International Mathematics (0607) | Angles

📚 Comprehensive IGCSE Mathematics (0607) Study Notes: Angles 📚

Hello future Geometry Master! This chapter is all about Angles, which is the foundational language of Geometry. Angles are everywhere—from designing buildings to navigating ships and even setting up a snooker shot. Mastering these rules will help you solve complex problems involving shapes and navigation.
Don't worry if geometry seems tricky at first; we will break down every rule into simple, easy-to-remember parts!

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1. Understanding Basic Angle Types and Notation

1.1 Geometrical Terms Review

Before calculating, we must use the correct language. Here are the basic terms required by the syllabus:

• Point: A specific location, usually marked by a dot.
• Line: A straight path that extends infinitely in both directions.
• Vertex (or Corner): The point where two lines or rays meet to form an angle.
• Perpendicular: Lines that meet exactly at a 90° angle.
• Parallel: Lines that run alongside each other and never meet (marked by arrows).

1.2 Classifying Angles by Size

We classify angles based on how many degrees they measure:

• Acute Angle: Less than 90°. (Think of a cute, small angle!)
• Right Angle: Exactly 90°. Usually marked with a small square box.
• Obtuse Angle: Greater than 90° but less than 180°.
• Straight Angle: Exactly 180°. This is simply a straight line.
• Reflex Angle: Greater than 180° but less than 360°.

1.3 Using Three-Letter Notation

When solving angle problems, especially when giving explanations, you must use the correct notation (C5.5 Notes).
If you are referring to the angle at vertex B, formed by lines AB and BC, you write:
Angle ABC or \(\angle ABC\). (The vertex must always be the middle letter.)

Key Takeaway 1

Geometry relies on precise language. Know your angle types and always use the three-letter notation correctly when explaining your answers.

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2. Angles at Intersections and on Straight Lines

2.1 Fundamental Angle Properties (C5.5.1)

These are the absolute basic rules you must know by heart:

Property 1: Sum of Angles at a Point
The sum of all angles around a single point is \(360^{\circ}\).
(Analogy: If you stand and turn all the way around, you have turned \(360^{\circ}\).)

Property 2: Angles on a Straight Line (Supplementary Angles)
The sum of angles on a straight line is \(180^{\circ}\).

Property 3: Vertically Opposite Angles
When two straight lines intersect (cross), the angles opposite each other are equal.

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3. Angles in Parallel Lines (C5.5.2)

These rules only apply when you have two or more lines that are parallel (and usually marked with arrows), cut by a third line called a transversal. These rules are essential for navigation (bearings)!

3.1 The F, Z, and C Rules (Mnemonics)

1. Corresponding Angles (The 'F' Rule)

If you draw a letter 'F' along the parallel lines and the transversal, the angles in the corresponding positions (under the top parallel line and under the bottom parallel line) are equal.

Reasoning: Corresponding angles are equal.

2. Alternate Angles (The 'Z' Rule)

If you draw a letter 'Z' (or a backward 'Z') along the lines, the angles in the "corners" of the Z are equal.

Reasoning: Alternate angles are equal.

3. Co-Interior Angles (The 'C' or 'U' Rule)

The angles stuck inside the parallel lines on the same side of the transversal (forming a 'C' or 'U' shape) are supplementary (they sum to \(180^{\circ}\)).

Reasoning: Co-interior angles are supplementary (or sum to \(180^{\circ}\)).

⚠ Common Mistake to Avoid: You must ensure the lines are parallel before you can use the F, Z, or C rules! If the question doesn't state they are parallel, you cannot use these theorems.

Key Takeaway 2

Parallel line rules (F, Z, C) are vital. Remember: F and Z angles are EQUAL. C angles SUM to 180°.

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4. Angles in Polygons (C5.5.1 & E5.5.3)

A polygon is any closed 2D shape with three or more straight sides.

4.1 Triangles and Quadrilaterals (C5.5.1)

The total sum of interior angles changes based on the number of sides (n):

• Triangle (n=3): Angle sum = \(180^{\circ}\).
• Quadrilateral (n=4): Angle sum = \(360^{\circ}\).

4.2 General Polygons: Interior Angles

For any polygon with \(n\) sides, the total sum of the interior angles is given by the formula:

Sum of Interior Angles = \((n-2) \times 180^{\circ}\)

Did you know? This formula works because you can always divide an n-sided polygon into \(n-2\) triangles by drawing diagonals from one vertex!

