Question

can $\triangle abd$ be proved to be congruent to $\triangle cbd$ by the hypotenuse-leg triangle congruency theorem?

image of a triangle with vertices a, b, c and d on ac, bd perpendicular to ac, ab and bc marked as congruent

if yes, select the appropriate statements and pairs of congruent corresponding parts to show that the triangles are congruent using the hypotenuse-leg triangle congruency theorem. if no, select \cannot be proved.\

• $\overline{bd} \cong \overline{bd}$
• $\overline{bc} \cong \overline{ad}$
• $\overline{ab} \cong \overline{cd}$
• $\overline{ab} \cong \overline{bc}$
• $\triangle abd$ and $\triangle cbd$ are right triangles.
• $\triangle abd$ and $\triangle abc$ are right triangles.

Explanation:

Step1: Recall Hypotenuse-Leg (HL) Theorem

The HL Theorem states that if the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and one leg of another right triangle, then the triangles are congruent. So first, we need to confirm the triangles are right triangles.

Step2: Identify Right Triangles

From the diagram, $\angle ADB$ and $\angle CDB$ are right angles (marked with the right angle symbol), so $\triangle ABD$ and $\triangle CBD$ are right triangles. This satisfies the first condition of HL (both are right triangles).

Step3: Identify Congruent Hypotenuse and Leg

We see that $AB \cong BC$ (marked with the same tick marks, so they are congruent hypotenuses), and $BD$ is a common leg, so $\overline{BD} \cong \overline{BD}$ (reflexive property, same segment).

Now, check the options:

• $\overline{BD} \cong \overline{BD}$: This is the common leg, correct for HL.
• $\overline{BC} \cong \overline{AD}$: Not relevant, $BC$ is hypotenuse, $AD$ is a leg of $\triangle ABD$, not congruent in general.
• $\overline{AB} \cong \overline{CD}$: $AB$ is hypotenuse of $\triangle ABD$, $CD$ is leg of $\triangle CBD$, not congruent.
• $\overline{AB} \cong \overline{BC}$: This is the congruent hypotenuses (from the diagram's tick marks), correct.
• $\triangle ABD$ and $\triangle CBD$ are right triangles: Correct, as $\angle ADB$ and $\angle CDB$ are right angles.
• $\triangle ABD$ and $\triangle ABC$ are right triangles: $\triangle ABC$ is not a right triangle (the right angles are at $D$), so incorrect.

So the correct statements are $\overline{BD} \cong \overline{BD}$, $\overline{AB} \cong \overline{BC}$, and $\triangle ABD$ and $\triangle CBD$ are right triangles. But since we are to select the appropriate ones for HL: first, confirm they are right triangles ($\triangle ABD$ and $\triangle CBD$ are right triangles), then the common leg $\overline{BD} \cong \overline{BD}$, and hypotenuse $\overline{AB} \cong \overline{BC}$. But from the options, the key ones are the right triangle statement, the common leg, and the congruent hypotenuse. However, the question is about proving congruence via HL, so we need to check if the conditions are met.

Since $\triangle ABD$ and $\triangle CBD$ are right triangles (right angles at $D$), $BD$ is common (so $\overline{BD} \cong \overline{BD}$), and $AB \cong BC$ (from the tick marks), so by HL, they are congruent. So the correct options to select are: $\overline{BD} \cong \overline{BD}$, $\overline{AB} \cong \overline{BC}$, and $\triangle ABD$ and $\triangle CBD$ are right triangles. But among the given checkboxes, the relevant ones are:

• $\overline{BD} \cong \overline{BD}$ (checked)
• $\overline{AB} \cong \overline{BC}$ (checked, since $AB$ and $BC$ have the same tick marks)
• $\triangle ABD$ and $\triangle CBD$ are right triangles (checked)

But the question is to select the appropriate statements. So first, confirm the triangles are right: $\triangle ABD$ and $\triangle CBD$ are right triangles. Then the leg: $\overline{BD} \cong \overline{BD}$, and hypotenuse: $\overline{AB} \cong \overline{BC}$.

Answer:

The correct statements to select are:

• $\boldsymbol{\overline{BD} \cong \overline{BD}}$
• $\boldsymbol{\overline{AB} \cong \overline{BC}}$
• $\boldsymbol{\triangle ABD}$ and $\boldsymbol{\triangle CBD}$ are right triangles

(Note: If the task is to answer whether it can be proved, the answer is yes, and the above statements are the ones to select. If the final answer is a selection, but based on the options, the key is that $\triangle ABD$ and $\triangle CBD$ are right triangles, $\overline{BD} \cong \overline{BD}$, and $\overline{AB} \cong \overline{BC}$.)

But since the question might expect the conclusion, the triangles can be proved congruent by HL. So the answer is that it can be proved, with the selected statements being $\overline{BD} \cong \overline{BD}$, $\overline{AB} \cong \overline{BC}$, and $\triangle ABD$ and $\triangle CBD$ are right triangles.