Chapter 5, Problem 19CUR

To determine

To complete: the given statement about the diagram.

Answer to Problem 19CUR

  $\overline{DF}$ is an angle bisector of $\angle ADC$ .

Explanation of Solution

Given information:

Given figure,

  

Given:

  $ABCD$ is a parallelogram;

  $\overline{AD}\cong \overline{AC}$ ; $\overline{AE}\cong \overline{EC}$

  $\angle ADF\cong \angle CDF$ ; $m\angle DAC=36$ .

Given statement,

  $\angle ADF\cong \angle CDF$ , So $\overline{DF}$ is a -------- of $\angle ADC$ .

Calculation:

Observe the parallelogram $ABCD$ , see that $\overline{AD}\cong \overline{AC}\cong \overline{AE}\cong \overline{EC}$ , $\angle ADF\cong \angle CDF$

As theorem 4-7 states that “if a point lies on the bisector of an angle, then the point is equidistant from the sides of the angle”.

Observe that $\angle ADF\cong \angle CDF$ , so segment $\overline{DF}$ is an angle bisector.

Thus, $\angle ADF\cong \angle CDF$ , so $\overline{DF}$ is an angle bisector of $\angle ADC$ .