Overview
What you’ll learn
- Use angle properties of parallel lines cut by a transversal with named reasons.
- Prove triangles congruent by SSS, SAS, ASA/AAS and RHS, and use CPCTC.
- Prove triangles similar by AA (with side ratios) and use linear and area scale factors.
- State and apply the midpoint theorem.
- Use the properties of parallelograms, rhombuses, rectangles, squares, trapeziums and kites.
- Apply the circle theorems: angle at centre, same segment, semicircle, cyclic quadrilateral, tangent ⊥ radius, tangents from an external point, and the alternate segment theorem.
- Write structured geometric proofs with a reason for every step.
In 30 seconds
- 1Parallel lines + transversal: alternate angles equal, corresponding angles equal, co-interior (allied) angles sum to 180°.
- 2Congruence tests: SSS, SAS, ASA (or AAS), RHS. Angle–angle–angle alone is not a congruence test.
- 3Corresponding sides of congruent triangles are equal — quote it as CPCTC (corresponding parts of congruent triangles).
- 4Similar triangles: equal angles with proportional sides; linear ratio k gives area ratio k².
- 5Midpoint theorem: the segment joining the midpoints of two sides of a triangle is parallel to the third side and half its length.
- 6Circle: the angle at the centre is twice the angle at the circumference on the same arc; angles in the same segment are equal; the angle in a semicircle is 90°.
- 7Cyclic quadrilateral: opposite angles sum to 180°.
- 8A tangent meets the radius at 90°; tangents from an external point are equal; the tangent–chord (alternate segment) angle equals the angle in the alternate segment.
- 9Every proof line needs a reason: a theorem name, a given fact, or CPCTC.
Notes · 1 of 5
Parallel lines and transversals
Three named angle pairs do all the work on a parallel-lines diagram.
| Pair | Fact | Example (63° given) |
|---|---|---|
| Alternate angles (Z-shape) | equal | 63° |
| Corresponding angles (F-shape) | equal | 63° |
| Co-interior angles (C/U-shape) | sum to 180° | 117° |
| Vertically opposite angles | equal | 63° |
| Angles on a straight line | sum to 180° | 117° |
Two bisectors also appear constantly. The angle bisector splits an angle into two equal halves; the perpendicular bisector of a segment crosses it at 90° through its midpoint, and every point on it is equidistant from the segment’s endpoints. In an isosceles triangle, the bisector of the vertex angle coincides with the perpendicular bisector of the base — a fact worth proving once with congruent triangles and then using freely.
Notes · 2 of 5
Congruent triangles, similar triangles and the midpoint theorem
Four congruence tests, one similarity test in daily use, and the midpoint shortcut.
| Test | You need |
|---|---|
| SSS | three pairs of equal sides |
| SAS | two pairs of equal sides with the equal included angle |
| ASA / AAS | two pairs of equal angles with one equal corresponding side |
| RHS | right angle, hypotenuse and one other equal side |
AAA is not a congruence test — equal angles only fix the shape, so infinitely many sizes fit. Once congruence is established, use CPCTC to transfer any corresponding side or angle. For similarity, the daily driver is AA (two equal angles), checked against side ratios when lengths are given: matching sides must be in one constant ratio — the scale factor. If the linear scale factor is k, areas scale by k².
The midpoint theorem is the one similarity result hiding inside every triangle: joining the midpoints of two sides gives a segment parallel to the third side and exactly half its length. It pairs naturally with questions about medians and trapeziums built from triangles.
Notes · 3 of 5
Special quadrilaterals
The property list each named shape carries into a proof.
| Shape | Defining properties |
|---|---|
| Parallelogram | opposite sides parallel and equal; opposite angles equal; diagonals bisect each other |
| Rhombus | all the parallelogram facts, plus all sides equal and diagonals perpendicular, each bisecting the vertex angles |
| Rectangle | parallelogram with four right angles; diagonals equal |
| Square | rhombus + rectangle: everything above |
| Trapezium | exactly one pair of parallel sides |
| Kite | two pairs of adjacent equal sides; one diagonal is the perpendicular bisector of the other |
In proofs these facts are stepping stones, not conclusions: "ABCD is a rhombus, so AB = BC" is a legal line; "the diagonals are perpendicular, so ABCD is a rhombus" needs the rest of the definition checked first. Midpoint and bisector language ("M is the midpoint of BC means BM = MC") is quotable straight from the given information.
