Question

if you wanted to explain why △abd≅cbd by the sas congruence theorem, which of the following would be helpful to know? (1 point)
○ bd is the angle - bisector of ∠abc
○ bd is the angle - bisector of ∠bcd
○ ac is the angle - bisector of ∠bad
○ ac is the angle - bisector of ∠bcd

Explanation:

Step1: Recall SAS Congruence Theorem

The SAS (Side - Angle - Side) Congruence Theorem 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.

Step2: Analyze $\triangle ABD$ and $\triangle CBD$

In $\triangle ABD$ and $\triangle CBD$, we have the common side $\overline{BD}$. For the SAS theorem to be applicable, we need to know that the angles between the pairs of corresponding sides are equal. If $\overline{BD}$ is the angle - bisector of $\angle ABC$, then $\angle ABD=\angle CBD$. Also, we have $\overline{BD}=\overline{BD}$ (common side) and we can consider the other sides $\overline{AB}$ and $\overline{CB}$. This gives us two sides and the included angle equal for the two triangles, which helps in proving $\triangle ABD\cong\triangle CBD$ by SAS.

Answer:

The first option: $\overline{BD}$ is the angle - bisector of $\angle ABC$