Chapter 6, Problem 9CUR

To determine

To prove: $PQRS\text{ is a quadrilateral, then}$

  $\angle Q,\angle R$ and $\angle S$ are not all ${120}^{\circ }$ angles.

Answer to Problem 9CUR

  $\angle Q,\angle R$ and $\angle S$ are not all ${120}^{\circ }$ angles is proved.

Explanation of Solution

Given information:

The system information is:

  $PQRS\text{ is a quadrilateral, then}$

  $\angle Q,\angle R$ and $\angle S$ are not all ${120}^{\circ }$ angles.

Calculation:

Assume temporarily that $\angle Q,\angle R$ and $\angle S$ are all ${120}^{\circ }$ angles.

Then $m\angle P>0$ and $\angle Q+\angle R+\angle S+\angle P>{360}^{\circ },\angle Q+\angle R+\angle S>{120}^{\circ }\times 4={480}^{\circ }$

This contradicts the theorem that states that the sum of the interior angles of a quadrilateral is ${360}^{\circ }$ .

Therefore, the temporary assumption must be false.

It follows that $\angle Q,\angle R$ and $\angle S$ are not all ${120}^{\circ }$ angles.

Hence, the above is proved.