Write a proof.

Question

Given: AB * BE = CB * BD
Prove: ΔABC ~ ΔDBE

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Answer 1

Proof:

In triangle ABC and DBE, we have:

angle ABC = angle DBE (both are right angles as AB and BD are perpendicular to BC and DE respectively)

angle A = angle D (given that AB and BD are parallel lines and angles A and D are corresponding angles)

Using the Angle-Angle similarity criterion, we can conclude that triangles ABC and DBE are similar.

To prove that they are in fact similar triangles, we need to show that their corresponding sides are proportional.

We are given that: AB * BE = CB * BD

Dividing both sides of this equation by BD*BE, we get:

AB/BD = CB/BE

This means that AB/BD = DE/BE (as AB is parallel to BD and DE is the transversal)

Hence, we have found that the corresponding sides AB/BD and DE/BE are proportional.

Therefore, we can conclude that ΔABC ~ ΔDBE by the Angle-Angle similarity criterion.