Question

Prove that all four angles of a quadrilateral cannot be obtuse angles.

Answer 1

Hint: Use the fact that obtuse angle is greater than 90 degrees. Also, apply the concept that the sum of all interior angles of the quadrilateral is exactly 360 degrees.

Complete step-by-step answer:
In the question, we have to prove that all four angles of a quadrilateral cannot be obtuse angles.
Now, it is already known that the quadrilateral has four sides. Bow, these four sides can be equal or can’t be equal, or can be equal to any one side of two sides. So, here we will take all the four sides are not equal, since nothing is mentioned about the side lengths of the quadrilateral.
Now it is also known that the sum of all the interior angles of the quadrilateral is exactly 360 degrees.
Also, it is known that the obtuse angle has the angle measure of greater than 90 degrees. So in order to prove that all four angles of a quadrilateral cannot be obtuse angles, we will use the method of contradiction here, in order to prove the above statement.
So, we will assume that all the interior angles of the quadrilateral are obtuse.
So let the angles of the quadrilateral are \[\angle A\], \[\angle B\], \[\angle C\]and \[\angle D\] . So all the angles are obtuse which means that;
\[\angle A\,>{{90}^{\circ }}\], \[\angle B\,>{{90}^{\circ }}\], \[\angle C\,>{{90}^{\circ }}\]and \[\angle D\,>{{90}^{\circ }}\]
Now, we will add all the angles as follows:
\[\begin{align}
  & \Rightarrow \angle A\,+\angle B+\angle C+\angle D>{{90}^{\circ }}+{{90}^{\circ }}+{{90}^{\circ }}+{{90}^{\circ }} \\
 & \Rightarrow \angle A\,+\angle B+\angle C+\angle D>{{360}^{\circ }} \\
\end{align}\]
So, by adding all the angles we get the sum of all the interior angles as greater than 360 degrees, which contradicts that fact that that sum of all the interior angles of the quadrilateral is exactly 360 degrees. So, our assumption that all the interior angles of the quadrilateral are obtuse is wrong. Hence, we have proven that all four angles of a quadrilateral cannot be obtuse angles.

Note: Here we will take the interior angles of the quadrilateral only and not the exterior angles of the quadrilateral. Also, for a four sided figure only the sum of interior angles is 360 degrees, and not for more than four sided figures.