Euclidean Geometry Riders: Proving It, With Reasons (Grade 10)

A rider is a geometry question that gives you a diagram and asks you to work something out — or prove something — by applying theorems one after another. In a paper it looks like “Determine, with reasons, the size of x” or “Prove that ABCD is a parallelogram”.

Riders carry more marks than any other part of Paper 2, and they are where most marks are lost — not because learners cannot see the answer, but because they cannot justify it. This lesson is about the justification.

1The one rule that decides your mark

Every statement you write needs a reason. A correct angle with no reason earns the statement mark and loses the reason mark — usually half the question.

Write it as a two-part line, every time:

• AD̂E = 70°   (corresponding ∠s; DE ∥ BC)  — correct, both marks.
• AD̂E = 70°  — correct answer, half the marks.
• AD̂E = 70°   (they look the same)  — no reason mark. Measuring or eyeballing is never a reason.

2The reasons you may quote in Grade 10

You may only use a result that has been proved. These are the ones available to you, with the exact wording to write in brackets:

• Lines: ∠s on a str line (add to 180°) · ∠s round a pt (360°) · vert opp ∠s
• Parallel lines: corresp ∠s; AB ∥ CD · alt ∠s; AB ∥ CD · co-int ∠s; AB ∥ CD (add to 180°)
• Triangles: ∠s of a △ (add to 180°) · ext ∠ of a △ · ∠s opp equal sides · sides opp equal ∠s · Pythagoras
• Congruence: SSS · SAS · AAS · RHS — then △ABC ≡ △PQR, and matching parts follow by ≡△s
• Similarity: AAA (equiangular) · sides in proportion — then △ABC ||| △PQR
• Quadrilaterals: opp sides of a parm · opp ∠s of a parm · diags of a parm · the five tests for proving a parallelogram
• Midpoint theorem: the line joining the midpoints of two sides is parallel to the third side and half its length — and its converse

3How to attack a rider

• Mark the diagram. Write every given size onto it. Put arrows on parallel lines and ticks on equal sides. A rider you have marked up is half solved.
• Find what you are asked for and look at what touches it. Which angle sits on a straight line with it? Which triangle is it inside?
• Work from the given, not from the answer. Start at a size you know and take one justified step at a time.
• Never assume what the diagram looks like. Lines that appear equal, parallel or perpendicular are not, unless you are told or you prove it.
• Use your earlier answers. A rider is built in parts on purpose — 4.1 is nearly always the way into 4.2.

4Rider 1 — angles with reasons

Notice how short the working is once the reasons are in place. That is what a full-mark rider looks like — not more writing, the right writing.

5Rider 2 — using congruence

6Rider 3 — proving a quadrilateral is a parallelogram

There are five accepted ways. You need one of them, proved — not asserted:

• Both pairs of opposite sides are parallel.
• Both pairs of opposite sides are equal.
• Both pairs of opposite angles are equal.
• One pair of opposite sides is both equal AND parallel.
• The diagonals bisect each other.

7Rider 4 — the midpoint theorem

The midpoint theorem gives you two conclusions from one condition, and a rider will usually want both.

The converse is examined too, and it works the other way: if a line through the midpoint of one side is parallel to a second side, then it bisects the third side. Use it when you are given a midpoint and a parallel line, and asked to prove something is a midpoint.

8What a full-mark answer looks like

• One statement per line, with its reason in brackets on the same line.
• Every parallel-line reason names the parallel lines.
• Congruent triangles written with matching vertices in order.
• The final line answers the question that was actually asked, in the units asked for.
• Nothing assumed from the look of the diagram.