Question

Solve the following question:
In the given figure, it being given that \[\angle 1 = {65^ \circ }\], find all other angles.

Answer 1

Hint: In this question we have to use the given figure. By using that we are going to find the other angle of the geometric figure. For that, write the angles given and find the relationship between each angle. The substitute the given one angle and find the other angles using that one angle to get all the angles.

Complete step-by-step solution:
Given angle,
\[\angle 1 = {65^ \circ }\]
We have to find the other angles.
From the given figure above, we can say that;
\[\angle 1 = \angle 3\] and
\[\angle 2 = \angle 4\]
Since these two pairs of angles are vertically opposite angles.
Since, we already know that \[\angle 1 = {65^ \circ }\], we can conclude that;
\[\angle 3 = {65^ \circ }\]
Now, from the given figure, we know that;
\[\angle 3\] and \[\angle 4\] are linear pair of angles.
\[ \Rightarrow \angle 3 + \angle 4 = {180^ \circ }\]
We already know \[\angle 3\]. Substituting this value in the above equation, we get;
\[ \Rightarrow {65^ \circ } + \angle 4 = {180^ \circ }\]
Now, solving the above equation, we get;
\[ \Rightarrow \angle 4 = {180^ \circ } - {65^ \circ }\]
Subtracting the terms on the right-hand side, we get;
\[ \Rightarrow \angle 4 = {115^ \circ }\]
We know \[\angle 4\]. Since we already know that \[\angle 2 = \angle 4\] since vertically opposite angles,
We have,
\[ \Rightarrow \angle 2 = {115^ \circ }\]

Therefore, the other angles are equal to
\[\angle 1 = {65^ \circ }\]
\[\angle 2 = {115^ \circ }\]
\[\angle 3 = {65^ \circ }\]
\[\angle 4 = {115^ \circ }\]

Note: A linear pair of angles is formed when two lines intersect each other at a point. The two angles are said to be linear if they are adjacent angles and they are formed by two intersecting lines at a particular point. The measure of angle of a straight line is \[{180^ \circ }\]. Therefore, the sum of the linear pair of angles should be equal to \[{180^ \circ }\]. Vertically opposite angles are the angles at a point of intersection of two lines, which are opposite to each other. These angles are said to be equal because they form at a point and form the same angle at the point of intersection.