Question

de ∥ ac given
∠d ≅ ∠a
∠b ≅ ∠b
△abc ~ △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: Justify ∠D ≅ ∠A

Since \( DE \parallel AC \), when a transversal (like \( AB \)) cuts these parallel lines, corresponding angles are congruent. So \( \angle D \cong \angle A \) by the "When parallel lines are cut by a transversal, corresponding angles are congruent" reason.

Step2: Justify ∠B ≅ ∠B

An angle is congruent to itself, which is the reflexive property. So \( \angle B \cong \angle B \) by "Reflexive property".

Step3: Justify \( \triangle ABC \sim \triangle DBE \)

We have two pairs of congruent angles (\( \angle D \cong \angle A \) and \( \angle B \cong \angle B \)), so by the AA (Angle - Angle) similarity criterion, \( \triangle ABC \sim \triangle DBE \). So the reason is "AA~".

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

For similar triangles, the corresponding sides are proportional. Since \( \triangle ABC \sim \triangle DBE \), the ratio of corresponding sides \( \frac{AB}{DB} \) (corresponding sides of \( \triangle ABC \) and \( \triangle DBE \)) and \( \frac{CB}{EB} \) (corresponding sides of \( \triangle ABC \) and \( \triangle DBE \)) are equal. 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.