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 angle of triangle ABC

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

Step2: Prove similarity

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

Step3: Set up proportion

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

Step4: Solve for \(x\)

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

Answer:

\(5\)