Secondary 4 · Mathematics · Space
Geometric reasoning and proof
Write logical arguments using congruence, similarity and angle properties to prove geometric results, including simple circle properties.
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These lessons were written around Australian Curriculum strands and matched to Singapore levels by age (Primary 1 is the first year of primary school). They are not aligned to MOE syllabuses or the PSLE, so check them against the syllabus your child is working towards.
Learning goals
Learners will
- Use angle properties of parallel lines, triangles and isosceles triangles to justify unknown angles with reasons.
- Prove that two triangles are congruent or similar using the standard tests, and use this to deduce further results.
- Apply simple circle properties, including the angle in a semicircle and the angle at the centre, in logical arguments.
Success looks like
- I can give a correct reason for every step when finding an unknown angle.
- I can write a congruence proof using SSS, SAS, AAS or RHS, setting out each statement and reason.
- I can use the angle in a semicircle and the angle at the centre to find unknown angles in a circle.
The big idea
In geometry, it is not enough to measure an angle and say 'it looks like 90°'. A proof shows that a result must be true, using facts we already know. Every statement needs a reason.
Key angle facts you can use as reasons:
- Angles on a straight line add to 180°; angles at a point add to 360°; vertically opposite angles are equal.
- When parallel lines are cut by a transversal: alternate angles are equal, corresponding angles are equal, and co-interior angles add to 180°.
- Angles in a triangle add to 180°; the base angles of an isosceles triangle are equal; the exterior angle of a triangle equals the sum of the two opposite interior angles.
Congruence tests prove two triangles are identical in shape and size: SSS (three pairs of equal sides), SAS (two sides and the included angle), AAS (two angles and a corresponding side) and RHS (right angle, hypotenuse and one other side). Note that AAA is not a congruence test – it only proves triangles are similar.
Similarity tests: AA (two pairs of equal angles, so the third pair is equal too), SSS with all sides in the same ratio, SAS with two sides in the same ratio and the included angle equal, and RHS with hypotenuse and one other side in the same ratio.
Once triangles are proved congruent, all their corresponding parts are equal. This is how we prove that a line bisects another, or that two lengths are equal.
Simple circle properties: radii of a circle are all equal (so a triangle formed by two radii is isosceles); the angle in a semicircle is 90° (an angle at the circumference standing on a diameter); the angle at the centre is twice the angle at the circumference standing on the same arc; and angles at the circumference standing on the same arc are equal.
Set out a proof clearly: a diagram, what is given, what you need to prove, then numbered statements each with a reason, ending with a conclusion.
Worked examples
A congruence proof
Segments AB and CD cross at M, and M is the midpoint of both segments. Prove that triangle AMC is congruent to triangle BMD, and hence that AC = BD.
- AM = BM (M is the midpoint of AB).
- CM = DM (M is the midpoint of CD).
- ∠AMC = ∠BMD (vertically opposite angles are equal).
- So triangle AMC ≅ triangle BMD (SAS).
- Therefore AC = BD (corresponding sides of congruent triangles are equal).
Answer: Triangle AMC ≅ triangle BMD by SAS, so AC = BD.
Angle reasoning with parallel lines
Lines AB and CD are parallel, and a transversal crosses them. The co-interior angles between the parallel lines are (2x + 10)° and (3x + 20)°. Find x.
- Co-interior angles between parallel lines add to 180°.
- So 2x + 10 + 3x + 20 = 180.
- Simplify: 5x + 30 = 180, so 5x = 150 and x = 30.
- Check: 2(30) + 10 = 70° and 3(30) + 20 = 110°, and 70° + 110° = 180°.
Answer: x = 30 (reason: co-interior angles on parallel lines are supplementary)
Using the angle in a semicircle
AB is a diameter of a circle and C is a point on the circumference. If ∠CAB = 35°, find ∠ABC.
- ∠ACB = 90° (angle in a semicircle).
- Angles in triangle ABC add to 180°: ∠ABC = 180° − 90° − 35°.
- ∠ABC = 55°.
Answer: ∠ABC = 55°
Practice check
Have a go, then check your answers. Each answer comes with an explanation.
Answer guide for parents
Which of these is NOT a valid test for congruent triangles?
AAA — Triangles with three equal angles have the same shape but can be different sizes, so AAA only proves similarity.
Two angles of a triangle are 48° and 67°. What is the size of the third angle, in degrees?
65° — Angles in a triangle add to 180°: 180 − 48 − 67 = 65.
When a transversal crosses two parallel lines, co-interior angles are…
Supplementary (they add to 180°) — Co-interior angles between parallel lines add to 180°.
In a circle, an arc subtends an angle of 110° at the centre. What angle does the same arc subtend at the circumference, in degrees?
55° — The angle at the centre is twice the angle at the circumference, so the angle at the circumference is 110 ÷ 2 = 55°.
AB is a diameter of a circle and C is any other point on the circumference. What is the size of ∠ACB?
90° — The angle in a semicircle is a right angle.
An isosceles triangle has an apex angle (between its two equal sides) of 40°. What is the size of each base angle, in degrees?
70° — The base angles are equal and the angles add to 180°: (180 − 40) ÷ 2 = 70.
Two straight lines cross at Q, forming ∠PQR and ∠SQT opposite each other. What is the correct reason for ∠PQR = ∠SQT?
Vertically opposite angles are equal — Angles directly opposite each other where two lines cross are vertically opposite and equal.
Two triangles have all three pairs of corresponding angles equal. What can you conclude?
They are similar — Equal angles guarantee the same shape (similar), but not necessarily the same size.
Watch out for
- Using AAA to prove congruence. Equal angles only prove similarity – a small and a large equilateral triangle have equal angles but different sizes.
- Writing the correct angle without a reason, or with a vague reason like 'because of the lines'. Name the property exactly, such as 'alternate angles, AB ∥ CD'.
- Using SSA (two sides and a non-included angle) as a congruence test. It can produce two different triangles, so it is not valid unless the angle is a right angle (RHS).
Tips for parents
- Ask 'How do you know?' after every step. A reason is required each time, just like in a proof.
- Cut out two triangles with the same three angles but different sizes to show why AAA proves similarity, not congruence.
- Encourage a large, clear, labelled diagram before any proof – most errors come from a cramped sketch.
Go further
- Prove the angle in a semicircle is 90°: join the centre O to point C, notice the two isosceles triangles formed by radii, and use angle sums.
- Use a pair of compasses to draw a circle and a diameter, mark several points on the circumference and measure the angles in the semicircle. What do you notice?
- Prove that the diagonals of a parallelogram bisect each other using congruent triangles.
About 60 minutes
You'll need
- Paper, pencil and ruler
- A protractor and a pair of compasses
- Scissors (for testing congruence by cutting out triangles)
Key words
- Proof
- A logical argument in which each statement follows from given facts or known results, with a reason given for every step.
- Congruent
- Exactly the same shape and size; corresponding sides and angles are equal. Symbol: ≅.
- Similar
- The same shape but not necessarily the same size; corresponding angles are equal and corresponding sides are in the same ratio.
- Co-interior angles
- Angles between two parallel lines on the same side of a transversal; they add to 180°.
- Subtend
- An arc or chord subtends an angle when lines from its endpoints meet at a point to form that angle.
- Circumference
- The boundary (edge) of a circle.
Still tricky?
For parents
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