Question

Sum and difference of two angle of triangle are \[{{128}^{\circ }}\]and \[{{22}^{\circ }}\]respectively, then third angle is
A.\[{{52}^{\circ }}\]
B.\[{{62}^{\circ }}\]
C.\[{{15}^{\circ }}\]
D. none of these

Answer 1

Hint: This question is from the topic of triangles. In this problem, we have to find the third angle. We can solve it by making equations with respect to questions. As we know that the sum of the angles in a triangle is \[{{180}^{\circ }}\], by following that, we can find out the third angle in a triangle.

Complete step-by-step solution:
Let us solve this question.
Given a, b c are the angles in a triangle
Let ‘a’ be the first angle
‘b’ be the second angle
‘c’ be the third angle
In the given problem, sum of the two angles in a triangle is \[{{128}^{\circ }}\]
\[a+b={{128}^{\circ }}\]………………. (1)
Difference of two angles in a triangle is \[{{22}^{\circ }}\]
\[a-b={{22}^{\circ }}\]……………….. (2)
We know that
Sum of all angles in a triangle is \[{{180}^{\circ }}\]
i.e.,
\[a+b={{128}^{\circ }}\]
\[a-b={{22}^{\circ }}\]
on cancelling the ‘b’ term, we get
\[2a={{150}^{\circ }}\]
\[\Rightarrow a=\dfrac{{{150}^{\circ }}}{2}\]
\[\therefore a={{75}^{\circ }}\]
First angle is \[{{75}^{\circ }}\].
Substituting the first angle in equation (1)
\[a+b={{128}^{\circ }}\]
\[\Rightarrow {{75}^{\circ }}+b={{128}^{\circ }}\]
\[\begin{align}
  & \Rightarrow b={{128}^{\circ }}-{{75}^{\circ }} \\
 & \therefore b={{53}^{\circ }} \\
\end{align}\]
 Hence, the second angle is \[{{53}^{\circ }}\].
Substituting a and b values in
\[a+b+c={{180}^{\circ }}\]
\[\Rightarrow {{75}^{\circ }}+{{53}^{\circ }}+c={{180}^{\circ }}\]
\[c={{180}^{\circ }}-{{128}^{\circ }}={{52}^{\circ }}\].
\[\therefore c={{52}^{\circ }}\]
Therefore, the third angle is \[{{52}^{\circ }}\].
Correct answer is option (A).

Note: Students made the mistakes on naming the triangles incorrectly. And also, error in taking the corresponding sides of the triangles. In solving this, we need to know the concept of supplementary and complementary angles. Supplementary angle is defined as the sum of all angles in a triangle is \[{{180}^{\circ }}\]whereas, the sum of angles in a triangle is \[{{90}^{\circ }}\], known as ‘complementary angle’.