Notes · 4 of 5
Circle theorems
Seven facts that unlock every circle diagram — including the alternate segment theorem.
| Theorem | Statement |
|---|---|
| Angle at centre | The angle at the centre is twice the angle at the circumference standing on the same arc. |
| Same segment | Angles at the circumference standing on the same arc (same segment) are equal. |
| Semicircle | The angle in a semicircle is 90° (the diameter subtends a right angle). |
| Cyclic quadrilateral | Opposite angles of a cyclic quadrilateral sum to 180°. |
| Tangent ⊥ radius | A tangent meets the radius at the point of contact at 90°. |
| Equal tangents | The two tangents from an external point are equal in length. |
| Alternate segment | The angle between a tangent and a chord equals the angle in the alternate segment. |
Reading the diagram is half the battle: before quoting a theorem, identify which arc the angle stands on. The alternate segment theorem needs care — the angle between the tangent and the chord matches the inscribed angle on the far side of the chord, never the angle in the segment the tangent touches. When a chord runs through the centre, remember it is a diameter, so the semicircle theorem is available.
Notes · 5 of 5
Structuring a geometric proof
A proof is a chain: every claim needs a reason, and no step may assume what it is proving.
A full-reason proof reads as a two-column habit written in prose: claim, then reason. Legal reasons are theorems by name, facts given in the question, and CPCTC after a congruence. Markers award the reason marks as generously as the answer marks — an unreasoned chain of equalities is treated as an unproven guess.
Three disciplines keep proofs honest. First, never use the result as a reason: if asked to prove DE ∥ BC, the midpoint theorem may not appear in your own argument before you have earned it. Second, mark up the diagram with everything you deduce — equal sides, equal angles, radii — so the next theorem is visible. Third, when stuck, look for congruent triangles: most quadrilateral and circle proofs collapse once one pair of congruent triangles is on the table.
Revise
Key terms
The words examiners expect you to use precisely.
- Transversal
- A line crossing two or more lines, creating alternate, corresponding and co-interior angle pairs.
- Congruent
- Identical in shape and size; provable by SSS, SAS, ASA/AAS or RHS.
- Similar
- Same shape with all angles equal and sides in a constant ratio (the scale factor).
- Cyclic quadrilateral
- A quadrilateral whose four vertices all lie on one circle; its opposite angles sum to 180°.
- Subtended
- Said of an angle "standing on" an arc: the arc between the two arms of the angle.
- Alternate segment
- The segment of the circle on the far side of a chord from the tangent–chord angle; its inscribed angle equals that angle.
Show 2 more key termsShow fewer key terms
- Midpoint theorem
- The segment joining midpoints of two sides of a triangle is parallel to the third side and half as long.
- CPCTC
- Corresponding Parts of Congruent Triangles are equal — the reason quoted after establishing congruence.
Revise
Common mistakes
Where students most often lose marks in this chapter — and what to do instead.
Mistake: Quoting alternate angles when the pair is corresponding (or vice versa).
The two facts look alike on awkward diagrams: alternate angles sit on opposite sides of the transversal, corresponding angles on the same side.
Instead: Trace the Z for alternate and the F for corresponding before naming the reason.
Mistake: Treating co-interior angles as equal.
Allied angles on parallel lines are supplementary, not equal.
Instead: Co-interior angles sum to 180°: with 63° given, the partner is 117°.
Show 8 more mistakesShow fewer mistakes
Mistake: Halving an angle at the centre that stands on a different arc from the circumference angle.
