Chapter 9, Problem 23CR

To determine

Toprove:The sideCF bisects angleBCD .

Explanation of Solution

Given information:

  $\begin{array}{l}\angle A\cong \angle E\\ \overline{FA}\cong \overline{FE}\end{array}$

Formula used:

The below properties are used:

If two sides are congruent, then there corresponding angles are also congruent.

If two angles are congruent, then there corresponding sides are also congruent.

Corresponding parts of congruent triangles are congruent.

If a ray divides an angle into two congruent angles, it bisects the angle.

Proof:

  

As two points determine a line.Draw $\overline{AE}$

  $\overline{FA}\cong \overline{FE}$

If two sides are congruent, then there corresponding angles are also congruent.

  $\angle FAE\cong \angle FEA$

  $\angle A\cong \angle E$

By Addition property, we get

  $\angle CAE\cong \angle CEA$

If two angles are congruent, then there corresponding sides are also congruent.

  $\overline{CA}\cong \overline{CE}$

Two triangles are congruent by SAS congruence rule.

  $\Delta CAF\cong \angle CEF$

Corresponding parts of congruent triangles are congruent.

  $\angle ACF\cong \angle ECF$

If a ray divides an angle into two congruent angles, it bisects the angle.

  $\overrightarrow{CF}$ bisects $\angle BCD$ .