Parallel lines inside triangles

A line parallel to one side of a triangle creates equal corresponding angles and a smaller similar triangle. Prove the angle correspondence before writing ratios; distinguish a whole side from one part of it.

Example

In \(\triangle ABC\), \(D\) lies on \(AB\), \(E\) lies on \(AC\), and \(DE\parallel BC\). Then \(\angle ADE=\angle ABC\), \(\angle AED=\angle ACB\) (corresponding angles), so \(\triangle ADE\sim\triangle ABC\) (AA).

If \(AD=4\text{ cm}\), \(DB=6\text{ cm}\), and \(AC=15\text{ cm}\), then \(AB=10\text{ cm}\) and \(\dfrac{AE}{15}=\dfrac4{10}\). Hence \(AE=6\text{ cm}\) and \(EC=9\text{ cm}\).

The sketch illustrates correspondence and is not a scale drawing. Using \(AD:DB\) in place of \(AD:AB\) mixes part-to-part with part-to-whole ratios.