A geometry diagram is a sketch, not a measurement. You can only use facts the question states or the marks show. These are stated lengths and angles, equal-side ticks, parallel arrows and right-angle squares. Anything else needs proof, however obvious it looks.
This is the final lesson in angles and geometric reasoning. It protects every earlier lesson, because each of them starts from a fact you are allowed to use.
What counts as a stated fact?
| Shown in the diagram | What it tells you |
|---|---|
| Same number of tick marks on two sides | The sides are equal |
| Paired arrows on two lines | The lines are parallel |
| Small square at a corner | The angle is 90° |
| A number or algebraic label on an angle | The angle has that size |
| Text such as “AB is a diameter” | AB passes through the centre |
Things that only look true are not facts: lines that seem parallel, angles that seem right, sides that seem equal or a point that seems to be a midpoint.
How do you keep assumptions out of your working?
- List the given facts before you start, from the text and the marks.
- Underline what you need to find.
- Use a fact only if it is on your list or follows from it by a rule.
- If you need a fact that is missing, see whether another rule produces it. If not, the question may be asking for something that cannot be found.
- Never measure with a ruler or protractor when the diagram is not drawn accurately.
Worked example
Lines AB and CD are crossed by a line meeting them at X (on AB) and Y (on CD), with X above Y.
B and D are on the right. ∠BXY = 70°. The diagram is not drawn accurately.
(a) The lines look parallel, but no arrows are marked and the question does not say so. ∠XYD cannot be found, because without parallel lines there is no rule linking the two angles.
(b) Now the question adds “AB is parallel to CD”, shown by arrows. ∠BXY and ∠XYD lie on the same side of the line, between the parallel lines, so they are co-interior angles. They add to 180°, so ∠XYD = 180° − 70° = 110°.
Check: 70° + 110° = 180°, and ∠XYD is obtuse, which matches a wide angle on the same side as the acute 70°.
The only change between (a) and (b) is one stated fact. The diagram looked the same both times.
The mistake to watch for
A common slip is to use something because it looks right.
Mistaken answer: In part (a), ∠XYD = 110°, because the lines are parallel.
The student trusted the drawing. No parallel marks were given, so the reason is not supported.
The correction is to ask “where does the question tell me this?” If the answer is “it looks like it”, the step cannot be used. Two similar traps are a corner that looks like a right angle but has no square marked, and a point that looks like a midpoint but is not described as one.
Check yourself
Try these, then open each answer.
1. A diagram shows a small square at one corner, two sides with one tick each and a line that looks vertical. Which of these three are facts you can use?
Show answer
The small square (a right angle of 90°) and the tick marks (equal sides) are facts. The line that only looks vertical is not a fact.
2. In triangle ABC, AB and AC each have one tick mark, and ∠BAC = 50°. Find ∠ABC, with a reason.
Show answer
The ticks show AB = AC, so the base angles are equal. ∠ABC = (180° − 50°) ÷ 2 = 130° ÷ 2 = 65°, because base angles of an isosceles triangle are equal and angles in a triangle add to 180°.
3. A four-sided figure looks like a rectangle. The question gives only ∠ABC = 90° and ∠BCD = 90°. Can you say ∠CDA = 90°?
Show answer
No. Only two right angles are given, so the figure could be a right-angled trapezium. You can say AB is parallel to CD, because the co-interior angles ∠ABC and ∠BCD add to 180°, but nothing states the fourth angle.
Where this leads next
After this lesson, try the whole module in the mixed practice set, where each question needs a stated fact and a reason. Reasoning from given facts also supports the next module, similarity, congruence and scale.
Some students lose marks because they trust the drawing rather than the question. Our teachers build the habit of listing given facts in online one-to-one Mathematics tuition.