Question

the diagram shows \\( \triangle abe \\) and \\( \triangle acd \\). in the diagram, \\( \frac { a b } { a e } = \frac { a c } { a d } \\).

complete the proof that \\( \overline { e b } \parallel \overline { d c } \\).

Explanation:

Step1: Determine similarity of triangles

We have \(\frac{AB}{AE}=\frac{AC}{AD}\) (given) and \(\angle A\cong\angle A\) (reflexive property). By the Side - Angle - Side (SAS) similarity criterion, if two sides of one triangle are proportional to two sides of another triangle and the included angles are congruent, then the triangles are similar. So, \(\triangle ABE\sim\triangle ACD\) by the SAS (Side - Angle - Side) similarity theorem.

Step2: Find congruent angles

Since \(\triangle ABE\sim\triangle ACD\), corresponding angles of similar triangles are congruent. The angle corresponding to \(\angle ABE\) in \(\triangle ACD\) is \(\angle ACD\). So, \(\angle ABE\cong\angle ACD\) by the property of similar triangles (corresponding angles of similar triangles are congruent).

Answer:

1. SAS (Side - Angle - Side) similarity theorem; 4. \(\angle ACD\), Corresponding angles of similar triangles are congruent.