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prove that a cyclic parallelogram must be a rectangle .
Let's break down the problem step by step:Cyclic Parallelogram: A cyclic parallelogram is a quadrilateral whose opposite angles are supplementary and all its vertices lie on a single circle.Parallelogram: A parallelogram is a quadrilateral with opposite sides parallel.To prove that a cyclic parallelogram must be a rectangle, we need to show that all its angles are right angles (90 degrees).Here's the proof:In a cyclic quadrilateral, opposite angles are supplementary, meaning the sum of the measures of the opposite angles is 180 degrees.Let's label the angles of the cyclic parallelogram as A, B, C, and D.In a parallelogram, opposite angles are equal. So, let's say angle A is equal to angle C, and angle B is equal to angle D.Now, if we add the measures of angle A and angle C, and the measures of angle B and angle D, the sum should be 180 degrees, because the opposite angles in a cyclic quadrilateral are supplementary.So, ( A + C = 180^\circ ) and ( B + D = 180^\circ ).Since A = C and B = D, we can substitute:[ A + A = 180^\circ ] and [ B + B = 180^\circ ].Thus, ( A = 90^\circ ) and ( B = 90^\circ ), which implies all angles in the parallelogram are right angles.Therefore, the cyclic parallelogram is a rectangle.
To prove that a cyclic parallelogram must be a rectangle, we need to use the properties of cyclic quadrilaterals and parallelograms.
### Definitions and Properties:
1. **Cyclic Parallelogram**: A parallelogram is cyclic if all its vertices lie on a common circle.
2. **Parallelogram**: A parallelogram is a quadrilateral with opposite sides parallel and equal in length.
3. **Cyclic Quadrilateral**: A quadrilateral is cyclic if its vertices lie on a single circle. In a cyclic quadrilateral, opposite angles sum up to 180 degrees.
### Proof:
1. **Properties of Cyclic Quadrilaterals**:
- For a cyclic quadrilateral, the sum of the opposite angles is \(180^\circ\). That is, if \(ABCD\) is a cyclic quadrilateral, then:
\[
\angle A + \angle C = 180^\circ
\]
\[
\angle B + \angle D = 180^\circ
\]
2. **Properties of a Parallelogram**:
- In a parallelogram, opposite angles are equal. So, if \(ABCD\) is a parallelogram, then:
\[
\angle A = \angle C
\]
\[
\angle B = \angle D
\]
3. **Combining the Properties**:
- Since \(ABCD\) is both a parallelogram and a cyclic quadrilateral, we use both sets of properties.
- From the property of the cyclic quadrilateral, we have:
\[
\angle A + \angle C = 180^\circ
\]
- Since opposite angles in a parallelogram are equal, we also have:
\[
\angle A = \angle C
\]
- Substitute \(\angle C\) from the parallelogram property into the cyclic quadrilateral property:
\[
\angle A + \angle A = 180^\circ
\]
\[
2 \angle A = 180^\circ
\]
\[
\angle A = 90^\circ
\]
- Thus, each angle in the parallelogram \(ABCD\) is \(90^\circ\), which means all the angles in the parallelogram are right angles.
4. **Conclusion**:
- Since a parallelogram with all angles equal to \(90^\circ\) is a rectangle, we conclude that a cyclic parallelogram must be a rectangle.
### Summary:
In a cyclic parallelogram, the property of the cyclic quadrilateral (opposite angles sum to \(180^\circ\)) combined with the property of the parallelogram (opposite angles are equal) shows that each angle in the parallelogram is \(90^\circ\). Therefore, the parallelogram must be a rectangle.
