
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.
A) ${75^ \circ }$
B) ${30^ \circ }$
C) ${60^ \circ }$
D) ${45^ \circ }$

Answer
475.5k+ views
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.
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.
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