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This section discusses cyclic quadrilaterals, defining them and exploring their unique properties, particularly that the sum of the opposite angles is 180°. Additionally, the converse is also established, emphasizing the significance of these quadrilaterals in circle geometry.
A cyclic quadrilateral is defined as a quadrilateral whose vertices all lie on a circle. This section illustrates key properties of cyclic quadrilaterals, particularly focusing on angle relationships. When measured, the opposite angles of a cyclic quadrilateral always sum to 180 degrees. Thus, for any cyclic quadrilateral ABCD, we find that: $$\angle A + \angle C = 180^\circ \quad \text{and} \quad \angle B + \angle D = 180^\circ$$
Additionally, the converse is proven: if the sum of a pair of opposite angles equals 180 degrees, the quadrilateral can be classified as cyclic. Several examples and exercises are presented to help solidify these concepts.
Cyclic Quadrilaterals: A quadrilateral whose vertices lie on a circle, allowing unique angle properties.
Sum of Opposite Angles: For cyclic quadrilaterals, the sum of opposite angles is always 180°.
In a cyclic quadrilateral drawing, opposite angles are 180° bonding.
Imagine a wheel representing a circle; where every spoke meets a point, those points form angles with love, giving 180° joy.
Two Opposite Angles = 180 (You can remember: 'Two Ovens Are Hot' - OAH).
{'example': 'In cyclic quadrilateral ABCD, if \( \angle A = 70° \), find \( \angle C \).', 'solution': '\( \angle C = 180° - 70° = 110° \).'}
{'example': 'Prove that quadrilateral ABCD is cyclic if \( \angle A + \angle C = 180° \).', 'solution': 'By the converse theorem, this means ABCD is cyclic.'}
Term: Cyclic Quadrilateral
Definition: A quadrilateral with all four vertices lying on a circle.
A quadrilateral with all four vertices lying on a circle.
Term: Opposite Angles
Definition: Angles that are across from each other in a quadrilateral (e.g., \( \angle A \) and \( \angle C \)).
Angles that are across from each other in a quadrilateral (e.g., \( \angle A \) and \( \angle C \)).
Term: Converse Theorem
Definition: The principle stating that if the sum of a pair of opposite angles equals 180°, the quadrilateral is cyclic.
The principle stating that if the sum of a pair of opposite angles equals 180°, the quadrilateral is cyclic.