Question

Can you have a triangle with two right angles? Can you have a triangle with two obtuse angles?

Answer 1

Hint: We have to understand and use the definition of a triangle and the angle sum property of a triangle to obtain the solution to this question.

Complete step-by-step answer:
The definition of a triangle is
A triangle is defined as a closed figure having three sides and three vertices where the angles corresponding to the vertices are greater than zero. …………………………………..(1.1)
The angle sum property of a triangle states that:
The total sum of the three angles of a triangle is ${{180}^{\circ }}$ …………..(1.2)
 Now, coming to our question, in the first question it is asked whether a triangle can have two right angles. As the right angles are equal to ${{90}^{\circ }}$, by the angle sum property (equation 1.2), if the third angle of this triangle is x, then
$\begin{align}
  & \text{Sum of the angles of the triangle= }{{90}^{\circ }}+{{90}^{\circ }}+x={{180}^{\circ }} \\
 & \Rightarrow {{180}^{\circ }}+x={{180}^{\circ }}\Rightarrow x={{0}^{\circ }} \\
\end{align}$
Thus, if two angles in a triangle are right angles, the third angle should be of ${{0}^{\circ }}$. However, from the definition of a triangle (equation 1.1), all the angles should be greater than zero and hence it is not possible.
Now, coming to our question, in the first question it is asked whether a triangle can have two obtuse angles.
By the angle sum property (equation 1.2), if the third angle of this triangle is x, then
$\begin{align}
  & \text{Sum of the angles of the triangle= Sum of the obtuse angles}+x={{180}^{\circ }} \\
 & \Rightarrow x={{180}^{\circ }}\text{-Sum of the obtuse angles}\Rightarrow x<{{0}^{\circ }} \\
\end{align}$
Because as the obtuse angles are equal to ${{90}^{\circ }}$, their sum will be greater than ${{180}^{\circ }}$.
Thus, if two angles in a triangle are obtuse angles, the third angle should be less ${{0}^{\circ }}$. However, from the definition of a triangle (equation 1.1), all the angles should be greater than zero and hence this kind of a triangle is not possible.

Note: We note that even if two obtuse angles are not possible in a triangle, a triangle having two acute angles is possible as it will not violate the angle sum property.