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.