Sum of interior angles of cyclic hexagon

In summary, the problem is to prove that the sum of interior angles at A, C, and E in a cyclic hexagon is equal to the sum of interior angles at B, D, and F. The vertices of the hexagon are labelled A to F and are all on the circumference of a circle. The question also asks for a generalization to other cyclic polygons. The person seeking help has attempted the problem without much success and is open to hints or further guidance.
  • #1
Natasha1
493
9
Right I have been given the following problem and cannot resolve it. I have had an attempt but without much success. Could anyone help me with this exercise, please? Hints or a little more welcome :-)

A cyclic hexagon is a hexagon whose vertices all lie on the circumference of a circle.

The vertices of a cyclic hexagon are labelled in order A to F. Prove that the sum of the interior angles at A, C and E is equal to the sum of the interior angles at B, D and F.

Generalise (concisely) to other cyclic polygons?
 
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  • #2
what have you done so far?
 

1. How do you find the sum of interior angles of a cyclic hexagon?

The sum of interior angles of a cyclic hexagon is always equal to 720 degrees.

2. What is a cyclic hexagon?

A cyclic hexagon is a six-sided polygon that can be inscribed in a circle, with all of its vertices touching the circumference of the circle.

3. Is there a formula for finding the sum of interior angles of a cyclic hexagon?

Yes, the formula is (6-2) x 180 = 720 degrees. This formula can be generalized for any regular polygon with n sides: (n-2) x 180.

4. How can you prove that the sum of interior angles of a cyclic hexagon is always 720 degrees?

This can be proven using the fact that the sum of exterior angles of any polygon is always 360 degrees. Since a cyclic hexagon has 6 sides, the sum of exterior angles is 6 x 360 = 2160 degrees. And since the exterior angle and interior angle are supplementary angles, the sum of interior angles is equal to 2160 - 360 = 720 degrees.

5. Can the sum of interior angles of a cyclic hexagon be less than 720 degrees?

No, the sum of interior angles of a cyclic hexagon will always be equal to 720 degrees, regardless of the size or shape of the hexagon.

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