Question

Can a triangle have two acute angles – justify your answers.

Answer 1

Hint: We first use the condition that the sum of three angles of any triangle is ${{180}^{\circ }}$. We find the combination for the angles of any arbitrary triangle. We also show that it will have at least two acute angles and never can have two obtuse angles.

Complete answer:
We know that the sum of three angles of any triangle is also ${{180}^{\circ }}$.
So, if $a,b,c$ are three angles then $a+b+c={{180}^{\circ }}$.
The definition of acute angle is that the range for the angle has to be ${{0}^{\circ }}<\alpha <{{90}^{\circ }}$.
Therefore, a triangle can have two acute angles as the sum of those two angles becomes less than ${{180}^{\circ }}$.
The combination for the angles can be two acute angles and one obtuse angle.
For example, we can have ${{20}^{\circ }},{{30}^{\circ }},{{130}^{\circ }}$ as the angles. We get ${{20}^{\circ }}+{{30}^{\circ }}+{{130}^{\circ }}={{180}^{\circ }}$.
It also can be all three being acute angles.
For example, we can have ${{50}^{\circ }},{{60}^{\circ }},{{70}^{\circ }}$ as the angles. We get ${{50}^{\circ }}+{{60}^{\circ }}+{{70}^{\circ }}={{180}^{\circ }}$.
The final conclusion for the angles of a triangle is that it must have at least two acute angles.

Note:
The definition of obtuse angle is that the range for the angle has to be ${{90}^{\circ }}<\alpha <{{180}^{\circ }}$.
Therefore, a triangle cannot have two obtuse angles as the sum of those two angles become greater than ${{180}^{\circ }}$. For example, we can’t have ${{100}^{\circ }},{{90}^{\circ }}$ as two the three angles.
We get ${{100}^{\circ }}+{{90}^{\circ }}={{190}^{\circ }}>{{180}^{\circ }}$.