Hard looking Easy Math Puzzle | Find angle inscribed in semicircle

  This question is uploaded on Instagram on page "@mymathssolutions" on 1st of may 2020. I provided Instagram link of this question. Click here to go to that question.

Take a look at figure given below. Give it a try and when you are ready then watch the solution. 

This puzzle may seem hard to some people but trust that it's very easy if you know meaning of cyclic quadrilateral.

Here no any construction is needed.

Let assume radius of semicircle be 'r'.

As we know O is center of semicircle

⇒ AO = BO = r

But we can observe, OD is also radius

⇒ DO = r

But AD = DO

⇒ AD = r

⇒ AO = OD = AD = r

⇒ Triangle ADO is equilateral triangle.

⇒ Angle DAO is 60°

From This Point I will use Two Methods, you will get same answer:

Method 1:

Now as we can observe,

Quadrilateral ABCD is Cyclic Quadrilateral (Inscribed in circle)

Sum of Angle of Opposite site will be 180°

⇒ Sum of angles DAB and BCD is 180°

⇒ A° + C° = 180°

⇒ 60° + θ = 180°

⇒ θ = 120°

Method 2:

Construct line AC

Triangle ADB is inscribed in semicircle

⇒ Angle ADB = 90°

We already calculated Angle DAB = 60°

⇒ Triangle ADB is 30-60-90 triangle.

⇒ x° = 30°

Now Observe Triangle ABC is inscribed in semicircle

⇒ Angle ADB = 90°

As we can see,

Angle ABD and Angle ACD is inscribed in same arc

⇒ Angle ABD = Angle ACD = x° = 30°

Now,

θ = Angle ACD + Angle ACB

⇒ θ = 30 + 90

⇒ θ = 120°