Theorem 10.7: In a cyclic quadrilateral, the opposite angles are supplementary.



Explanation:
A cyclic quadrilateral is a quadrilateral whose all four vertices lie on a circle. The theorem states that the opposite angles of such a quadrilateral are supplementary, which means that their sum is equal to 180 degrees.

Proof:
Let's consider a cyclic quadrilateral ABCD with vertices A, B, C, and D lying on a circle.

1. Angle A + Angle C = 180 degrees:
The opposite angles in the cyclic quadrilateral are angle A and angle C. To prove that they are supplementary, we can use the fact that the sum of angles in a straight line is 180 degrees.

Since angle BCD is an exterior angle of triangle ABC, it is equal to the sum of the opposite interior angles, angle A and angle C. Therefore, angle A + angle C = angle BCD.

2. Angle B + Angle D = 180 degrees:
Similarly, the opposite angles in the cyclic quadrilateral are angle B and angle D. To prove that they are supplementary, we can use the same reasoning as above.

Since angle CAD is an exterior angle of triangle BCD, it is equal to the sum of the opposite interior angles, angle B and angle D. Therefore, angle B + angle D = angle CAD.

Conclusion:
In a cyclic quadrilateral, the opposite angles are supplementary. This means that the sum of angle A and angle C is equal to 180 degrees, and the sum of angle B and angle D is also equal to 180 degrees. This property holds true for all cyclic quadrilaterals.

Note:
The concept of cyclic quadrilaterals and their properties is important in geometry. They have various applications, such as in proving theorems related to angles, sides, diagonals, and circumcircles of quadrilaterals. Understanding and applying these properties can help in solving problems and proving other geometric theorems.

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