10.1 Angles & Parallel and Perpendicular Lines

Angles sit inside almost every Space & Geometry question in ICAS Mathematics. From Paper D (Year 6) onwards you are expected to know parallel and perpendicular lines, and by Papers E–F the examiners build multi-step angle chases using triangles and transversals. Because no personal calculator is allowed — and angle questions rarely need one anyway — these marks reward clear reasoning, not arithmetic speed.

The Five Angle Types

An angle is the amount of turn between two lines (called arms) that meet at a point (called the vertex). Angles are measured in degrees (°), and a full turn is 360°.

A right angle is marked on diagrams with a small square, not an arc. If you see that square, the angle is exactly 90° even if the drawing looks slightly off.

Naming Angles and Using a Protractor

An angle is named with three letters, with the vertex in the middle: ∠ABC means the angle at point B formed by the arms BA and BC. In single-angle diagrams you may also see just ∠B.

To measure with a protractor:

1. Place the centre point of the protractor exactly on the vertex.
2. Line up the zero baseline with one arm of the angle.
3. Read the scale that starts at 0° on that arm, and follow it round to the other arm.

The classic trap is reading the wrong scale. Defeat it by estimating first: decide whether the angle is acute or obtuse before you measure. If the angle is clearly smaller than a right angle and your protractor says 130°, you have read the outer scale — the true value is 50°.

Angles on a Straight Line and at a Point

Two facts solve a large share of ICAS angle questions:

• Angles on a straight line add to 180° (they are supplementary).
• Angles around a point add to 360°.

Worked example. Three angles meet at a point: 110°, 95° and x. Find x.

• Sum at a point = 360°
• x = 360 − 110 − 95 = 360 − 205 = 155°

Check by estimation: the missing sector is clearly the biggest, and 155° is bigger than the other two — sensible.

Worked example. A straight line is split into two angles, one of which is 67°. The other is 180 − 67 = 113°. A quick mental method: 180 − 67 = 180 − 70 + 3 = 113.

Vertically Opposite Angles

When two straight lines cross, they make two pairs of vertically opposite angles, and vertically opposite angles are equal. If one angle at the crossing is 54°, the angle opposite it is also 54°, and each of the other two is 180 − 54 = 126°. Notice the check: 54 + 126 + 54 + 126 = 360°, as it must be around a point.

Parallel and Perpendicular Lines

• Perpendicular lines meet at exactly 90°. On diagrams this is shown with the right-angle square.
• Parallel lines run in the same direction and never meet, however far they are extended. They are marked with matching arrowheads on the lines.

Paper D expects you to identify parallel and perpendicular lines in shapes and real objects (the opposite sides of a rectangle are parallel; the adjacent sides are perpendicular). From Paper E upwards you must use parallel lines to find angles.

Corresponding, Alternate and Co-Interior Angles

When a third line (a transversal) crosses a pair of parallel lines, eight angles are formed and they come in predictable patterns:

• Corresponding angles (an F-shape) are equal.
• Alternate angles (a Z-shape) are equal.
• Co-interior angles (a C- or U-shape) add to 180°.

Worked example. A transversal cuts two parallel lines. One angle is 118°. Find the angle co-interior to it, and the angle alternate to it.

• Co-interior: 180 − 118 = 62° (they are supplementary).
• Alternate: 118° (alternate angles are equal).

Trap: corresponding and alternate angles are equal, but co-interior angles are not — they only sum to 180°. Mixing these up is the most common error in this topic.

Worked Multi-Step Angle Chase

ICAS loves a diagram where a triangle sits between two parallel lines. Strategy: write every angle you find directly onto the diagram before looking at the options.

Problem. Two parallel horizontal lines are cut by a sloping transversal. The angle above the top line, to the left of the transversal, is 70°. A triangle is drawn between the lines using the transversal as one side; the triangle's angle on the bottom line (to the right of its base vertex) is 45°. Find the angle of the triangle at its top vertex.

1. By alternate angles, the angle inside the triangle at the top line, on the right of the transversal, equals 70°.
2. By alternate angles the other way, the triangle's angle at the bottom line, on the left of its vertex, equals 45°... which tells you the triangle's interior angle there is 45°.
3. Triangle angle sum: 180 − 70 − 45 = 65°.

Each step used one fact. That is the whole game: ICAS angle chases are chains of single facts, so if you are stuck, ask which of the four facts (straight line 180°, point 360°, vertically opposite equal, parallel-line patterns) you have not used yet.