Full solution

Q. Which of the following relationships proves why

\triangle \mathrm{ABD}

and

\triangle \mathrm{CBD}

are congruent?

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ASA

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\mathrm{HL}

\newline

SAS

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Overlapping Triangles

\newline

Segments

A B

and

C B

are congruent.

Identify Given Information: Identify the given information about the triangles $ABD$ and $CBD$ . We know that segments $AB$ and $CB$ are congruent. This gives us one pair of congruent sides.
Establish Congruence: Determine if there is any information about angles or other sides that can help us establish congruence between the two triangles. Since the triangles share vertex $B$ , they also share angle $B$ , which means angle $ABD$ is congruent to angle $CBD$ by the reflexive property.
Search for Additional Data: Look for additional information that can help us establish a congruence theorem. If we know that another pair of angles or sides are congruent, we can use ASA, SAS, or SSS congruence theorems. However, HL (Hypotenuse-Leg) is only applicable to right triangles, and there is no information given that suggests these triangles are right triangles.
Apply ASA Theorem: Since we have one pair of congruent sides $AB \cong CB$ and a pair of congruent angles $\angle ABD \cong \angle CBD$ that are included between the congruent sides, we need one more piece of information to prove congruence. If we know that the other pair of included angles are congruent, we can use the ASA (Angle-Side-Angle) congruence theorem.
Check for Additional Info: Check if the problem statement or diagram provides information about the other pair of included angles or the third side of the triangles. If there is no additional information, we cannot conclusively determine the congruence of the triangles using ASA, SAS, or SSS. However, the question prompt suggests that we are to assume the triangles are congruent and select the theorem that would prove it.
Conclusion: Since we have two pairs of congruent angles (angle $ABD \cong angle CBD$ and angle $B$ by reflexive property) and a pair of congruent sides ( $AB \cong CB$ ) included between the angles, the ASA (Angle-Side-Angle) congruence theorem is the correct choice to prove the triangles are congruent.