Question

given: \\( \overline { a e } \parallel \overline { d c } \\) prove: \\( \triangle b f e \sim \triangle c f d \\) 1. \\( \overline { a e } \parallel \overline { d c } \\) 1. given 2. \\( \triangle d c f \sim \triangle b e f \\) 2. alternate interior angle theorem 3. \\( \angle d f c \cong \angle b f e \\) 3. vertical angle theorem

Explanation:

Step1: Use Alternate Interior Angle Theorem

Since \( \overline{AE}\parallel\overline{DC}\), we have \( \angle DCF=\angle BEF\) and \( \angle CDF = \angle EBF\) (Alternate Interior Angles).

Step2: Use Vertical Angle Theorem

\(\angle DFC\) and \( \angle BFE\) are vertical angles, so \( \angle DFC\cong\angle BFE\) (Vertical Angle Theorem).

Step3: Apply AA (Angle - Angle) Similarity Criterion

In \( \triangle BFE\) and \( \triangle CFD\), we have two pairs of congruent angles (\( \angle BEF=\angle DCF\) and \( \angle BFE=\angle DFC\)). By the AA similarity criterion, if two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar.

Answer:

\(\triangle BFE\sim\triangle CFD\) (by AA similarity criterion)