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Question: In the given figure, ABCD is a square, BCF is an equilateral triangle and AEDF is a rhombus. Find \(...

In the given figure, ABCD is a square, BCF is an equilateral triangle and AEDF is a rhombus. Find EAF\angle EAF

A. 3030^\circ
B. 120120^\circ
C. 150150^\circ
D. None of these

Explanation

Solution

From the figure, it is shown that AFB\angle AFB and CFD\angle CFD are right angles. As BCF is an equilateral triangle, so calculate BFC\angle BFC. As we know that angles around a point add up to 360360^\circ . From this find the value of AFD\angle AFD. From the properties of the rhombus, the sum of adjacent angles of the rhombus is supplementary. From this property, we will get the value of EAF\angle EAF.

Complete step by step answer:
From the figure, it is clear that AFB\angle AFB and CFD\angle CFD is right angles. So,
AFB=CFD=90\Rightarrow \angle AFB = \angle CFD = 90^\circ
As BCF is an equilateral triangle, we know that all angles of an equilateral triangle are equal. Then,
BFC=60\Rightarrow \angle BFC = 60^\circ
Now, we know that all angles around a point add up to 360360^\circ . Thus at point F,
BFC+AFB+AFD+CFD=360\Rightarrow \angle BFC + \angle AFB + \angle AFD + \angle CFD = 360^\circ
Substitute the values,
60+90+AFD+90=360\Rightarrow 60^\circ + 90^\circ + \angle AFD + 90^\circ = 360^\circ
Add the terms on the left side,
240+AFD=360\Rightarrow 240^\circ + \angle AFD = 360^\circ
Move the angle value on the right side,
AFD=360240\Rightarrow \angle AFD = 360^\circ - 240^\circ
Subtract the value on the right side,
AFD=120\Rightarrow \angle AFD = 120^\circ
Now, apply the property of rhombus states that the sum of adjacent angles of a rhombus is supplementary. So,
AFD+EAF=180\Rightarrow \angle AFD + \angle EAF = 180^\circ
Substitute the value,
120+EAF=180\Rightarrow 120^\circ + \angle EAF = 180^\circ
Move the angle value on the right side,
EAF=180120\Rightarrow \angle EAF = 180^\circ - 120^\circ
Subtract the value on the right side,
EAF=60\Rightarrow \angle EAF = 60^\circ
Thus, the value of EAF\angle EAF is 6060^\circ .

Hence, option (D) is the correct answer.

Note: A square is a quadrilateral in which all the four sides are equal in length and all the angles are equal. All the angles are equal to 90 degrees i.e. they are right angles.
A rhombus is a type of quadrilateral in which all four sides are of equal length. Also, the diagonals are perpendicular to one another and bisect each other too.
A triangle whose all three sides are of equal length is called an equilateral triangle. The measure of each angle of a triangle is 60 degrees.