One of the angles of triangle is 65 find the remaining two angles if their difference is 25

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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.

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