Question

if ad = bd, which of the following relationships can be proved and why? a. △acd = △bcd, because of asa. b. there is not enough information to prove a relationship. c. △acd = △bcd, because of sas. d. △acd = △bcd, because of as.

Explanation:

Step1: Identify given information

We know that $AD = BD$ and $\angle ADC=\angle BDC = 90^{\circ}$ (from the right - angle symbol at $D$), and $CD$ is common to both $\triangle ACD$ and $\triangle BCD$.

Step2: Apply congruence criterion

The Side - Angle - Side (SAS) congruence criterion states that if two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, the two triangles are congruent. Here, in $\triangle ACD$ and $\triangle BCD$, $AD = BD$, $\angle ADC=\angle BDC$, and $CD = CD$. So, $\triangle ACD\cong\triangle BCD$ by SAS.

Answer:

C. $\triangle ACD\cong\triangle BCD$, because of SAS.