Question

geometry - kim - yr - 2026 (lms)
4 - 6: mathxl for school: mixed review (lms not graded)
part 2 of 3
given: \\( \angle a \cong \angle c \\) and \\( \overline { b x } \cong \overline { d x } \\)
prove: \\( \overline { a x } \cong \overline { c x } \\)
complete the proof below.

Explanation:

Step1: Recall the AAS (Angle - Angle - Side) Congruence Criterion

In two triangles, if two angles and a non - included side of one triangle are congruent to two angles and the corresponding non - included side of the other triangle, then the two triangles are congruent.

Step2: Identify the congruent parts for AAS

We are given that \(\angle A\cong\angle C\) (one pair of angles), \(\angle AXD\cong\angle CXB\) (vertical angles, second pair of angles), and \(BX\cong DX\) (side). Since in \(\triangle AXD\) and \(\triangle CXB\), the side \(BX\) in \(\triangle CXB\) and \(DX\) in \(\triangle AXD\) are non - included sides with respect to the given angles \(\angle A,\angle AXD\) and \(\angle C,\angle CXB\) respectively.
So, \(\triangle AXD\cong\triangle CXB\) by the AAS (Angle - Angle - Side) congruence criterion.

Answer:

AAS (Angle - Angle - Side)