A tricky geometric problem

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

Today, I have cone up with another intersting and tricky problem. This a unique case where angles 75°, 60° and 45° are at different point of base of a triangle and difference between are same. You can see , it is 15°.

Ohh sorry! We haven't yet find the required angle. Yeah, it will be 75°. Before heading towards solution and proof, give it try if you want. All the best. Whenever I see a problem for the first time, I always try it out by myslef. It is a fun for me and most of the time I find a solution. If you have habit of doing so, you soon gonna master of it. I don't know if I am a master of solving problem but when I solve them, it satisfies me.

If you are done, let's solve it with the help of some construction.

We have drawn a point AD such that DE and CD become equal. Thus we find the above figure. The other agnles mentined in the above figure come form a postulate which you must be aware of and it is given by as follows:

The external is equals to the sum of opposite two angles of right angle triangle.

As we have drawn DE =DC and we have the avobe poatulate, we can find ∠CED = ∠ECD = 30° And from angle some property we can find ∠EAC = ∠ECA= 15°

And so thus we can find AE = CE

Now notice the ∆BDE. It will be a right angle triangle. Why!!??
let's checkt it first in a separate figure with our eye-capturing veiw

From the above figure(∆) we just needed to know the value of ∠EBD and wchich is 30°

If you have doubt tap pythagorean TRIPLET to know details.

Now again get to the original problem figure with a new edited vesion . Try to write all the acquired angles derived from amgle sum property and relation of external angle and inter angle of a trinagle.

Now let's check the final figure to conclude what I was trying to prove.

What properties / prostulate(s) /axiom(s) have used are as follows:

1. Contruction.
2. Angle sum property of triangle.
3. Realation interior and exterior angle of trange.
4. Ralation of angles and thier opposite sides.

It were very easy if there were a given construction but without construction it is hard to get to the solution if you are facing it for the first time.

All the pictures(figures) are made my me using android app.There may be some silly misatke , please ignore it if you can.

I hope you liked my present to you. If you have doubt or anything to say please feel free to make a comment. I am waiting for you reply.🧐

Thanks you so much for visiting.

Have a nice day.

All is well

Regards: @meta007