### ABCD is a square. Two arcs are drawn with A and B as centers, and radius equal to the side of square. If arcs intersects at point E, find the angle ∠ACE.

30°

Step by Step Explanation:
1. We are given, ABCD is a square with a diagonal AC.

We know,
• All the angles of a square is equal to 90°.
• The diagonal of a square bisects its angles.

So, ∠A = 90°

and ∠BAC = 45°   (AC is a diagonal)

2. Now, join A and B to E.

We know,

AE is the radius of the circle with center A.

BE is the radius of the circle with center B.

AB is the side of the square.

We are given that radius is equal to the side of the square.

So, ΔABE is an equilateral triangle

⇒ Each angle of ΔABE is 60°.   (All the angles of an equilateral triangle are 60°.)

⇒BAE = 60°

3. We can see,

∠CAE = ∠BAE − ∠BAC = 60° − 45° = 15°

4. As, ∠B = 90°

Reflex ∠B = 360° − 90° = 270°   (Angle at a point is 360°)

Now, ∠CEA = 135°   (The angle subtended by a chord at the center of the circle is double the angle subtended by the same chord at the circumference.)

5. Consider ΔCEA:

∠CEA = 135°

∠CAE = 15°

∠CEA + ∠CAE + ∠ACE = 180°   (Sum of angles of a triangle are 180°)
135° + 15° + ∠ACE = 180°
∠ACE = 180° − (135° + 15°)
∠ACE = 180° − (150°)
∠ACE = 30°