What are giving reasons?

Every step in a geometry answer needs a reason - the name of the rule used. For example: "x = 110^ (corresponding angles)." Marks are awarded for the reason as well as the value, so never leave it out.

SEAB Original-style practice: Three angles meet on a straight line. Two of them are 65^ and 80^. Find the third angle, giving a reason.

Angles on a straight line add up to 180^. Third angle = 180^ - 65^ - 80^ = 35^. What markers reward: stating the rule "angles on a straight line sum to 180^", the correct subtraction, and the answer with its reason. In geometry, the reason is part of the mark, so always name the rule you used.

SingaporeMathsSyllabus dot point

What angle rules let us find unknown angles on lines, at points, and between parallel lines?

Use angle properties on a straight line, at a point, and between parallel lines to find unknown angles

A focused answer to the N(A)-Level Mathematics outcome on angles. Angles on a line and at a point, vertically opposite angles, and corresponding, alternate and co-interior angles on parallel lines.

Generated by Claude Opus 4.88 min answerUpdated 2026-06-06

Reviewed by: AI editorial process; not yet individually human-reviewed

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What this dot point is asking
The answer
Examples in context
Try this

What this dot point is asking

SEAB wants you to find unknown angles using the basic angle rules: angles on a straight line, angles at a point, vertically opposite angles, and the angles formed when a line crosses parallel lines. In geometry you must always give a reason, so learning the name of each rule matters as much as the arithmetic.

The answer

Angles on a line and at a point

Two essential facts:

Angles on a straight line add up to $180^\circ$ .
Angles around a point add up to $360^\circ$ .

So if angles on a line are $x$ and $130^\circ$ , then $x = 180^\circ - 130^\circ = 50^\circ$ .

Vertically opposite angles

When two straight lines cross, the angles directly opposite each other are equal. These are called vertically opposite angles. If one angle at a crossing is $70^\circ$ , the angle opposite it is also $70^\circ$ .

Parallel lines and a transversal

When a straight line (a transversal) crosses two parallel lines, three named relationships appear:

Corresponding angles (in matching positions, like an "F" shape) are equal.
Alternate angles (on opposite sides of the transversal, between the parallel lines, like a "Z" shape) are equal.
Co-interior angles (on the same side, between the parallel lines, like a "C" shape) add up to $180^\circ$ .

Giving reasons

Every step in a geometry answer needs a reason - the name of the rule used. For example: " $x = 110^\circ$ (corresponding angles)." Marks are awarded for the reason as well as the value, so never leave it out.

Building up several steps

Harder questions chain the rules together: find one angle with vertically opposite angles, use it as an alternate angle, then finish on a straight line. Work one angle at a time, writing the reason for each.

Worked example

A transversal crosses two parallel lines. At the upper crossing, one angle is $75^\circ$ . Find the alternate angle at the lower crossing, and then the angle on a straight line next to it, giving reasons.

Step 1: Find the alternate angle

Alternate angles between parallel lines are equal, so the alternate angle at the lower crossing is:

75^\circ \quad \text{(alternate angles)}.

Step 2: Use angles on a straight line

The angle next to it on the same straight line must add with it to $180^\circ$ :

180^\circ - 75^\circ = 105^\circ \quad \text{(angles on a straight line)}.

Step 3: State the results

The alternate angle is $75^\circ$ and the adjacent angle on the line is $105^\circ$ , each justified by its rule.

Examples in context

Example 1. A zig-zag path. A path bends between two parallel walls, forming a "Z" shape. The two angles the path makes with the walls are alternate angles, so they are equal. Spotting the Z shape immediately tells you the angles match without measuring.

Example 2. A ladder against a wall. A ladder leaning across a horizontal floor and a horizontal ceiling (parallel lines) makes equal angles with both, by alternate angles. Many real arrangements of parallel surfaces reduce to these standard angle pairs.

Try this

Cue. Angles on a straight line are $x$ and $115^\circ$ . Then $x = 180^\circ - 115^\circ = 65^\circ$ .
Cue. Two lines cross; one angle is $48^\circ$ . The vertically opposite angle is also $48^\circ$ .
Cue. A co-interior angle pairs with $130^\circ$ between parallel lines. The other is $180^\circ - 130^\circ = 50^\circ$ .

Exam-style practice questions

Practice questions written in the style of SEAB exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.

Original3 marksThree angles meet on a straight line. Two of them are

65^\circ

and

80^\circ

. Find the third angle, giving a reason.

Show worked answer →

Angles on a straight line add up to $180^\circ$ .

Third angle $= 180^\circ - 65^\circ - 80^\circ = 35^\circ$ .

What markers reward: stating the rule "angles on a straight line sum to $180^\circ$ ", the correct subtraction, and the answer with its reason. In geometry, the reason is part of the mark, so always name the rule you used.

Original3 marksTwo parallel lines are cut by a straight line. One of the angles formed is

112^\circ

. State, with reasons, the size of (a) the corresponding angle and (b) the co-interior angle on the same side.