Question

here is a diagram with triangles abc and def.
find the length of df by setting up a proportion.
df = units

Explanation:

Step1: Find the third angle of triangle ABC

In triangle \(ABC\), using the angle - sum property of a triangle (\(\angle A+\angle B+\angle C = 180^{\circ}\)). Given \(\angle A = 85^{\circ}\) and \(\angle B=56^{\circ}\), then \(\angle C=180^{\circ}-(85^{\circ}+56^{\circ})=39^{\circ}\).

Step2: Prove similarity of triangles

In \(\triangle ABC\) and \(\triangle DFE\), \(\angle B=\angle F = 56^{\circ}\) and \(\angle C=\angle E = 39^{\circ}\). By the AA (angle - angle) similarity criterion, \(\triangle ABC\sim\triangle DFE\).

Step3: Set up the proportion

Since \(\triangle ABC\sim\triangle DFE\), the ratios of corresponding sides are equal. \(\frac{BC}{FE}=\frac{AC}{DE}\). We know \(BC = 24\), \(FE = 6\), \(AC = 20\). Let \(DE=x\). The proportion is \(\frac{24}{6}=\frac{20}{x}\). Cross - multiply: \(24x=6\times20\).

Step4: Solve the proportion

\(24x = 120\), then \(x=\frac{120}{24}=5\).

Answer:

\(5\)