Question

triangle abc with right angle bdc is shown below
which additional fact is needed to prove $\triangle abd \cong \triangle cbd$ by side-angle-side (sas)?
a. $\overline{ad} \cong \overline{cd}$
b. $\overline{ab} \cong \overline{cb}$
c. $\angle bad \cong \angle bcd$
d. $\angle abd \cong \angle cbd$

Explanation:

Step1: Identify shared side

$\overline{BD} \cong \overline{BD}$ (reflexive property, shared side)

Step2: Identify right angles

$\angle ADB = \angle CDB = 90^\circ$, so $\angle ADB \cong \angle CDB$

Step3: Match SAS congruence

SAS requires two sides and included angle. We have one side ($\overline{BD}$) and included right angle. Need the other pair of adjacent sides to the right angle: $\overline{AD} \cong \overline{CD}$.

Answer:

A. $\overline{AD} \cong \overline{CD}$