Question

1. use a paragraph, flow chart, or two - column proof to prove the angle congruency.

given: \\( \angle cxy \cong \angle bxy \\)
\\( \angle cax \cong \angle bax \\)
\\( \overline { a c } \cong \overline { a b } \\)
prove: \\( \angle xcy \cong \angle xby \\)

Explanation:

Step1: Prove $\triangle ACX\cong\triangle ABX$

Since $\angle CAX\cong\angle BAX$, $AC\cong AB$, and $AX = AX$ (common side), by the Side - Angle - Side (SAS) congruence criterion, $\triangle ACX\cong\triangle ABX$.

Step2: Get $CX = BX$

From $\triangle ACX\cong\triangle ABX$, corresponding parts of congruent triangles are congruent, so $CX\cong BX$.

Step3: Prove $\angle XCY\cong\angle XBY$

Given $\angle CXY\cong\angle BXY$, and $XY = XY$ (common side). In $\triangle CXY$ and $\triangle BXY$, we have $CX\cong BX$, $\angle CXY\cong\angle BXY$, $XY = XY$. By the Angle - Side - Angle (ASA) congruence criterion, $\triangle CXY\cong\triangle BXY$. Then, corresponding angles of congruent triangles are congruent, so $\angle XCY\cong\angle XBY$.

Answer:

$\angle XCY\cong\angle XBY$ is proved as above.