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If two lines intersect such that four vertical angles are equal then each angle is
A) $45^\circ $
B) $100^\circ $
C) $180^\circ $
D) $90^\circ $

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Last updated date: 13th Jul 2024
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Answer
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Hint: First of all let each angle be x and then apply the angle sum property of to find the value of x. Since each angle is equal therefore the value of all four angles will be x.
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Angle sum property is given by:
\[{\text{A + B + C + D}} = {360^\circ }\]

Complete step by step solution:
Let each angle be $x$.
Then according to the angle sum property of vertical angles, the sum of all four angles is equal to \[{360^ \circ }\].
i.e, if A,B,C,D are four vertical angles of two intersecting lines then according to the angle sum property:
$A+B+C+D=360^\circ $
Now here since all four angles are equal therefore,
\[{\text{A = B = C = D = x}}\]
Now , putting in the value of A,B,C,D in the above equation we get:
\[ x + x + x + x = {360^ \circ } \]
On simplifying the above equation,
$\Rightarrow 4x = {360^ \circ } $
On further calculations,
$\Rightarrow x = {90^ \circ } $

$\therefore$ Each angle is equal to $90^\circ $. Hence, the correct option is option (D).

Note: When two lines intersect then the sum of angles between them is equal to \[{360^ \circ }\] by angle sum property. Just observe the image and relate the equal angles on both sides. It will be easy to understand.