4.6-4.8: Angles and Circle Theorems Notes:

Angles

Key Angle Properties

• Angles at a point → Sum = 360°

• Angles on a straight line → Sum = 180°

• Vertically opposite angles → Are equal

• Angle sum of a triangle → Sum = 180°

• Angle sum of a quadrilateral → Sum = 360°

Angles in Parallel Lines

• When a transversal cuts parallel lines:

  • Corresponding angles → Equal → Same relative position (e.g. top right of each intersection)

  • Alternate angles → Equal → Z-shape pattern across the lines

  • Co-interior angles → Sum to 180° (supplementary angles) → C-shape pattern inside the parallel lines

Angle Properties of Regular Polygons

• Interior angle of a regular polygon: → [(n − 2) × 180°] ÷ n, where n = number of sides

• Exterior angle of a regular polygon: → 360° ÷ n

• Example: 

• Regular hexagon (n = 6) 

→ Interior angle = [(6 − 2) × 180°] ÷ 6 = 120° 

→ Exterior angle = 360° ÷ 6 = 60°

Notation and Reasoning

• Use three-letter notation for angles, e.g. ∠ABC where the angle you're talking about is the middle letter

• Always give geometric reasons when solving angle problems: → e.g. “∠ABC = 120° because it’s an interior angle of a regular hexagon”

Circle Theorems

Key Theorems

• Angle in a semicircle = 90° 

  • If a triangle is drawn inside a semicircle with its base as the diameter, the angle opposite the diameter is a right angle.

• Angle between a tangent and a radius = 90° 

  • A line that just touches the circle (tangent) is always perpendicular to the radius at the point of contact.

How to Use These Theorems

• Identify the semicircle or tangent–radius pair in the diagram

• Apply the correct theorem to calculate unknown angles

• Always give a geometrical reason for your answer

• Example 1: 

  • In a triangle inscribed in a semicircle, if the diameter is AB and point C lies on the arc, then: → ∠ACB = 90° (angle in a semicircle)

• Example 2: 

  • If a tangent touches a circle at point T and OT is the radius: → ∠OTX = 90° (angle between tangent and radius)

More Theorems:

Key Theorems:

• Angle at the center is twice the angle at the circumference. If two angles subtend the same arc, the one at the center is double the one at the circumference.

• Angles in the same segment are equal. Angles subtended by the same chord on the same side of the chord are equal.

• Cyclic quadrilateral: opposite angles sum to 180° A quadrilateral inscribed in a circle has opposite angles that are supplementary.

• Angle between chord and tangent = angle in the alternate segment Known as the Alternate Segment Theorem.

NOTES DONE BY FARIDA SABET

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