Mathematics (0580) | Angles

Study Notes: Geometry – Angles (Cambridge IGCSE 0580 / 0607)

Welcome to the chapter on Angles! Geometry can look intimidating, but angles are the building blocks of all shapes and structures around us. Mastering these rules will help you solve complex problems and understand why shapes behave the way they do. Don't worry if some concepts seem tricky at first; we’ll break everything down using simple rules and memory aids!

1. The Fundamentals: Types of Angles and Basic Facts

Before diving into calculations, let's quickly review the types of angles you need to recognize:

• Acute angle: Less than 90°. (Think of a cute, small angle!)
• Right angle: Exactly 90°. Marked with a small square. (The corner of a book.)
• Obtuse angle: Greater than 90° but less than 180°.
• Reflex angle: Greater than 180° but less than 360°. (This is the ‘outside’ angle.)

1.1 Key Angle Rules (The Building Blocks)

These four rules are absolutely essential. You must know these and be ready to state them as reasons in your explanations.

1. Angles at a Point

• The sum of angles around a point is always \(360^\circ\).
• Analogy: A full turn, like spinning 360 degrees.

2. Angles on a Straight Line

• The sum of angles on a straight line is always \(180^\circ\).
• These angles are called supplementary.
• Analogy: Half a full turn.

3. Vertically Opposite Angles

• When two straight lines intersect, the angles opposite each other are equal.
• Did you know? These angles are sometimes called the "X-angles" because they form the shape of an 'X'.

4. Angles in Triangles and Quadrilaterals

• The sum of interior angles in any triangle is \(180^\circ\).
• The sum of interior angles in any quadrilateral (a four-sided shape) is \(360^\circ\).

2. Angles and Parallel Lines

This section deals with the special relationships created when a straight line (called a transversal) crosses two or more parallel lines. Parallel lines are marked with arrows on the diagram.

We have three crucial rules here, often remembered using letters (Z, F, C).

2.1 The Parallel Line Rules

1. Alternate Angles (The ‘Z’ Rule)

• Alternate angles are found inside the parallel lines and on opposite sides of the transversal.
• Alternate angles are equal.
• Memory Aid: Look for the letter 'Z' drawn across the parallel lines. The angles in the corners of the Z are equal.
• Reason to state: Alternate angles are equal.

2. Corresponding Angles (The ‘F’ Rule)

• Corresponding angles are in the same relative position at each intersection (e.g., top-left).
• Corresponding angles are equal.
• Memory Aid: Look for the letter 'F' drawn along the parallel lines. The angles under the arms of the F are equal.
• Reason to state: Corresponding angles are equal.

3. Co-interior Angles (The ‘C’ Rule)

• Co-interior angles (or consecutive interior angles) are found inside the parallel lines and on the same side of the transversal.
• Co-interior angles sum to \(180^\circ\) (they are supplementary).
• Memory Aid: Look for the letter 'C' (or a backward 'C'). The angles inside the C add up to 180°.
• Reason to state: Co-interior angles sum to \(180^\circ\).

3. Angles in Polygons (Shapes with Many Sides)

A polygon is any closed shape with straight sides. We focus on calculating the sum of the interior angles and the size of the exterior angles.

3.1 Interior Angles

The sum of the interior angles changes depending on the number of sides, \(n\).

Formula for Interior Angle Sum:
Sum \( = (n - 2) \times 180^\circ\)

Step-by-step Explanation:

1. Count the number of sides (\(n\)).
2. Subtract 2. This is the number of non-overlapping triangles you can form inside the polygon from one vertex.
3. Multiply this number by \(180^\circ\) (the sum of angles in one triangle).

Example: A hexagon has \(n=6\) sides.
Sum \(= (6 - 2) \times 180^\circ = 4 \times 180^\circ = 720^\circ\).

3.2 Exterior Angles

An exterior angle is formed when one side of the polygon is extended. The exterior angle and the adjacent interior angle always form a straight line, so they sum to \(180^\circ\).

