Question

**#11.) given: $overline{df} perp overline{ce}$ and $g$ is the midpoint of $overline{ce}$ prove: $overline{cf} cong overline{ef}$

Explanation:

Step1: Identify Given Information

We know \( \overline{DF} \perp \overline{CE} \), so \( \angle CGF = \angle EGF = 90^\circ \) (definition of perpendicular lines). Also, \( G \) is the midpoint of \( \overline{CE} \), so \( \overline{CG} \cong \overline{EG} \) (definition of midpoint).

Step2: Identify Common Side

\( \overline{GF} \) is common to both \( \triangle CGF \) and \( \triangle EGF \), so \( \overline{GF} \cong \overline{GF} \) (reflexive property of congruence).

Step3: Prove Triangle Congruence

By the Side - Angle - Side (SAS) congruence criterion, in \( \triangle CGF \) and \( \triangle EGF \):

• \( \overline{CG} \cong \overline{EG} \) (from step 1)
• \( \angle CGF \cong \angle EGF \) (from step 1)
• \( \overline{GF} \cong \overline{GF} \) (from step 2)

So, \( \triangle CGF \cong \triangle EGF \) (SAS).

Step4: Corresponding Parts of Congruent Triangles

Since \( \triangle CGF \cong \triangle EGF \), their corresponding sides \( \overline{CF} \) and \( \overline{EF} \) are congruent (CPCTC - Corresponding Parts of Congruent Triangles are Congruent).

Answer:

By proving \( \triangle CGF \cong \triangle EGF \) using SAS (with \( \overline{CG} \cong \overline{EG} \), \( \angle CGF=\angle EGF = 90^\circ \), and \( \overline{GF}\cong\overline{GF} \)) and then using CPCTC, we conclude \( \overline{CF}\cong\overline{EF} \).