Euclidean Geometry: Lines, Angles, Polygons, and Circles

1. Properties of Angles and Lines

Geometry begins with understanding the relationship between lines and angles. Key concepts include:

• Complementary Angles: Two angles that sum up to 90°.
• Supplementary Angles: Two angles that sum up to 180°.
• Angles on a Straight Line: Sum to 180°.
• Angles at a Point: Sum to 360°.
• Vertically Opposite Angles: Formed by two intersecting lines; they are always equal.
• Parallel Lines: When a transversal cuts two parallel lines, alternate angles are equal, corresponding angles are equal, and co-interior angles sum to 180°.

2. Polygons

A polygon is a closed plane figure bounded by straight lines.

• Triangles: The sum of interior angles is 180°. Types include Equilateral (all sides/angles equal), Isosceles (two sides/angles equal), and Scalene (no sides equal).
• Interior Angles of a Polygon: For a polygon with n sides, the sum of interior angles is (n - 2) × 180°.
• Exterior Angles: The sum of the exterior angles of any convex polygon is 360°. For a regular polygon, each exterior angle = 360/n.
• Quadrilaterals: Sum of angles is 360°. Includes Parallelograms, Rectangles, Rhombuses, Squares, and Trapeziums.

3. Circle Theorems

Circles have specific geometric properties essential for JAMB:

• Angle at Center: The angle subtended by an arc at the center is twice the angle subtended at the circumference.
• Angle in a Semi-circle: Any angle subtended by the diameter at the circumference is always 90°.
• Angles in the Same Segment: These are always equal.
• Cyclic Quadrilaterals: A quadrilateral with all four vertices on a circle. Opposite angles are supplementary (sum to 180°). The exterior angle is equal to the interior opposite angle.
• Tangent Properties: A tangent is perpendicular to the radius at the point of contact. Tangents from an external point to a circle are equal in length.
• Alternate Segment Theorem: The angle between a tangent and a chord is equal to the angle in the alternate segment.

4. Geometric Constructions

Construction involves drawing shapes accurately using a ruler and a pair of compasses.

• 60°: Constructed by drawing an equilateral triangle.
• 90°: Constructed by bisecting a straight angle (180°).
• 30°: Bisecting a 60° angle.
• 45°: Bisecting a 90° angle.
• 75°: Bisecting the angle between 60° and 90°.
• Locus: The set of points that satisfy a specific condition (e.g., a circle is the locus of points equidistant from a fixed center).