Question

i) Can a triangle have all the angles less than $60^\circ $? Give reason for your answer.
ii) How many triangles can be drawn having its angles $45^\circ ,64^\circ ,72^\circ $? Give reason for your answer.

Answer 1

Hint: Triangle is a three sided polygon with three edges and three vertices. The sum of all these three vertices must always be equal to 180. It cannot be less than or more than 180. This is known as the angle sum property.

Complete step-by-step solution:
i)Can a triangle have all the angles less than $60^\circ $? Give reason for your answer.
In this question, we are asked if a triangle has all its angles less than $60^\circ $ or not and we need to justify our answer.
First of all, let us see the definition of triangle.
So, a triangle is a three sided polygon having three edges and three vertices or we can say angles. If ABC is a triangle, then it can be denoted by $\Delta ABC$, where A, B and C are the vertices. The sum of these three angles must always be equal to 180. This is the most important property of a triangle and it is known as the angle sum property.
So, if we take all the angles of a triangle less than 60 (Let us say we take all the angle 59, 59 and 59) then their sum $\left( {59 + 59 + 59 = 177} \right)$ will not be equal to 180 and will be less than 180. So, we cannot take all the angles of a triangle less than 180.
ii) How many triangles can be drawn having its angles $45^\circ ,64^\circ ,72^\circ $? Give reason for your answer.
The answer is 0. We cannot draw any triangle having angles $45^\circ ,64^\circ ,72^\circ $. Because, if we add them $\left( {45 + 64 + 72 = 181} \right)$ then their sum exceeds 180 and will not be equal to 180. So, we cannot draw any triangle having angles $45^\circ ,64^\circ ,72^\circ $.

Note: A triangle can never have any one of the angles as 0.It has to be greater than 0. Also, angles can never be negative. Another important property of a triangle is that the sum of all the exterior angles of a triangle is equal to 360.