Question

Can a triangle have two obtuse angles? If true enter 1, else 0.

Answer 1

Hint: The sum of interior angle of a triangle is always ${180}^{\circ }$ . There are three types of angles namely obtuse, acute angle and right angle triangle respectively. So basically obtuse angle means an angle more than ${90}^{\circ }$ , acute angle means angle less than ${90}^{\circ }$ , then we have the right angle which is equal to ${90}^{\circ }$ . So taking these concepts we will analyse whether the given statement is possible or not.

Complete step by step solution:
We have the property that the sum of the angles of a triangle is always ${180}^{\circ }$ .
Obtuse angle is an angle which has magnitude more than ${90}^{\circ }$ . So adding that two angles only we will get ${180}^{\circ }$ or more than that.
Lets see how it happens
Let's take two obtuse angle
Let it be ${90}^{\circ }+x$ and ${90}^{\circ }+y$ where, $x,y>0$
So now we will sum up these angle and see what happens
Now after adding theses angle we get that sum as
 ${90}^{\circ }+x+{90}^{\circ }+y={180}^{\circ }+x+y$
 ${180}^{\circ }+x+y>{180}^{\circ }$
Hence, this contradicts the property of the triangle that the sum of three angles should be ${180}^{\circ }$ .
So there is no possibility of a third angle and without three angles we can’t have a triangle.
Hence having two angle obtuse, construction of a triangle is not at all possible.
Therefore we conclude that a triangle can only have one obtuse angle and not more than that.
 So to verify this let’s take a case where we have one obtuse angle and two acute angles.
 So let the obtuse angle be ${90}^{\circ }+x$ and another angle be ${90}^{\circ }-y$ then we have another angle that is
 ${180}^{\circ }-({90}^{\circ }+x+{90}^{\circ }-y)$
 $={180}^{\circ }-({90}^{\circ }+x+{90}^{\circ }-y)$
 $={180}^{\circ }-{90}^{\circ }-x-{90}^{\circ }+y$
 $=y-x$
For such a triangle $y-x>0$ because angle is never negative.
 $\Rightarrow y>x$
Hence, the other angle is also an acute angle if this property is satisfied.
Hence it is verified
And we reach the conclusion that a triangle cannot have two obtuse angles.
So, according to the question we should enter $0$
So, we should enter “Option 0”.

Note: Always remember that sum of angles of a triangle is always ${180}^{\circ }$ .You can verify the fact that when you choose two angles the obtuse angle be ${90}^{\circ }+x$ and another angle be ${90}^{\circ }-y$ then for all $y-x>0$ a triangle can be formed.