Question

1. which pair of triangles can be proved congruent by the sas postulate? given: $overline{ax}congoverline{ex}$, $overline{bx}congoverline{dx}$

$\triangle abx$ and $\triangle edx$
$\triangle ade$ and $\triangle abe$
$\triangle acd$ and $\triangle ecb$

Explanation:

Step1: Recall SAS Postulate

The SAS (Side - Angle - Side) postulate 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 given congruent sides

We are given that $\overline{AX}\cong\overline{EX}$ and $\overline{BX}\cong\overline{DX}$. The included angle for these sides in $\triangle ABX$ and $\triangle EDX$ is the vertical - angle pair $\angle AXB$ and $\angle EXD$. Vertical angles are congruent.

Step3: Check each option

• For $\triangle ABX$ and $\triangle EDX$: We have $\overline{AX}\cong\overline{EX}$, $\angle AXB\cong\angle EXD$ (vertical angles), and $\overline{BX}\cong\overline{DX}$. So, $\triangle ABX$ and $\triangle EDX$ can be proved congruent by SAS.
• For $\triangle ADE$ and $\triangle ABE$, there is no information given about the sides and angles that would satisfy SAS.
• For $\triangle ACD$ and $\triangle ECB$, there is no information given about the sides and angles that would satisfy SAS.

Answer:

$\triangle ABX$ and $\triangle EDX$