Question

given: x is midpoint of \\( \overline{ag} \\) and \\( \overline{nr} \\) prove: \\( \triangle anx \cong \triangle grx \\)

Explanation:

Step1: Use mid - point property

Since \(X\) is the mid - point of \(\overline{AG}\), \(AX = GX\). Since \(X\) is the mid - point of \(\overline{NR}\), \(NX=RX\).

Step2: Use vertical angles

\(\angle AXN=\angle GXR\) (vertical angles are equal).

Step3: Apply SAS (Side - Angle - Side) congruence criterion

In \(\triangle ANX\) and \(\triangle GRX\), we have \(AX = GX\), \(\angle AXN=\angle GXR\), \(NX = RX\). By the SAS congruence criterion, \(\triangle ANX\cong\triangle GRX\).

Answer:

\(\triangle ANX\cong\triangle GRX\) (by SAS congruence criterion)