Question

Math

Posted 4 months ago

```
Segment $\overline{A D}$ is an altitude of $\triangle A B C$.
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 $\overline{A B} \cong \overline{A C}$.
```

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Answer from Sia

Posted 4 months ago

Solution by Steps

step 2

If $\overline{CD} \cong \overline{BD}$, then by the Hypotenuse-Leg (HL) congruence theorem for right triangles, $\triangle ABD \cong \triangle ACD$, 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, $\triangle ABD \cong \triangle ACD$, 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.

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