🟡 Concept 1: Central Angle and Minor vs Major Arc

Let \( \angle AOC = \theta \), where \( O \) is the center of the circle. The arc \( \overset{\frown}{AC} \) has:

• Minor arc: \( m(\overset{\frown}{AC}) = \theta \) if \( \theta < 180^\circ \)
• Major arc: \( m(\overset{\frown}{AC}) = 360^\circ - \theta \)

🔹 Concept 2: Angles Can't Lie on Arcs

Angles like \( \angle ACB \) are defined at points, not over arcs. Use arc notation like \( \overset{\frown}{ADB} \) to refer to curved paths.

🔹 Concept 3: Arc Determines Angle Type

The size of the arc determines the angle at the circle. Larger arc → smaller peripheral angle and vice versa.

🔹 Concept 4: Chords vs Secants

• Chord: Segment connecting two points on the circle (e.g., \( AB \))
• Secant: A line cutting through the circle at two points (e.g., \( ACL \))

🔹 Concept 5: Angle Subtended by Arc

\( \angle ADB \) is subtended by arc \( \overset{\frown}{ACB} \). As arc length increases, the peripheral angle decreases.

🔹 Concept 6: Towards the Center = Minor Arc

If an angle opens towards the center (like \( \angle ADB \)), it is subtending the minor arc \( \overset{\frown}{AB} \).

🔹 Concept 7: Cyclic Quadrilateral Properties

• Opposite angles in a cyclic quadrilateral sum up to \( 180^\circ \)
• If point D is near center: \( \angle D < 90^\circ \); away from center: \( \angle C > 90^\circ \)