The factor of 2 only holds for angles on the same arc; mixing minor and major arcs breaks it.
Instead: Identify the arc each angle subtends; halve only when the arcs match (and use the reflex centre angle when needed).
Mistake: Subtracting adjacent angles in a cyclic quadrilateral.
Opposite angles of a cyclic quadrilateral are supplementary; adjacent ones have no fixed sum.
Instead: angle A + angle C = 180° and angle B + angle D = 180°, with A opposite C.
Mistake: Using AAA as a congruence test.
Equal angles fix shape but not size — scaled copies share all three angles.
Instead: Congruence needs SSS, SAS, ASA/AAS or RHS; AAA proves similarity only.
Mistake: Matching similar-triangle sides by length order instead of corresponding vertices.
The ratio must pair the same positions in both triangles; sorting all six lengths gives a meaningless ratio.
Instead: Write the similarity statement (e.g. triangle ADE ∼ triangle ABC) and pair sides in that order.
Mistake: Using the linear scale factor for areas.
Area scales with the square of the ratio: a factor of 3 multiplies areas by 9.
Instead: Area ratio = k² for linear scale factor k.
Mistake: Matching the alternate-segment angle with the angle in the near segment.
The theorem pairs the tangent–chord angle with the inscribed angle on the far side of the chord.
Instead: Pick the inscribed angle standing on the same chord from the alternate (opposite) segment.
Mistake: Writing a chain of equalities with no reasons.
In proof questions the named reason carries marks of its own; an unreasoned chain is not a proof.
Instead: Attach a theorem name, a given fact or CPCTC to every line.
Mistake: Using the midpoint theorem inside a proof whose aim is that same parallelism.
Assuming the conclusion as a reason is circular reasoning and scores zero for the proof.
Instead: Build the result from congruent triangles or angle facts, then quote the theorem only once it is available.
Check yourself
Quick quiz
18 questions with instant explanations. About 5 minutes.
Question 1 of 3
- Question 1: current question
- Question 2: not answered
- Question 3: not answered
Use parallel-line angle facts
Parallel lines are cut by a transversal. One angle is 65°. Its alternate angle is
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Practise
Topical questions
Exam-style practice in three tiers, from recall to data questions.
- 4Foundation
- 4Standard
- 2Challenging
10 questions worth 58 marks, each with a model answer, mark scheme and examiner tips.
Practise topical questionsExam practice
Practice paper
A 30-mark paper set out like a school paper, with a separate mark scheme. Give yourself 40 min.
Question paper
Chapter 2 Practice Paper
30 marks · 40 min · 5 pages
5 multiple-choice questions and structured questions with answer lines. Chapter 2 (G3 Proofs in plane geometry).
Mark scheme
PROChapter 2 Practice Paper — mark scheme
Answers · Marking points · 3 pages
Model answers with a mark for each creditworthy point — for marking at home or in class.
Where this chapter is tested
Each school sets its own scope — check your assessment outline.
Questions
Frequently asked questions
What does Sec 4 A-Math Chapter 2 cover?
It covers syllabus sub-topic G3 of O-Level 4049 / SEC G3 K341: parallel-line angle facts, congruent and similar triangles, the midpoint theorem, special quadrilaterals, all the circle theorems including the alternate segment theorem, and full-reason geometric proofs.
How much working does a geometry proof need?
Every claim needs a reason: a theorem name, a fact from the given information, or CPCTC after a congruence. A bare chain of equal angles usually earns the answer marks but loses the reasoning marks.
Which similarity facts are worth memorising?
AA (two pairs of equal angles) is the workhorse; equal ratios of corresponding sides about a common angle is the second; and remember areas scale with the square of the linear scale factor.
How do I stop confusing the circle theorems?
Always ask which arc the angle stands on before quoting anything: centre-versus-circumference pairs must stand on the same arc, same-segment angles must share that arc, and the alternate segment angle sits on the far side of the chord from the tangent.
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Chapter 3: Linear Law