Question

question
given: \\( \overline { g h } \cong \overline { e h } , \overline { d h } \cong \overline { f h } \\) and \\( \angle a d h \cong \angle b f h \\).
prove: \\( \overline { c d } \cong \overline { c f } \\).
answer

Explanation:

Step1: Prove \(\triangle D H G \cong \triangle F H E\)

Since \(\overline{G H} \cong \overline{E H}\), \(\overline{D H} \cong \overline{F H}\), and \(\angle D H G=\angle F H E\) (vertical angles are congruent), by the Side - Angle - Side (SAS) congruence criterion, \(\triangle D H G \cong \triangle F H E\).

Step2: Get \(DG = FE\)

Because \(\triangle D H G \cong \triangle F H E\), corresponding parts of congruent triangles are congruent (CPCTC), so \(DG = FE\).

Step3: Prove \(\triangle A D G \cong \triangle B F E\)

We know that \(\angle A D H \cong \angle B F H\), so \(\angle A D G = 180^{\circ}-\angle A D H\) and \(\angle B F E=180^{\circ}-\angle B F H\), then \(\angle A D G \cong \angle B F E\). Also, \(\angle A=\angle B\) (if two angles of one triangle are equal to two angles of another triangle, the third angles are equal. Here, \(\angle A D H+\angle A+\angle A G D = 180^{\circ}\) and \(\angle B F H+\angle B+\angle B E F = 180^{\circ}\), and \(\angle A G D=\angle B E F\) (corresponding angles from \(\triangle D H G \cong \triangle F H E\)). And \(DG = FE\) (from step 2). By the Angle - Side - Angle (ASA) congruence criterion, \(\triangle A D G \cong \triangle B F E\).

Step4: Get \(AD = BF\)

Since \(\triangle A D G \cong \triangle B F E\), by CPCTC, \(AD = BF\).

Step5: Prove \(\triangle A D C \cong \triangle B F C\)

We know that \(\angle A=\angle B\) (from step 3), \(AD = BF\) (from step 4), and \(\angle A D C = 180^{\circ}-\angle A D H\), \(\angle B F C = 180^{\circ}-\angle B F H\), so \(\angle A D C \cong \angle B F C\). By the Angle - Side - Angle (ASA) congruence criterion, \(\triangle A D C \cong \triangle B F C\).

Step6: Prove \(\overline{C D} \cong \overline{C F}\)

Since \(\triangle A D C \cong \triangle B F C\), by CPCTC, \(\overline{C D} \cong \overline{C F}\).

Answer:

\(\overline{C D} \cong \overline{C F}\) because of the Corresponding Parts of Congruent Triangles are Congruent (CPCTC) after proving \(\triangle A D C \cong \triangle B F C\) (using ASA congruence criterion with \(\angle A=\angle B\), \(AD = BF\), \(\angle A D C \cong \angle B F C\)).