Question

# If a triangle $\Delta ABC$ is an obtuse-angled triangle in which $\angle C = {110^ \circ}$ then which one of the following is true-A $AB = AC$B $AB < AC$C $AB > AC$D $AB < BC$

Hint: In this question we have given that in $\Delta ABC$, $\angle C = {110^ \circ}$. So, we need to show which of the above statement is true for that as we know the angle sum property of triangle which is that the sum of all the angles of a triangle is ${180^ \circ}$. And in this case, we have been given a triangle in which the measure of $\angle C = {110^ \circ }$ therefore, we are left with angle A and B, keeping in mind that the sum of these two angles should be such that it must not violate the angle sum property of a triangle. So, from here we will get that the angle A and B are acute angles. And after that, we will be using the property which is the sum of the two angles other than the obtuse angle is less than ${90^ \circ}$. The side opposite to the greater angle is longer.

Complete step-by-step solution:
We have been provided in $\Delta ABC$, $\angle C = {110^ \circ}$. So, we need to show which of the above statement is true for that as we know the angle sum property of triangle which is that the sum of all the angles of a triangle is ${180^ \circ}$.
$\angle A + \angle B + \angle C = {180^ \circ}$
We have been given that $\angle C = {110^ \circ}$
Now we will be keeping the value of C in the above equation, from which we would get
$\angle A + \angle B + {110^ \circ} = {180^ \circ}$
$\angle A + \angle B = {70^ \circ}$
As we know the angle measuring less than ${90^ \circ}$ is called an acute angle. Since, here we have the sum of both the angles $\angle A$ and $\angle B$ is ${70^ \circ }$, which is less than ${90^ \circ }$, we can surely say that both angles $\angle A$and $\angle B$ must be acute angles.
Now we will be using the property of the triangle that is the side opposite to the greater angle is longer.
As $\angle C$ is the greatest so the side opposite to $\angle C$ is AB
So, we can say that the side $AB > AC$.
So, from this we can conclude option (c) is correct.

Note: In this question we should remember the properties of the triangle for finding the relation according to the conditions given in the question. We can also solve this question by the direct method as we are given an obtuse-angled triangle so we can easily find out that the other 2 would be acute by the angle sum property of the triangle. So, we can directly use the property that the side opposite to the greatest angle is the largest.