Question

complete the proof that $\triangle abd \sim \triangle cbe$.
(there is a diagram of two right triangles with right angles at a and c respectively, and a common vertex at b with two angles marked equal at b. below the diagram is a table with three rows for statements and reasons:

1. statement: pick statement, reason: all right angles are congruent.
2. statement: pick statement, reason: given.
3. statement: $\triangle abd \sim \triangle cbe$, reason: pick criterion similarity)

Explanation:

Step1: Identify right angles

From the diagram, $\angle A$ and $\angle C$ are right angles. So the statement for step 1 should be $\angle A \cong \angle C$ (since all right angles are congruent).

Step2: Identify given angle congruence

Looking at the diagram, the two non - right angles at $B$ (let's say $\angle ABD$ and $\angle CBE$ or the other pair of angles formed) - typically, we have a pair of angles given as congruent (from the "Given" reason). Let's assume the given congruent angles are $\angle ABD \cong \angle CBE$ (this is a common setup for similar triangles with right angles).

Step3: Apply similarity criterion

We have two angles of $\triangle ABD$ congruent to two angles of $\triangle CBE$ (right angles and the given congruent angles). 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 criterion for step 3 is AA.

Answer:

1. Statement: $\boldsymbol{\angle A \cong \angle C}$; Reason: All right angles are congruent.
2. Statement: $\boldsymbol{\angle ABD \cong \angle CBE}$; Reason: Given.
3. Criterion: $\boldsymbol{AA}$ (Angle - Angle); Reason: AA similarity (two angles congruent).