Question

complete the proof that △abd ~ △cbe.

statement reason
1 ∠a ≅ ∠c all right angles are congruent.
2 ∠abd ≅ ∠cbe given
3 △abd ~ △cbe pick criterion similarity
(options: angle - angle, side - angle - side, side - side - side)

Explanation:

Step1: Recall Similarity Criteria

To prove two triangles are similar, we can use Angle - Angle (AA), Side - Angle - Side (SAS), or Side - Side - Side (SSS) criteria. The AA criterion states that if two angles of one triangle are congruent to two angles of another triangle, the triangles are similar.

Step2: Analyze Given Angles

We know that $\angle A\cong\angle C$ (both right angles) and $\angle ABD\cong\angle CBE$ (given). So we have two pairs of congruent angles.

Step3: Apply AA Criterion

Since two angles of $\triangle ABD$ are congruent to two angles of $\triangle CBE$, by the Angle - Angle (AA) similarity criterion, $\triangle ABD\sim\triangle CBE$.

Answer:

The criterion is Angle - angle (AA) similarity. So we pick "Angle - angle" as the criterion for the similarity of $\triangle ABD$ and $\triangle CBE$.