HOW ANGLE SOME PROPERTY OF ∆ COMES HANDY

Hello math bugs(🐞) & hivers(🐝)
I hope you are strong and stout and doing good in life.

Today there is another beautiful geometric problem as you can see it in the following figure.Before checking solution you must give it a try to test your geometric understanding.We gonna use two geometric property. One is angle sum property of triangle and the other is vertically opposite angle created by two intersecting straight lines.
Easiest approach:

Let's try giving a name to every required angle.In the core vertically opposite angles are (x & x) , (y & y) and (z & z) as you can see in the following figure. Sum of 2(X + y + z) = 360° and so sum of (x + y + z) becomes 180°. The sum of three ∆ in the figure must be 540° and if 180° is subtracted from 540°, we get 360° which is the required answer for the problem.
Check it below:

Constructive approach:

We can solve it in many ways. Let me try it a little different way. For this we need to draw some lines as you can see in the following figure.
Now the ∆s becomes six. Sum of all the angles of those ∆s are (6×180°). Don't try to multiply it cause it is the answer. Yeah we will calculate when it reduced to answer. It saves time 😀😀.

The point on which the triangles have same vertex conceive and 360° angle. If this 360° angle is substracted from6×180° ,we reach double the answer. So after subtracting we need to divide it by 2. And thus we get the required answeragain.check the figure below:
Approach if it is a regular hexagon:

The problem today is a easy one but there are some important considerations. The figure you have seen above is a hexagon after the construction of three dot dot line segment.It will always be a regular hexagon if the perimeter of the outside line is kept equal and the diagonals are changed in their length, still the figure would be a regular hexagon and thus all the the angles of each Triangle will be equal. You have seen the angles are a , b & c. There would be equal check the figure below and here numbers 1 , 2 & 3 replacing a , b & c for better understanding.

I hope you liked the problem and the different solutions.Thank you so much for stoping by and spending your valuable time.

Have a great day.

All is well

Regards : @meta007