Question

1. which pair of triangles can be proved congruent by the sas postulate? given: x is the mid - point of both $overline{ad}$ and $overline{be}$.

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

Explanation:

Step1: Recall mid - point property

Since \(X\) is the mid - point of \(AD\), then \(AX = DX\). Since \(X\) is the mid - point of \(BE\), then \(BX=EX\).

Step2: Identify vertical angles

\(\angle AXB\) and \(\angle DXE\) are vertical angles, so \(\angle AXB=\angle DXE\) (vertical angles are equal).

Step3: Apply SAS postulate

In \(\triangle ABX\) and \(\triangle EDX\), we have \(AX = DX\), \(\angle AXB=\angle DXE\), and \(BX = EX\). So, by the Side - Angle - Side (SAS) postulate, \(\triangle ABX\cong\triangle EDX\).

Answer:

\(\triangle ABX\) and \(\triangle EDX\)