Chapter 3.2, Problem 17WE

a.

To determine

To find: the measures of $\angle DAB,\angle KAB$ and $\angle DKA$

Explanation of Solution

Given information:

  $\overrightarrow{AB}\text{ is parallel to }\overrightarrow{CD}$ , $m\angle D=116$

  $\overrightarrow{AK}$ Bisects $\angle DAB$

Calculation:

Consider, $\overrightarrow{AB}\text{ is parallel to }\overrightarrow{CD}$ and $\overrightarrow{AK}$ Bisects $\angle DAB$

The figure is shown below,

  

Since the line AB and line DC are parallel, by the consecutive interior angles theorem,

  $\angle DAB$ and $\angle ADC$ are supplementary.

  $\angle DAB+\angle ADC=180$

  $\angle DAB=180-116$

  $\angle DAB=64$

  $\overrightarrow{AK}$ Bisects $\angle DAB$ .So, $\angle DAK$ is half of the $\angle DAB$ .

  $\angle KAB=\frac{\angle DAB}{2}$

  $\angle KAB=32$

And $\angle KAD=32$

In $\Delta DAK$

  $\begin{array}{l}\angle DAK+\angle ADK+\angle AKD=180\\ 32+116+\angle AKD=180\\ \angle AKD=32\end{array}$

Therefore, the angles $\angle DAB,\angle KABAND\angle DKA$ are $64,32\text{and }\text{32}$ respectively.

b.

To determine

To find: The angle $\angle C$ and check the given information is enough or not.

Explanation of Solution

Given:

The angles $\angle DAB,\angle KAB$ and $\angle DKA$ are $64,32$ and $32$ respectively.

Calculation:

Consider,

  $\overrightarrow{AK}$ Bisects $\angle DAB$ .So that, angle is half of the $\angle DAB$ .

But the line $\overrightarrow{AK}$ does not bisect the ABCD quadrilateral into two identical triangles.

Therefore, the given information is not enough for the calculation of $\angle C$ .