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Grade 10 Euclidean Geometry — How to Identify the Question Type

Grade 10 Euclidean geometry questions test angle relationships in triangles, on straight lines, and with parallel lines. Every answer must be supported by a reason in brackets — the reason is part of the mark.

Type 1: Angle relationships on a straight line and at a point

Trigger words: "Find the value of x", "Calculate the size of"

Trigger structure: Angles are formed by lines meeting at a point or on a straight line. The diagram shows supplementary, co-interior, or vertically opposite angles.

Do not confuse with: Triangle angle sum — only use 180° for triangle interior angles, not for angles on a straight line.

Method (no numbers — just the steps)

  1. Identify the angle relationship from the diagram
  2. Write the equation (e.g. x + 40° = 180°)
  3. Solve for x
  4. State the reason in brackets: (angles on a straight line), (vertically opposite angles), etc.
Grade 10Paper 2Level 1 — Knowledge
3.1 Two lines intersect. One angle is 65°. Find the value of the vertically opposite angle.

Source: DBE Mathematics Paper 2, November 2021, Question 3.1

Common mistake

Stating the correct answer but omitting the reason — in Euclidean geometry, the reason earns a separate mark.

See the progression — same type, increasing difficulty

Easy
Angles on a straight line: one angle is 3x and the other is 42°. Find x.

Practice question — not sourced from a past paper.

Medium
Three angles meet at a point: 120°, 2x, and x + 15°. Find x.

Practice question — not sourced from a past paper.

Hard
AB ∥ CD. A transversal cuts AB at P and CD at Q, forming angle APQ = 3x + 10° and angle PQD = 5x − 20°. Find x and name the angle pair.

Practice question — not sourced from a past paper.

Type 2: Triangle properties — angle sum and exterior angle

Trigger words: "Find x", "Calculate the size of", "Prove that"

Trigger structure: A triangle is shown. Unknown angles are labelled with x or letters.

Do not confuse with: Properties of parallel lines — do not apply the triangle angle sum when angles are formed by a transversal, not inside a triangle.

Method (no numbers — just the steps)

  1. Interior angle sum: the three angles inside any triangle add to 180°
  2. Exterior angle theorem: an exterior angle equals the sum of the two non-adjacent interior angles
  3. Set up the equation and solve for the unknown
  4. State the reason in brackets
Grade 10Paper 2Level 1 — Knowledge
In triangle ABC, Â = 55° and B̂ = 70°. Calculate Ĉ.

Practice question — not sourced from a past paper.

See the progression — same type, increasing difficulty

Easy
Triangle PQR has angles P̂ = 90° and Q̂ = 38°. Find R̂.

Practice question — not sourced from a past paper.

Medium
An exterior angle of a triangle is 110°. One of the non-adjacent interior angles is 45°. Find the other non-adjacent interior angle.

Practice question — not sourced from a past paper.

Hard
In triangle ABC, Â = 2x + 10°, B̂ = x + 30°, and Ĉ = 3x − 20°. Find the value of x and hence determine all three angles.

Practice question — not sourced from a past paper.

Type 3: Congruency and similarity

Trigger words: "Prove that triangles are congruent", "Prove that triangles are similar", "Find the unknown side"

Trigger structure: Two triangles are described or shown. The question asks you to prove a relationship or use it to find a measurement.

Do not confuse with: Proving triangles are equal area — that is a different relationship than congruence or similarity.

Method (no numbers — just the steps)

  1. Congruence: show three matching parts using SSS, SAS, AAS, or RHS
  2. Similarity: show two pairs of equal angles (AA is sufficient)
  3. For unknown sides in similar triangles: set up a proportion and solve
Grade 10Paper 2Level 2 — Routine Procedures
Triangle ABC and Triangle DEF have AB = DE, BC = EF, and AC = DF. State the criterion and conclude whether they are congruent.

Practice question — not sourced from a past paper.

Common mistake

Using SSA (two sides and a non-included angle) to prove congruence — this is NOT a valid congruence criterion.

See the progression — same type, increasing difficulty

Easy
△PQR and △XYZ have P̂ = X̂ = 50° and Q̂ = Ŷ = 70°. Are the triangles similar? State the reason.

Practice question — not sourced from a past paper.

Medium
△ABC ||| △DEF. AB = 6, DE = 9, and BC = 8. Calculate EF.

Practice question — not sourced from a past paper.

Hard
In the diagram, triangles ABD and CBD share the side BD. AD = CD, AB = CB. Prove that triangle ABD ≅ triangle CBD.

Practice question — not sourced from a past paper.

Words like determine and hence appear throughout this topic — see the instruction word glossary for full definitions.