Question

write a proof.
given: \\( \angle a \cong \angle c \\) and \\( \overline{bx} \cong \overline{dx} \\)
prove: \\( \overline{ax} \cong \overline{cx} \\)
complete the proof below.
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$$\begin{tabular}{l|l} statement & \\\\ \\hline \\( \\angle a \\cong \\angle c \\) and \\( \\overline{bx} \\cong \\overline{dx} \\) & \\\\ \\( \\angle axd \\cong \\angle cxb \\) & \\\\ \\( \\triangle axd \\cong \\triangle cxb \\) & \\\\ \\( \\overline{ax} \\cong \\overline{cx} \\) & \\\\ \\end{tabular}$$

Explanation:

Step1: Justify $\angle AXD\cong\angle CXB$

By the Vertical Angles Theorem (which is a case of the Reflexive Property in angle - congruence context).

Step2: Justify $\triangle AXD\cong\triangle CXB$

We have $\angle A\cong\angle C$ (given), $BX\cong DX$ (given), and $\angle AXD\cong\angle CXB$ (from Step 1). By the AAS (Angle - Angle - Side) congruence criterion for triangles.

Step3: Justify $\overline{AX}\cong\overline{CX}$

Since $\triangle AXD\cong\triangle CXB$, by CPCTC (Corresponding Parts of Congruent Triangles are Congruent).

Answer:

CPCTC