4.3 General Polygons: Exterior Angles

The exterior angle of a polygon is the angle formed between a side and the extension of the adjacent side.

Crucial Rule: The sum of the exterior angles of any convex polygon (regular or irregular) is always \(360^{\circ}\).

4.4 Regular Polygons (C5.5.3)

A regular polygon has all sides equal and all interior angles equal.

If a polygon has \(n\) sides:

1. Each Exterior Angle: \(\frac{360^{\circ}}{n}\)
2. Each Interior Angle: \(180^{\circ} - (\text{Exterior Angle})\)
   OR: \(\frac{(n-2) \times 180^{\circ}}{n}\)

Example: A regular hexagon has 6 sides (n=6).
Exterior Angle = \(360^{\circ}/6 = 60^{\circ}\).
Interior Angle = \(180^{\circ} - 60^{\circ} = 120^{\circ}\).

Key Takeaway 3

When dealing with polygons, the simplest angle to find is always the Exterior Angle using \(360/n\). Find the exterior angle first, then use it to find the interior angle!

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5. Bearings (C5.2.2)

Bearings are used in navigation to describe direction using angles. They have three crucial rules:

1. Always measure from North: North is considered the starting line (\(000^{\circ}\)).

2. Always measure clockwise: Turn right from the North line.

3. Always use three figures: Even if the angle is small, like 25°, it must be written as \(025^{\circ}\).

Example: If the bearing from point A to point B is \(050^{\circ}\). This means you start at A, face North, and turn \(50^{\circ}\) clockwise to face B.

Calculating the Reverse Bearing

To find the bearing of A from B (the return journey), you use the rules of parallel lines, because the North lines at A and B are always parallel.

Step 1: Recognize that the line AB cuts two parallel North lines. The forward bearing angle and the angle inside the parallel lines at B are co-interior (C-rule), so they sum to \(180^{\circ}\).

Step 2: The bearing of A from B is the angle measured clockwise from B's North line.

• If the original bearing is less than \(180^{\circ}\): Add \(180^{\circ}\).
• If the original bearing is greater than \(180^{\circ}\): Subtract \(180^{\circ}\).

Example: Bearing of B from A is \(100^{\circ}\).
Reverse bearing (A from B) = \(100^{\circ} + 180^{\circ} = 280^{\circ}\).

Key Takeaway 4

Bearings are 3-figure, Clockwise, and start from the North line. Reverse bearings usually involve adding or subtracting \(180^{\circ}\).

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6. Circle Theorems (C5.6, E5.6, E5.7)

These are special angle rules that apply only to circles.

6.1 Core Content Circle Theorems (C5.6)

Theorem 1: Angle in a Semicircle
The angle subtended by the diameter at any point on the circumference is always \(90^{\circ}\) (a right angle).

Theorem 2: Tangent and Radius
The angle between a tangent (a line that touches the circle at exactly one point) and the radius (or diameter) at the point of contact is always \(90^{\circ}\).

6.2 Extended Content Circle Theorems (E5.6 & E5.7)

If you are studying the Extended curriculum, you must know these additional properties.

Theorem 3: Angle at Centre vs. Circumference
The angle subtended by an arc at the centre is twice the angle subtended by the same arc at any point on the circumference.

Formula: \(\angle \text{Centre} = 2 \times \angle \text{Circumference}\)

Theorem 4: Angles in the Same Segment
Angles subtended by the same arc (or chord) in the same segment of a circle are equal.

Theorem 5: Cyclic Quadrilateral
A cyclic quadrilateral is a four-sided shape where all four vertices lie on the circumference of the circle.
Rule: Opposite angles of a cyclic quadrilateral are supplementary (they 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 one sounds complicated! Imagine a triangle inside a circle, touching a tangent line. The angle formed between the chord and the tangent equals the opposite internal angle of the triangle.)

6.3 Symmetry Properties of Circles (E5.7)

Symmetry 1: Equal Tangents
Tangents drawn to a circle from an external point are equal in length.

Symmetry 2: Perpendicular Bisector of a Chord
The perpendicular line drawn from the centre of the circle to a chord bisects (cuts in half) the chord.

Symmetry 3: Equal Chords
Equal chords are equidistant (the same distance) from the centre.

Key Takeaway 5

For circle theorems, always identify the key components first: Is there a diameter? Is there a tangent? Where is the angle measured (centre or circumference)? You must quote the correct theorem as your reason!