Question

One of the angles of a triangle is $65{}^\circ $. How do you find the remaining two angles if their difference is $25{}^\circ $?

Answer 1

Hint: To solve the given question first let us consider a triangle ABC. Assume that $\angle A=65{}^\circ $ and the difference of $\angle B$ and $\angle C$ will be $25{}^\circ $. Then we will use the angle sum property of the triangle to get the desired answer.

Complete step by step answer:
We have been given that one of the angles of a triangle is $65{}^\circ $and the difference of the remaining two angles is $25{}^\circ $.
We have to find the angles of the triangle.
First let us consider a triangle $\Delta ABC$ in which $\angle A=65{}^\circ $ and difference of $\angle B$ and $\angle C$ will be $25{}^\circ $.

Now, we have \[\angle B-\angle C=25{}^\circ \].
\[\Rightarrow \angle B=25{}^\circ +\angle C........(i)\]
Now, we know that by the angle-sum property of a triangle the sum of all angles of a triangle is $180{}^\circ $.
So we will get
\[\Rightarrow \angle A+\angle B+\angle C=180{}^\circ \]
Now, substituting the value $\angle A=65{}^\circ $ in the above obtained equation we will get
\[\Rightarrow 65{}^\circ +\angle B+\angle C=180{}^\circ \]
Now, simplifying the above obtained equation we will get
\[\begin{align}
  & \Rightarrow \angle B+\angle C=180{}^\circ -65{}^\circ \\
 & \Rightarrow \angle B+\angle C=115{}^\circ \\
 & \Rightarrow 25{}^\circ +\angle C+\angle C=115{}^\circ \\
 & \Rightarrow 2\angle C=115{}^\circ -25{}^\circ \\
 & \Rightarrow 2\angle C=90{}^\circ \\
 & \Rightarrow \angle C=\dfrac{90{}^\circ }{2} \\
 & \Rightarrow \angle C=45{}^\circ \\
\end{align}\]
Now, substituting the above obtained value in equation (i) we will get
\[\begin{align}
  & \Rightarrow \angle B=25{}^\circ +45{}^\circ \\
 & \Rightarrow \angle B=70{}^\circ \\
\end{align}\]

Hence we get the remaining two angles of a triangle as \[45{}^\circ \] and \[70{}^\circ \].

Note: We can verify the answer obtained by substituting the values in the angle-sum property of a triangle. The sum of all three angles must be $180{}^\circ $. We can also solve this question by assuming the remaining two angles as x and y and by forming and simplifying the equations we will get the desired answer.