED\), \(\triangle ABC\sim\triangle EDC\) (by the AA similarity criterion, as \(\angle ABC=\angle EDC\) and \(\angle BAC = \angle DEC\) because of parallel lines).

Step2: Set up the proportion for side lengths

For similar triangles \(\triangle ABC\) and \(\triangle EDC\), the ratios of corresponding sides are equal.
We have \(\frac{AB}{ED}=\frac{BC}{DC}=\frac{AC}{EC}\).
Given \(AB = 5\), \(ED=10\), \(EC = 16\), \(DC = 20\).
Using \(\frac{AB}{ED}=\frac{AC}{EC}\), substitute the values: \(\frac{5}{10}=\frac{n}{16}\).
Cross - multiply: \(10n=5\times16\).
Solve for \(n\): \(n=\frac{5\times16}{10}=8\).
Using \(\frac{AB}{ED}=\frac{BC}{DC}\), substitute \(AB = 5\), \(ED = 10\), \(DC=20\). Let \(BC = k\), then \(\frac{5}{10}=\frac{k}{20}\).
Cross - multiply: \(10k=5\times20\).
Solve for \(k\): \(k = 10\).

Answer:

\(n = 8\), \(k = 10\)