Question

given $de \parallel ac$
prove $\frac{ab}{db} = \frac{cb}{eb}$

statement	reason
$\angle d \cong \angle a$
$\angle b \cong \angle b$
$\triangle abc \sim \triangle dbe$
$\frac{ab}{db} = \frac{cb}{eb}$

when parallel lines are cut by a transversal, corresponding angles are congruent.
reflexive property aa~
corresponding sides of similar triangles are congruent. sas~
corresponding sides of similar triangles are proportional.

Explanation:

Step1: Reason for ∠D ≅ ∠A

Since \( DE \parallel AC \) and \( BA \) is a transversal, by the property of parallel lines cut by a transversal, corresponding angles are congruent. So \( \angle D \cong \angle A \) because of "When parallel lines are cut by a transversal, corresponding angles are congruent."

Step2: Reason for ∠B ≅ ∠B

The angle \( \angle B \) is common to both \( \triangle ABC \) and \( \triangle DBE \). By the reflexive property, an angle is congruent to itself. So the reason is "Reflexive property".

Step3: Reason for \( \triangle ABC \sim \triangle DBE \)

We have two angles of \( \triangle ABC \) congruent to two angles of \( \triangle DBE \) ( \( \angle D \cong \angle A \) and \( \angle B \cong \angle B \) ). By the AA (Angle - Angle) similarity criterion, if two angles of one triangle are congruent to two angles of another triangle, the triangles are similar. So the reason is "AA~".

Step4: Reason for \( \frac{AB}{DB}=\frac{CB}{EB} \)

Since \( \triangle ABC \sim \triangle DBE \), by the property of similar triangles, the corresponding sides of similar triangles are proportional. So the reason is "Corresponding sides of similar triangles are proportional."

Answer:

• For \( \angle D \cong \angle A \): When parallel lines are cut by a transversal, corresponding angles are congruent.
• For \( \angle B \cong \angle B \): Reflexive property
• For \( \triangle ABC \sim \triangle DBE \): AA~
• For \( \frac{AB}{DB}=\frac{CB}{EB} \): Corresponding sides of similar triangles are proportional.