A problem on supplementary angle

Hello math bugs(🐞) and hivers(🐝)
Well come to another sweet geometrical problem.I hope you are strong and stout and doing good in life.

You have already seen today's problem in cover photo. First try it yourself and then check the solution and the concept.

All of my post are made considering all kinds of readers. So I make sure to include every little detail as possible. Experts to whom the concepts are known , please escape to the solution to save your time.

Before heading to the solution, we need to know a few concepts of geometry as follows:

1️⃣) Opposite angles of a cyclic quadrilateral are supplementary. It means sum of them will be 180°.

Note: If all vertices of a quadrilateral are on the circumference of circle, it is called cyclic quadrilateral. In the problem figure ADEF is cyclic quadrilateral.

2️⃣) For a cyclic quadrilateral opposite exterior angle of an angle is equal to itself.Check it below:

3️⃣) Vertically opposite angles are also equal to each other. Check it below:

Note: When two straight line or line segment intersect each other, creates two pair of Vertically opposite angles

4️⃣) Before conclude the concept part one more we need to know. Which is the sum of two interior angles of a triangles is equals to opposite exterior angle. Check it below.

The same property related to our problem. Interior Angle EBF plus angle BEF is equal to exterior angle AFE and on the other hand sum of interior angle DCE and CED is equal to exterior angle ADE. Detail is in the figure given below:

The solution:

Now the work is done. You have seen in the above figure ∠ABE= a +45° and ∠ADE= a + 24°. They are also opposite angles of the cyclic quadrilateral. So sum of them will be 180°. We can have a equation as a+25° + a + 45°= 180°. After slicing for a we get a = 55° and that's is the required answer. Check it below:

I think you get it now how to solve the problem. If you know all of the properties I mentioned , the problem can be solved orally.

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I have shared different geometrical concepts , properties , postulates in my previous blog. Check it please , I am sure it will help you to get better in geometry. If you have any unsolved problem(s), send it in the comment section.

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