Question

segment \\(\overline{ad}\\) is an altitude of \\(\triangle abc\\).
the figure is not to scale.

which of the following additional statements would allow us to prove that \\(\overline{ab} \cong \overline{ac}\\)?

choose 1 answer:

\\(\boldsymbol{\text{a}}\\) \\(\overline{cd} \cong \overline{bd}\\) only

\\(\boldsymbol{\text{b}}\\) \\(\angle b \cong \angle c\\) only

\\(\boldsymbol{\text{c}}\\) either statement is sufficient.

\\(\boldsymbol{\text{d}}\\) we need both statements.

\\(\boldsymbol{\text{e}}\\) even with both statements, we still could not prove that \\(\overline{ab} \cong \overline{ac}\\).

Explanation:

Step1: Analyze given altitude property

Since $\overline{AD}$ is an altitude, $\angle ADC = \angle ADB = 90^\circ$, and $\overline{AD} \cong \overline{AD}$ (reflexive property).

Step2: Test $\overline{CD} \cong \overline{BD}$

If $\overline{CD} \cong \overline{BD}$, we use SAS congruence:
$\triangle ADC \cong \triangle ADB$ (SAS: $\overline{AD} \cong \overline{AD}$, $\angle ADC \cong \angle ADB$, $\overline{CD} \cong \overline{BD}$).
Corresponding sides $\overline{AB} \cong \overline{AC}$.

Step3: Test $\angle B \cong \angle C$

If $\angle B \cong \angle C$, we use AAS congruence:
$\triangle ADC \cong \triangle ADB$ (AAS: $\angle C \cong \angle B$, $\angle ADC \cong \angle ADB$, $\overline{AD} \cong \overline{AD}$).
Corresponding sides $\overline{AB} \cong \overline{AC}$.

Step4: Evaluate sufficiency

Both statements alone prove $\overline{AB} \cong \overline{AC}$.

Answer:

C. Either statement is sufficient.