Question

if \\( \overline{ad} \cong \overline{bd} \\), which of the following relationships can be proved and why?

a. \\( \triangle acd \cong \triangle bcd \\), because of asa.
b. \\( \triangle acd \cong \triangle bcd \\), because of as.
c. there is not enough information to prove a relationship.
d. \\( \triangle acd \cong \triangle bcd \\), because of sas.

Explanation:

Step1: Identify the given information

We are given that \( \overline{AD}\cong\overline{BD}\), and \( \angle ADC=\angle BDC = 90^{\circ}\) (right angles), and \( \overline{CD}\cong\overline{CD}\) (common side).

Step2: Recall the SAS (Side - Angle - Side) congruence criterion

The SAS congruence criterion states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.
In \( \triangle ACD\) and \( \triangle BCD\):

• \( \overline{AD}\cong\overline{BD}\) (given side)
• \( \angle ADC\cong\angle BDC\) (right angles, included angle)
• \( \overline{CD}\cong\overline{CD}\) (common side)

Answer:

D. \( \triangle ACD\cong\triangle BCD\), because of SAS.