Chapter 4.4, Problem 10PSA

a)

To determine

To Verify: $AB\perp DC.$

Explanation of Solution

Given information: $\angle EBC=(\sqrt{x}+{83}^{\circ }),\angle ABD={(2x-8)}^{\circ },\angle ABC={(x+41)}^{\circ }\}$ In both a & b.

Formula used: linear pair axiom

Verification:

  $\begin{array}{l}\angle ABD+\angle ABC={180}^{\circ }(\text{linear}\text{pair}\text{axiom})\\ \therefore 2x-8+x+41={180}^{\circ }\\ \therefore x={49}^{\circ }\\ \therefore \angle ABD=2\times 49-8={90}^{\circ }\\ \therefore AB\perp DC.\end{array}$

b)

To determine

To Verify: To draw conclusion AB and BE.

Explanation of Solution

Given information: $\angle EBC=(\sqrt{x}+{83}^{\circ }),\angle ABD={(2x-8)}^{\circ },\angle ABC={(x+41)}^{\circ }\}$ In both a & b.

Formula used: linear pair axiom

Verification:

As $x={49}^{\circ }$ , we have

  $\begin{array}{l}\angle EBC=\sqrt{49}+{83}^{\circ }\\ ={90}^{\circ }\end{array}$

Hence $EB\perp DC.$ since EB and AB are co initial rays and both are perpendicular to CD, we conclude that AB and BE are coincident.