Cyclic quadrilateral and alternate segment

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In summary, a cyclic quadrilateral is a four-sided polygon whose vertices lie on a single circle. Its properties include opposite angles being supplementary, opposite sides being equal, and the sum of any two opposite angles being 180 degrees. A quadrilateral can be proven to be cyclic if all four angles are inscribed angles, two opposite angles are equal, or two opposite sides are equal. The alternate segment theorem states that the angle between a tangent and a chord at the point of contact is equal to the angle in the alternate segment. This theorem is used in solving problems and proving other theorems related to circles, as well as in real-world applications such as engineering and architecture.
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grzz
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As the secant AE is moved downwards, the exterior angle remains equal to the same interior angle, with the result that as the secant becomes a tangent, the cyclic quadrilateral disappears and the exterior angle becomes equal to the angle in the alternate segment. pdf is attached.It is interesting to see a property of the cyclic quadrilateral being transformed into a property of the alternate segment.Any other geometrical examples like the above?
 

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The post included all I had to say.
Thanks.
 

1. What is a cyclic quadrilateral?

A cyclic quadrilateral is a four-sided polygon whose vertices lie on a single circle.

2. What are the properties of a cyclic quadrilateral?

The properties of a cyclic quadrilateral include:

  • Opposite angles are supplementary (sum up to 180 degrees).
  • Opposite sides are equal in length.
  • The sum of the measures of any two opposite angles is equal to 180 degrees.
  • The exterior angle is equal to the interior opposite angle.
  • The diagonals are perpendicular to each other.

3. How do you prove that a quadrilateral is cyclic?

A quadrilateral can be proven to be cyclic if one of the following conditions is met:

  • All four angles are inscribed angles.
  • Two opposite angles are equal.
  • Two opposite sides are equal.
In any of these cases, the quadrilateral's vertices must lie on a single circle, making it a cyclic quadrilateral.

4. What is alternate segment theorem?

The alternate segment theorem states that the angle between a tangent and a chord at the point of contact is equal to the angle in the alternate segment. In other words, if a tangent and a chord intersect at a point on the circle, the angle formed by the tangent and the chord is equal to the angle formed by the chord and a line segment connecting the point of intersection to any other point on the circle.

5. How is alternate segment theorem used in solving problems?

The alternate segment theorem can be used to find missing angles or lengths in a cyclic quadrilateral. It is also used in proving other theorems related to circles, such as the inscribed angle theorem. In real-world applications, it can be used in engineering and architecture for designing circular structures.

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