Geometry Revision Notes

Angles on a Straight Line and Around a Point

Angles on a straight line always add up to 180°. For example, if two angles share a vertex and form a straight line, their sum is 180°. Angles around a central point complete a full circle, so their sum is always 360°. This applies to any number of angles meeting at a single vertex with no gaps or overlaps. These principles are essential for finding missing angles in various diagrams.

Parallel Lines and Transversals

When a straight line (called a transversal) crosses two parallel lines, specific angle relationships are formed. 'Alternate angles' are equal; they are on opposite sides of the transversal and between the parallel lines (like a 'Z' shape). 'Corresponding angles' are also equal; they are in the same relative position at each intersection (like an 'F' shape). 'Interior angles' (or allied angles) are on the same side of the transversal and between the parallel lines, and they add up to 180° (like a 'C' or 'U' shape). Identifying these relationships accurately is key to solving problems involving parallel lines.

Vertically Opposite Angles

When two straight lines intersect, they create two pairs of vertically opposite angles. These angles are directly opposite each other at the point of intersection. A crucial property is that vertically opposite angles are always equal. For instance, if two lines cross to form angles A, B, C, and D, where A and C are opposite, and B and D are opposite, then A = C and B = D. This relationship is very useful for finding unknown angles quickly.

Angles in Polygons

The sum of the interior angles of a triangle is always 180°. This is a fundamental rule in geometry. For example, if you know two angles in a triangle, you can find the third by subtracting the sum of the known angles from 180°. The sum of the interior angles of a quadrilateral is always 360°. This can be thought of as two triangles joined together. Understanding these sums is vital for solving problems involving shapes with straight sides.

Polygon Angle Sums

For any polygon with 'n' sides: The sum of the interior angles is given by the formula (n - 2) * 180°. For example, a quadrilateral (n=4) has an interior angle sum of (4-2)*180° = 360°. A pentagon (n=5) has a sum of (5-2)*180° = 540°. The sum of the exterior angles of any convex polygon is always 360°. For a regular polygon, each interior angle is (n-2)*180°/n and each exterior angle is 360°/n. Remember that an interior angle and its adjacent exterior angle sum to 180°.

Circle Theorems

There are several key circle theorems: 1. Angle at the centre is twice the angle at the circumference subtended by the same arc. 2. Angles in the same segment are equal. 3. The angle in a semicircle is a right angle (90°). 4. Opposite angles of a cyclic quadrilateral sum to 180°. 5. The angle between a tangent and a radius drawn to the point of contact is 90°. 6. Tangents from an external point to a circle are equal in length. 7. The alternate segment theorem states that the angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment.

Parallel Lines and Transversals

When two parallel lines are intersected by a transversal line, several angle relationships arise: 1. Alternate angles are equal (they form a 'Z' shape). 2. Corresponding angles are equal (they form an 'F' shape). 3. Co-interior (or consecutive interior) angles sum to 180° (they form a 'C' shape). These rules are fundamental for determining unknown angles in diagrams involving parallel lines, often combined with other angle facts such as angles on a straight line or vertically opposite angles.

Reasoning and Problem Solving

Solving complex angle problems often requires a systematic approach. First, carefully examine the diagram and identify all given information. Look for parallel lines, tangents, radii, chords, and regular polygons. Next, identify which angle rules apply to the specific geometric features you've found. You may need to add auxiliary lines (e.g., a radius, a chord, or another parallel line) to reveal hidden angle relationships. Break down the problem into smaller, manageable steps, and deduce unknown angles one by one, stating the geometric reason for each step. Finally, check your work to ensure all conditions are met and the requested angle has been found and justified.

What is a Polygon?

A polygon is a closed 2D shape with straight sides. The sides meet at vertices. The number of sides determines the polygon's name: 3 sides = triangle, 4 sides = quadrilateral, 5 sides = pentagon, 6 sides = hexagon, 7 sides = heptagon, 8 sides = octagon, 9 sides = nonagon, 10 sides = decagon. A regular polygon has all sides equal in length and all interior angles equal in measure. An irregular polygon does not have all sides and angles equal.

Interior Angles of Polygons

The sum of the interior angles of a polygon with 'n' sides can be calculated using the formula: Sum = (n - 2) × 180°. For example, a quadrilateral (n=4) has an interior angle sum of (4-2) × 180° = 2 × 180° = 360°. For a regular polygon, each interior angle is found by dividing the sum by the number of sides: Each interior angle = ( (n - 2) × 180° ) / n. Interior and exterior angles at any vertex of a polygon always add up to 180°.

Exterior Angles of Polygons

The sum of the exterior angles of any convex polygon is always 360°. This is a constant value regardless of the number of sides. For a regular polygon, each exterior angle is found by dividing 360° by the number of sides: Each exterior angle = 360° / n. This relationship is very useful. Knowing one exterior angle of a regular polygon allows you to immediately find the number of sides (n = 360° / exterior angle) and thus all other properties.

What is a Polygon?

A polygon is a closed 2D shape with straight sides. The number of sides determines its name. Common names include: Triangle (3 sides), Quadrilateral (4 sides), Pentagon (5 sides), Hexagon (6 sides), Heptagon (7 sides), Octagon (8 sides), Nonagon (9 sides), Decagon (10 sides). A polygon must have at least 3 sides. The sides meet at vertices. A regular polygon has all sides equal in length and all interior angles equal in measure. An irregular polygon does not have all sides and angles equal.

Sum of Interior Angles

The sum of the interior angles of a polygon with 'n' sides is given by the formula: Sum = (n - 2) × 180°. For example, a quadrilateral (n=4) has a sum of interior angles of (4 - 2) × 180° = 2 × 180° = 360°. A pentagon (n=5) has a sum of interior angles of (5 - 2) × 180° = 3 × 180° = 540°. This formula is crucial for finding missing angles within polygons, especially when dealing with irregular shapes or when verifying regular shapes.

Exterior Angles of Polygons

The sum of the exterior angles of ANY convex polygon is always 360°. This is a very useful property. An exterior angle is formed by extending one side of the polygon. At any vertex, an interior angle and its corresponding exterior angle add up to 180°. For a regular polygon with 'n' sides, each exterior angle is 360° / n. From this, each interior angle of a regular polygon can be found as 180° - (360° / n) or using the formula: ( (n - 2) × 180° ) / n.