Question

Complete the proof. Help. Please.

Answer 1

The correct answers are:

Division Property of Equality;
Vertical angles; and
SAS Similarity Theorem.

Explanation:

We are given that 
AB·BE = CB·BD

To rewrite this, we can divide both sides by BD to cancel it on the right, and BE to cancel it on the left:
$\frac{AB \cdot BE}{BD \cdot BE}=\frac{CB \cdot BD}{BD \cdot BE} \\ \\\frac{AB}{BD}=\frac{CB}{BE}$

For the next step, the two angles that are listed, ∠ABC and ∠DBE, are opposite angles that share only a vertex.  These are vertical angles, and we know that vertical angles are congruent.

For the last step, we have two sides in one triangle that are proportional to two corresponding sides in the other triangle, with a congruet angle between them.  This is the SAS similarity theorem.

Answer 2

The reason for 2 is simply dividing both sides of the expression in 1 by BE*BD. In 3, angle ABC=DBE because they are vertical angles formed by two intersecting lines (AE and CD). In 3, with these two conditions, we can conclude that the two triangles are similar by SAS.