Question

In $\Delta MNK$, if $\angle M + \angle N = {126^ \circ }$ and $\angle M + \angle K = {113^ \circ }$ then find $\angle M, \angle N$ and $\angle K$ .
A) ${59^ \circ },{67^ \circ },{54^ \circ }$
B) ${62^ \circ },{63^ \circ },{50^ \circ }$
C) ${60^ \circ },{65^ \circ },{55^ \circ }$
D) ${63^ \circ },{63^ \circ },{50^ \circ }$

Answer 1

Hint: Recall the measure the sum of all the angles of a triangle. Thus, the sum of the angles in a triangle is ${180^ \circ }$ Now form the equations for the angles given above. Find the measure of each angle by making some arrangements in the obtained equations. Then we can calculate the respective angles using the given relation.

Complete step-by-step answer:
It is given that $\angle M + \angle N = {126^ \circ }$ and $\angle M + \angle K = {113^ \circ }$ .
We know that the sum of all the angles in a triangle is ${180^ \circ }$ .
Therefore, the same thing holds up for the given triangle as well.
Thus, we can write the following:
$\angle M + \angle N + \angle K = {180^ \circ }$ … (1)
It is given that \[\angle M + \angle N = {126^ \circ }\] .
Substitute this value of the addition of two angles in equation (1).
${126^ \circ } + \angle K = {180^ \circ }$
On subtracting ${126^ \circ }$ from both sides we get,
$\angle K = {54^ \circ }$ … (2)
We are also given that $\angle M + \angle K = {113^ \circ }$ .
Therefore, using equation (2) in the above we can write the following:
$\angle M + {54^ \circ } = {113^ \circ }$
On subtracting ${54^ \circ }$ from both sides we get,
$\angle M = {59^ \circ }$ … (3)
Similarly, we are also given that $\angle M + \angle N = {126^ \circ }$ .
Therefore, using equation (3) in the above equation we get:
${59^ \circ } + \angle N = {126^ \circ }$
On subtracting ${59^ \circ }$ from both sides we get,
$\angle N = {67^ \circ }$ … (4)
Therefore, from equations (2), (3) and (4) we get that the respective measures of the angle are $\angle M = {59^ \circ },\angle N = {67^ \circ },\angle K = {54^ \circ }$

So, the correct answer is “Option A”.

Note: Here the important fact to note is that the minor calculations should be done correctly. We always know the sum of all the angles of the triangle so we can verify our answer at any point. Also, we have given sums of two angles in the given data so keep verifying there.