Question

11/14 - triangle congruence schoology assignment
write a proof.
given: \\( \angle a \cong \angle c \\) and \\( \overline { b x } \cong \overline { d x } \\)
prove: \\( \overline { a x } \cong \overline { c x } \\)

Explanation:

Step1: Vertical angles are congruent

$\angle AXD\cong\angle CXB$ (Vertical angles are congruent)

Step2: AAS (Angle - Angle - Side) congruence

In $\triangle AXD$ and $\triangle CXB$, we have $\angle A\cong\angle C$ (given), $\angle AXD\cong\angle CXB$ (vertical angles), and $BX\cong DX$ (given). So, $\triangle AXD\cong\triangle CXB$ (AAS congruence)

Step3: Corresponding parts of congruent triangles

Since $\triangle AXD\cong\triangle CXB$, then $\overline{AX}\cong\overline{CX}$ (Corresponding parts of congruent triangles are congruent)

Answer:

1. Vertical angles are congruent; 3. AAS (Angle - Angle - Side); 4. Corresponding parts of congruent triangles are congruent