Segment $\overline{A D}$ is an altitude of $ riangle A B C$.
The figure is not to scale.

Which of the following additional statements would allow us to prove that $\overline{A B} \cong \overline{A C}$ ?

Choose 1 answer:
(A) $\overline{C D} \cong \overline{B D}$ only
(B) $\angle B \cong \angle C$ only
(C) Either statement is sufficient.
(D) We need both statements.
(E) Even with both statements, we still could not prove that $\overline{A B} \cong \overline{A C}$.

Question

Segment $\overline{A D}$ is an altitude of $	riangle A B C$.
The figure is not to scale.

Which of the following additional statements would allow us to prove that $\overline{A B} \cong \overline{A C}$ ?

Choose 1 answer:
(A) $\overline{C D} \cong \overline{B D}$ only
(B) $\angle B \cong \angle C$ only
(C) Either statement is sufficient.
(D) We need both statements.
(E) Even with both statements, we still could not prove that $\overline{A B} \cong \overline{A C}$.

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ To prove that $\overline{AB} \cong \overline{AC}$ in $ riangle ABC$ with altitude $\overline{AD}$, we need to establish that $ riangle ABD$ and $ riangle ACD$ are congruent. ‖ ‖ step 2 ⋮ If $\overline{CD} \cong \overline{BD}$, then by the Hypotenuse-Leg (HL) congruence theorem for right triangles, $ riangle ABD \cong riangle ACD$, since $\overline{AD}$ is a common side and is the altitude, making $\overline{AD}$ the perpendicular leg. ‖ ‖ step 3 ⋮ If $\angle B \cong \angle C$, then by the Angle-Side-Angle (ASA) congruence theorem, $ riangle ABD \cong riangle ACD$, since $\overline{AD}$ is a common side and is the altitude, and $\overline{BC}$ is the base for both triangles. ‖ ‖ step 4 ⋮ Therefore, either statement (A) $\overline{CD} \cong \overline{BD}$ or (B) $\angle B \cong \angle C$ alone is sufficient to prove that $\overline{AB} \cong \overline{AC}$. ‖ ∻[question number] Answer∻ ***C*** ∻Key Concept∻ ⚹Congruence of Triangles⚹ ∻Explanation∻ ⚹To prove two triangles congruent, we can use congruence theorems such as HL for right triangles or ASA for any triangles, given the appropriate congruent parts.⚹◊What are the necessary conditions to prove that line segment AB is congruent to line segment AC in triangle ABC?⍭ Generate me a similar question◊

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