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In the adjoining figure, it is given that
\[AB \bot BF\] and $EF \bot BF$,
$AC \bot BC$
$KD \bot BC$ and $DE$
Find the value of x.
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A) ${75^ \circ }$
B) ${30^ \circ }$
C) ${60^ \circ }$
D) ${45^ \circ }$

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Last updated date: 13th Jun 2024
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Answer
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Hint: From the given geometry diagram we have to find the angle value of $x$. For that, we are going to use the given geometrical relation in the exterior angle of a triangle. To solve this question we have to apply the formula of exterior angles of a triangle and then we need to apply the concept that the sum of all angles of a rectangle is ${360^ \circ }$

Complete step-by-step answer:
It is given in the question that $\angle DFZ = {25^ \circ }$
From the diagram we can say that $DZ \bot BF$
Therefore, $\angle DZF = {90^ \circ }$
Now by applying the concept of exterior angles of a triangle we get-
$\angle FDC = \angle DZF + \angle DFZ$$ = {90^ \circ } + {25^ \circ } = {115^ \circ }$
Further it is stated in the question that $\angle KBZ = {35^ \circ }$
Since $DZ \bot BF$ therefore we can write $\angle BZD = {90^ \circ }$
Again, $KD \bot BC$so $\angle BKD = {90^ \circ }$
Now by applying the concept that the sum of all the angles of a rectangle is ${360^ \circ }$ we get-
$\angle KDZ = {360^0} - [\angle BKD + \angle BZD + \angle KBZ] = {360^ \circ } - ({90^ \circ } + {90^ \circ } + {35^ \circ }) = {145^ \circ }$
Since $\angle ZDC$ is a straight angle so $\angle ZDC = {180^ \circ }$
Therefore $\angle KDC = {180^ \circ } - \angle KDZ = {180^ \circ } - {145^ \circ } = {35^ \circ }$
So the value of $x = \angle FDC + \angle KDC - \angle KDE$
Since we know that $KD \bot DE$ therefore $\angle KDE = {90^ \circ }$
Thus $x = {115^ \circ } + {35^ \circ } - {90^ \circ } = {60^ \circ }$
Hence we get the value of $x = {60^ \circ }$

So the correct option is $C$

Note: A triangle can be considered as a polygon which consists of three sides, three edges, three vertices and the sum of internal angles of a triangle equal to \[180^\circ \].
The sum of all the angles of a triangle (of all types) is equal to \[180^\circ \].
The sum of the length of the two sides of a triangle is always greater than the third side.
The side opposite to the greater angle is the longest side among all the sides of a triangle.
The exterior angle of a triangle is always equal to the sum of the interior opposite angles. This property of a triangle is called an exterior angle property.
Two triangles will be treated as similar if the corresponding angles of both triangles are congruent and lengths of their sides are proportional.