Question

Given $\overline{AB}\parallel \overline{CD};m\angle D=116;\overrightarrow{AK}$bisects $m\angle DAB$

1. Find the measures of $\angle DAB,\angle KAB\text{and}\angle DKA$
2. Is there enough information for you to conclude that$\angle D$ and$\angle C$are supplementary, or is more information needed?

   
   

   
   

Short answer

a. Measurement of angles are$m\angle DAB=64,m\angle KAB=32\text{and}m\angle DKA=32$

b.More information is needed to conclude that$\angle D$ and$\angle C$ are supplementary.

Step-by-step solution

Part a. Step 1. Find ∠DAB.

Line segment$\overline{DA}$ is transversal to the parallel lines$\overline{AB}$ and$\overline{CD}$

$\angle DAB$and$\angle CDA$ forms a pair of same-side interior angles

Theorem 3-3: If two parallel lines are cut by a transversal then the same-side interior angles are supplementary.

$$\begin{gathered} m\angle DAB+m\angle CDA=180 \\ m\angle DAB+116=180 \\ m\angle DAB=180-116 \\ m\angle DAB=64 \end{gathered}$$

Part a. Step 2. Find ∠KAB.

Since, $\overrightarrow{AK}$bisects $\angle DAB$. So $\angle KAB$is half of angle $\angle DAB$

$$\begin{gathered} \angle KAB=\frac{\angle DAB}{2} \\ \angle KAB=\frac{64}{2} \\ \angle KAB=32 \end{gathered}$$

Part a. Step 3. Find ∠DKA.

Line segment$\overrightarrow{AK}$ is transversal to the parallel lines$\overline{AB}$ and$\overline{CD}$

$\angle KAB$and$\angle DKA$ forms a pair of alternate interior angles

Theorem 3-2: If two parallel lines are cut by a transversal then the alternate interior angles are congruent.

$$\begin{gathered} \angle DKA\cong \angle KAB \\ m\angle DKA=m\angle KAB \\ m\angle DKA=32 \end{gathered}$$

Thus,$m\angle DAB=64,m\angle KAB=32\text{and}m\angle DKA=32$

Part b. Step 1. Observations from figure.

Line segments$\overline{DA}\text{and}\overline{BC}$ are transversal to the parallel lines$\overline{AB}$ and$\overline{CD}$

Part b. Step 2. Angles which can be find using given information.

Angles which can be find are$\angle DAB,\angle KAB,\angle KAD\text{,and}\angle DKA$

Part b. Step 3. Angles which cannot be find using given information.

Angles which cannot be find are $\angle AKB,\angle KBA,\angle KBC\text{,}\angle BKC,\text{and}\angle KCB$

Thus, to find $\angle C$more information is needed.

Hence, we cannot conclude that$\angle D$ and$\angle C$ are supplementary.