Chapter 9.5, Problem 10CE

(a)

To determine

To State: The contrapositive of corollary,“if a quadrilateral is inscribed in a circle, then the opposite angles are supplementary”.

Answer to Problem 10CE

Explanation of Solution

Concept used:

In a contrapositive statement both hypothesis and conclusion are interchanged and negated.

Calculations:

The contrapositive statement is.

“If the opposite angles in a quadrilateral are not supplementary then, thequadrilateral is not inscribed in a circle”.

To determine

(b)

To Find: whether it is possible to circumscribe a circle about quadrilateral PQRS such that $m\angle P=100\text{ and }m\angle R=90$ .

Answer to Problem 10CE

No, because opposite angles are not supplementary.

Explanation of Solution

Given:

  $m\angle P=100\text{ and }m\angle R=90$

No, it is not possible because in a inscribed quadrilateral opposite angles are supplementary which means their sum is ${180}^{\circ }$ .

In this case, sum is.

  $\begin{array}{l}m\angle P+m\angle R\\ =100+90\\ =180\end{array}$

Therefore, the answer is no.