Rule 1: Sum of Exterior Angles

• For any convex polygon (regular or irregular), the sum of the exterior angles is always \(360^\circ\).
• Analogy: If you walk around the edge of a polygon and turn a corner at each vertex, by the time you get back to where you started, you've completed a full \(360^\circ\) turn!

Rule 2: Regular Polygons

A regular polygon has all sides equal and all angles equal. This makes calculating individual angles very simple.

• Exterior Angle: \( = \frac{360^\circ}{n}\)
• Interior Angle: \( = 180^\circ - \text{Exterior Angle}\)

Example: Find the interior angle of a regular octagon (\(n=8\)).

1. Calculate Exterior Angle: \(\frac{360^\circ}{8} = 45^\circ\)
2. Calculate Interior Angle: \(180^\circ - 45^\circ = 135^\circ\)

4. Bearings: Angles for Navigation

Bearings use angles to describe direction, especially in navigation or map reading. They have strict rules you must follow:

4.1 The Three Rules of Bearings

1. Measured from North

• Always start measuring from the North line. The North line points vertically upwards.

2. Measured Clockwise

• The angle must be measured in a clockwise direction from the North line.

3. Use Three Figures

• The bearing must always be written using three figures (e.g., \(045^\circ\), not \(45^\circ\)).

Example: A bearing of \(090^\circ\) means due East. A bearing of \(270^\circ\) means due West.

4.2 Using Parallel Lines in Bearings

The North line at point A and the North line at point B are always parallel. This means you must use your parallel line rules (Z, F, C) to solve bearing problems involving two points.

If you know the bearing of B from A, you can find the bearing of A from B (the reverse bearing) by using co-interior angles (the C-rule, which sums to \(180^\circ\)).

• If Bearing (B from A) \( < 180^\circ \): Reverse Bearing \( = \text{Bearing} + 180^\circ \)
• If Bearing (B from A) \( > 180^\circ \): Reverse Bearing \( = \text{Bearing} - 180^\circ \)

5. Extended Content: Angles in Circles (Circle Theorems)

If you are studying for Extended Mathematics, you must master the Circle Theorems. These rules allow you to find unknown angles in diagrams involving circles, radii, chords, and tangents.

Remember to use the correct geometrical term (e.g., Angle at the centre is twice the angle at the circumference) when giving reasons.

5.1 Circle Theorems (I) – Core/Basic Extended

Theorem 1: Angle in a Semicircle

• The angle subtended by the diameter at any point on the circumference is \(90^\circ\).
• Reason: Angle in a semicircle is \(90^\circ\).

Theorem 2: Tangent and Radius

• A tangent (a line that touches the circle at exactly one point) meets the radius (or diameter) at the point of contact at \(90^\circ\).
• Reason: Angle between tangent and radius is \(90^\circ\).

5.2 Circle Theorems (II) – Full Extended Content

Theorem 3: Angle at the Centre vs. Circumference

• The angle at the centre is twice the angle at the circumference when both angles are subtended by the same arc.
• If Angle at Circumference is \(x\), then Angle at Centre is \(2x\).
• Reason: Angle at the centre is twice the angle at the circumference.

Theorem 4: Angles in the Same Segment

• Angles subtended by the same arc (or chord) in the same segment of a circle are equal.
• Reason: Angles in the same segment are equal.

Theorem 5: Cyclic Quadrilateral

• A cyclic quadrilateral is a four-sided shape where all four vertices lie on the circumference.
• The opposite angles of a cyclic quadrilateral sum to \(180^\circ\) (they are supplementary).
• Reason: Opposite angles of a cyclic quadrilateral sum to \(180^\circ\).

Theorem 6: Alternate Segment Theorem

• The angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment.
• This is often the hardest theorem to spot! Look for a triangle sitting on the point where the tangent touches the circle.
• Reason: Alternate segment theorem.

Theorem 7: Tangents from an External Point (Symmetry Property)

• If two tangents are drawn from an external point to a circle, their lengths from the external point to the points of contact are equal.