Chapter 11.4, Problem 27HP

To determine

To determine: If a quadrilateral may have two angles that are twice that of other two remaining angles. To explain its reasoning or giving an example.

Answer to Problem 27HP

Yes. A quadrilateral may have two angles that are twice that of other two remaining angles.

Explanation of Solution

Given information: Two angles of a quadrilateral are twice as large as the other two angles.

Concept used: By rule, the sum of all four angles of a quadrilateral is 360 degree.

Conclusion: Let any two angles of a quadrilateral are x and y , so its remaining other two angles are $2x°$ and $2y°$ . Now by above concept,

  $\begin{array}{l}x+y+2x+2y=360\\ \Rightarrow 3x+3y=360\\ \Rightarrow 3(x+y)=360\end{array}$

Or, $x+y=\frac{360}{3}=120$

So, there may be any two angles in a quadrilateral, having their sum as 120 and thus the sum of its two other angles will be 240, so that to have the sum of all the four angles as 360 degree.

Example: Let any two angles of a quadrilateral be 40 degree and 80 degree, so its remaining two angles will be 80 and 160 degree, so by above rule

  $40+80+80+160=360°$

Conclusion: Above example shows that there may be any two angles in a quadrilateral, that are as large as the other two angles, so that the sum of all four such angles is 360 degree.