Question

In $\Delta ABC$, \[\angle A + \angle B = {110^ \circ }\] and \[\angle C + \angle A = {135^ \circ }\] . Find $\angle A$.

Answer 1

Hint: The given question revolves around the concepts and properties related to triangles. So, we have to find a missing angle in the triangle ABC given to us. One must remember the angle sum property of the triangle in order to solve the problem given to us.

Complete step by step solution:
In the given question, we are given a triangle ABC and two relations between the angles of the triangle.

So, we are given that,
 \[\angle A + \angle B = {110^ \circ } - - - - - - - - \left( 1 \right)\]
 \[\angle C + \angle A = {135^ \circ } - - - - - - - - \left( 2 \right)\]
Also, according to the angle sum property of the triangle, we know that the sum of the angles of the triangle ABC is ${180^ \circ }$.
So, we have, \[\angle A + \angle B + \angle C = {180^ \circ }\] .
So, we can solve the three equations for all the three angles and get the value of the required missing angle.
Now, we add the first two equations $\left( 1 \right)$ and $\left( 2 \right)$ . So, we get,
 \[ \Rightarrow \angle A + \angle B + \angle C + \angle A = {110^ \circ } + {135^ \circ }\]
 \[ \Rightarrow \angle A + \angle B + \angle C + \angle A = {245^ \circ }\]
Now, we know that \[\angle A + \angle B + \angle C = {180^ \circ }\] .
Therefore, we have,
 \[ \Rightarrow {180^ \circ } + \angle A = {245^ \circ }\]
Shifting the constants to the right side of the equation, we get,
 \[ \Rightarrow \angle A = {245^ \circ } - {180^ \circ }\]
Doing the calculations and simplifying further, we get,
 \[ \Rightarrow \angle A = {65^ \circ }\]
So, the value of \[\angle A\] is \[{65^ \circ }\] .
So, the correct answer is “ \[{65^ \circ }\] ”.

Note: The sum of all the angles of a triangle is ${180^ \circ }$. One must take care while doing the calculations and should recheck it so as to be sure of the answer. All angles must be in the same unit so as to simplify the operations such as addition and subtraction.