Question

Can all the angles of a quadrilateral be acute angles?

• A. Yes
• B. No
• C. May be
• D. Cannot be determined

Answer 1

The correct option is B No

Let us take a qudrilateral $ABCD$ with $BD$ as diagonal.

In $\Delta ABD$ we have,

$\angle ADB+\angle ABD+\angle DAB={180}^O$ .......$\left(i\right)$

And in $\Delta DBC$ we have

$\angle DCB+\angle CBD+\angle BDC={180}^O$ ......$\left(ii\right)$

Adding $\left(i\right)$ & $\left(ii\right)$, we get,

$\angle ADB+\angle ABD+\angle DAB+\angle DCB+\angle CBD+\angle BDC={360}^O$

$⟹\angle A+\angle B+\angle C+\angle D={360}^O$

So, the sum of the angles of a quadrilateral is ${360}^O$ .....$\left(iii\right)$.

Now, if the angles are acute then $\angle A<{90}^O,\angle B<{90}^O,\angle C<{90}^O,\angle D<{90}^O$

$⟹\angle A+\angle B+\angle C+\angle D<{90}^O+{90}^O+{90}^O+{90}^O$

$⟹\angle A+\angle B+\angle C+\angle D<{360}^O.$

But this does not comply with $\left(iii\right)$ .

Therefore, all the angles of a quadrilateral cannot be acute angles.

Hence, option $B$ is correct.