Question

complete the proof that $\triangle abd \sim \triangle cbe$.

(image of two right triangles with right angles at a and c, sharing a common vertex b on the base line ac, with segment ab labeled 8 and bc labeled x. below the image is a two - column proof table with 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

We know that $\angle A$ and $\angle C$ are right angles (from the diagram, the right angle symbols at $A$ and $C$). So the first statement should be $\angle A \cong \angle C$ because all right angles are congruent.

Step2: Identify Vertical Angles or Given Angles

Looking at the diagram, $\angle ABD$ and $\angle CBE$ are related, but also, we can see that $\angle ABD + \angle DBE + \angle CBE = 180^\circ$, and since $\angle DBE$ is between them, but more importantly, we can use the fact that $\angle ABD$ and $\angle CBE$ are related through the right angles. Wait, actually, the second statement should be about the other pair of angles. Wait, no, let's think again. For triangle similarity (AA - Angle - Angle), we need two pairs of congruent angles. We already have the right angles ($\angle A \cong \angle C$). The second pair of angles: $\angle ABD$ and $\angle CBE$? Wait, no, actually, $\angle ABD + \angle DBC = 90^\circ$ and $\angle CBE + \angle DBC = 90^\circ$, so $\angle ABD \cong \angle CBE$ (complementary angles to the same angle). But maybe the given is that $\angle ABD \cong \angle CBE$? Wait, no, the diagram shows that $\angle A$ and $\angle C$ are right angles, so first statement: $\angle A \cong \angle C$ (all right angles are congruent). Second statement: $\angle ABD \cong \angle CBE$ (maybe given or by complementary angles). Then by AA similarity, the triangles are similar.

Wait, let's correct:

Step1: State Right Angles Congruent

Statement 1: $\angle A \cong \angle C$ (because $\angle A$ and $\angle C$ are right angles, and all right angles are congruent).

Step2: State Another Pair of Angles Congruent

Statement 2: $\angle ABD \cong \angle CBE$ (this can be because $\angle ABD + \angle DBE = 90^\circ$ and $\angle CBE + \angle DBE = 90^\circ$, so they are complementary to the same angle, hence congruent, or maybe it's given? Wait, the reason for statement 2 is "Given", so maybe the given is $\angle ABD \cong \angle CBE$? Or maybe the vertical angles? Wait, no, the diagram has $B$ as the common vertex. Wait, actually, $\angle ABD$ and $\angle CBE$: let's see, $\angle A$ and $\angle C$ are right angles, so $\triangle ABD$ and $\triangle CBE$ are right triangles. Then, if we can show that another pair of angles is congruent, we can use AA similarity.

So step 1: $\angle A \cong \angle C$ (right angles, all right angles congruent).

Step 2: $\angle ABD \cong \angle CBE$ (maybe given, or by the fact that $\angle ABD + \angle DBC = 90^\circ$ and $\angle CBE + \angle DBC = 90^\circ$, so they are congruent). Then by AA (Angle - Angle) similarity, $\triangle ABD \sim \triangle CBE$.

So for the first statement, the pick statement should be $\angle A \cong \angle C$. For the second statement, the pick statement should be $\angle ABD \cong \angle CBE$. Then the criterion is AA (Angle - Angle) similarity.

Answer:

1. $\angle A \cong \angle C$ (All right angles are congruent.)
2. $\angle ABD \cong \angle CBE$ (Given)
3. $\triangle ABD \sim \triangle CBE$ (AA criterion similarity)