Question
Is it possible to have a triangle, in which.
Each angle is greater than $60^\circ $?
Each angle is greater than $60^\circ $?
✓
Answer
Give reasons in support of your answer in. No, because if each angle is greater than $60^\circ $,
then the sum of all three angles will be greater than $180^\circ $, which is not possible.
Proof:
Let the three angles of the triangle be $\angle \text{A},\angle \text{B}$ and $\angle \text{C}.$
As per the given information,
$\angle \text{A}>60^\circ...(\text{i})$
$\angle \text{B}>60^\circ...(\text{ii})$
$\angle \text{C}>60^\circ...(\text{iii})$ On adding $(i), (ii)$ and $(iii)$,
we get: $\angle \text{A}+\angle \text{B}+\angle \text{C}>60^\circ+60^\circ+60^\circ$
$\angle \text{A}+\angle \text{B}+\angle \text{C}>180^\circ$
We can see that the sum of all three angles of the given triangle are greater than $180^\circ$ ,
which is not possible for a triangle.
Hence, we can say that it is not possible for each angle of a triangle to be greater than $60^\circ$
then the sum of all three angles will be greater than $180^\circ $, which is not possible.
Proof:
Let the three angles of the triangle be $\angle \text{A},\angle \text{B}$ and $\angle \text{C}.$
As per the given information,
$\angle \text{A}>60^\circ...(\text{i})$
$\angle \text{B}>60^\circ...(\text{ii})$
$\angle \text{C}>60^\circ...(\text{iii})$ On adding $(i), (ii)$ and $(iii)$,
we get: $\angle \text{A}+\angle \text{B}+\angle \text{C}>60^\circ+60^\circ+60^\circ$
$\angle \text{A}+\angle \text{B}+\angle \text{C}>180^\circ$
We can see that the sum of all three angles of the given triangle are greater than $180^\circ$ ,
which is not possible for a triangle.
Hence, we can say that it is not possible for each angle of a triangle to be greater than $60^\circ$
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