Write a proof.
Given: AB * BE = CB * BD
Prove: ΔABC ~ ΔDBE
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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.
To prove that ΔABC ~ ΔDBE, we need to show that their corresponding angles are congruent and their corresponding sides are proportional.
Proof:
1. Since AB * BE = CB * BD (given), we can rewrite this equation as AB/BD = CB/BE.
2. The ratios AB/BD and CB/BE represent the proportional lengths of the corresponding sides of ΔABC and ΔDBE.
3. We will show that the corresponding angles of the two triangles are congruent:
a. Angle ABC and angle DBE are vertical angles and are therefore congruent.
b. Angle BAC and angle BED are alternate interior angles formed by a transversal line and two parallel lines. By the alternate interior angles theorem, they are congruent.
c. Angle ACB and angle EBD are also alternate interior angles formed by a transversal line and two parallel lines. By the alternate interior angles theorem, they are congruent.
4. Since all corresponding angles are congruent and the corresponding sides are proportional, we can conclude that ΔABC ~ ΔDBE by the Angle-Angle Similarity Theorem.
Therefore, using the given information and the Angle-Angle Similarity Theorem, ΔABC ~ ΔDBE has been proven.