Question

1.Can a triangle have two obtuse angles? Give reason for your answer.
2. How many triangles can be drawn having its angles as ${45^ \circ }$ , ${64^ \circ }$ and ${72^ \circ }$? Give reason for your answer.

Answer 1

Hint: A triangle is a polygon with 3 vertices and 3 edges, hence it has 3 interior angles. The given two questions can be answered by using the fact that the sum of all the interior angles of a triangle is ${180^ \circ }$. Also since they have mentioned obtuse angle we should also be aware of the concept of obtuse angle.

Complete step-by-step answer:
Let us consider a triangle$ABC$,

I.Then by the property of the triangle the sum of all the interior angles is.
i.e. $\angle ABC + \angle BAC + \angle ACB = {180^ \circ }$.
The three different types of angles are,
Right angle: Its value is ${90^ \circ }$.
Acute angle: Its value is less than ${90^ \circ }$.
Obtuse angle: Its value is more than ${90^ \circ }$.
Suppose the triangle contains two obtuse angles then the sum of only those two angles will itself be more than ${180^ \circ }$
i.e. let the two obtuse angles be$\angle ABC = {(90 + {\theta _1})^ \circ }$ and $\angle BAC = {(90 + {\theta _2})^ \circ }$
Since we don’t know the nature of the third angle let it be $\angle ACB = {\theta _3}^ \circ $.
Then the sum of all the three angles becomes, $\angle ABC + \angle BAC + \angle ACB = {(90 + {\theta _1})^ \circ } + {(90 + {\theta _2})^ \circ } + {\theta _3}^ \circ = {180^ \circ } + {\theta _1} + {\theta _2} + {\theta _3}^ \circ $ which is more than ${180^ \circ }$, which is not possible for any triangle.
Therefore a triangle cannot have two obtuse angles.

II.For the second question the three interior angles given are ${45^ \circ }$ , ${64^ \circ }$ and ${72^ \circ }$.
Let us consider their sum, $${45^ \circ } + {64^ \circ } + {72^ \circ } = {181^ \circ }$$.
Since the sum of the given angles exceeds, we cannot have any triangle with these set of interior angles.
Therefore there is no triangle with angles as ${45^ \circ }$ ,${64^ \circ }$ and ${72^ \circ }$.

Note: The sum of all the interior angles of a triangle is ${180^ \circ }$, so in every question in which all the three angles values are given first check whether the sum is ${180^ \circ }$. If only two angles are given then the third angle can be easily obtained by using this property.