Module 5 of 8Lesson 1 of 11 · Geometrical Terms, Angles & Polygons
Geometry · Lesson 1 of 10

Geometrical Terms, Angles & Polygons

🔵 CorePapers 1, 2 & 3
1 Learn it

Angle facts you must quote by name:

• Angles on a straight line add to 180°.
• Angles around a point add to 360°.
• Vertically opposite angles are equal.
• Angles in a triangle add to 180°; in a quadrilateral, 360°.

Parallel lines cut by a transversal give: alternate angles equal (Z-shape), corresponding angles equal (F-shape), co-interior angles add to 180° (C/U-shape).

Special triangles: isosceles — two equal sides and two equal base angles; equilateral — all 60°. Special quadrilaterals: square, rectangle, parallelogram (opposite angles equal), rhombus, trapezium, kite — know their side, angle and diagonal properties.

Polygons. For a polygon with n sides:

💡 Sum of interior angles \(= (n-2)\times180^\circ\).
Sum of exterior angles \(= 360^\circ\) — always, for any polygon.
Regular polygon: each exterior angle \(= \dfrac{360^\circ}{n}\), and interior + exterior \(= 180^\circ\).

Similar shapes have the same angles and proportional sides; congruent shapes are identical in shape and size. Know solid names too: cube, cuboid, prism, cylinder, pyramid, cone, sphere — and their nets.

2 Watch it work
EXAMPLE 1 · Parallel lines

→ → 62° x y

Two parallel lines and a transversal. The 62° lies between the parallel lines at the upper crossing, to the right of the transversal; x lies between the lines at the lower crossing, to the left of the transversal (the Z-shape); y is below the lower line, next to x on the same side of the transversal.

The marked 62° and x are alternate angles (Z-shape) → \(x = 62^\circ\).

x and y are angles on a straight line → \(y = 180-62 = 118^\circ\). Quote the reasons — they carry marks.

EXAMPLE 2 · Isosceles triangle

In triangle PQR, PQ = PR and angle P = 48°. Find angle Q.

Step 1 — base angles equal: \(Q = R\).

Step 2 — \(48+2Q = 180 \Rightarrow 2Q = 132 \Rightarrow Q = \mathbf{66^\circ}\).

EXAMPLE 3 · Regular polygon

Find the size of each interior angle of a regular hexagon.

Method (exterior first): exterior \(= \dfrac{360}{6} = 60^\circ\); interior \(= 180-60 = \mathbf{120^\circ}\).

Check (sum): \((6-2)\times180 = 720\); \(\dfrac{720}{6} = 120\) ✓

EXAMPLE 4 · Finding n

Each exterior angle of a regular polygon is 24°. How many sides has it?

\(n = \dfrac{360}{24} = \mathbf{15}\) sides.

3 Your turn
🚫 No calculator3 marks

Two angles on a straight line are \(3x\) and \(x+20\) (in degrees). Find x and the two angles.

Show solution

Step 1 — \(3x+x+20 = 180 \Rightarrow 4x = 160 \Rightarrow x = 40\).

Step 2 — angles: \(3(40) = \mathbf{120^\circ}\) and \(40+20 = \mathbf{60^\circ}\). Check: sum 180 ✓

🚫 No calculator3 marks

In a triangle, the angles are in the ratio 2 : 3 : 4. Find the largest angle.

Show solution

Step 1 — parts: 9; one part \(= \dfrac{180}{9} = 20^\circ\).

Step 2 — largest \(= 4\times20 = \mathbf{80^\circ}\).

Why: ratio sharing (Number lesson 5) applied to the 180° triangle sum — topics chain together.

🚫 No calculator4 marks

A regular polygon has interior angles of 156°. (a) Find its exterior angle. (b) How many sides does it have? (c) Find the sum of all its interior angles.

Show solution

(a) \(180-156 = \mathbf{24^\circ}\).

(b) \(n = \dfrac{360}{24} = \mathbf{15}\).

(c) \(156\times15 = \mathbf{2\,340^\circ}\) (or \((15-2)\times180 = 2340\) ✓).

4 Common mistakes
  • Alternate vs corresponding mixed up — sketch the Z or the F on the diagram.
  • Assuming a diagram is drawn to scale — “NOT TO SCALE” means calculate, never measure.
  • Using \((n-2)\times180\) for exterior angles — exterior angles always total 360°.
  • Writing answers without the reason when the question says “give a reason” — half the marks are the reasons.
5 In the exam
  • Angle-chasing with reasons is on every paper. Learn the exact phrases: “angles on a straight line”, “alternate angles”, “base angles of an isosceles triangle”.
  • Polygon formulas are quick marks — regular-polygon questions almost always route through the exterior angle.
6 Stretch & challenge
⭐ Challenge 1🚫 No calculator4 marks

Four angles of a pentagon are 100°, 110°, 120° and 130°. Find the fifth angle, x.

x 100° 110° 120° 130°

Not drawn to scale — calculate, don’t measure.
Show full solution

Step 1 — a pentagon has \(n = 5\) sides, so its interior angles total \((5-2)\times180 = 540^\circ\).

Step 2 — add the known angles: \(100+110+120+130 = 460^\circ\).

Step 3 — \(x = 540-460 = \mathbf{80^\circ}\).

Why this works: the \((n-2)\times180\) formula counts the triangles a polygon splits into. State the 540° with its formula — that line is a mark on its own.
⭐ Challenge 2🚫 No calculator5 marks

One angle of an isosceles triangle is 40°. Find ALL the possible sets of angles of the triangle, explaining why there is more than one answer.

Show full solution

Step 1 — an isosceles triangle has TWO equal angles. The 40° could be the odd one out (apex) or one of the equal pair — two cases to check.

Case 1 — 40° is the apex: the equal base angles share \(180-40 = 140^\circ\), so they are 70° each. Angles: 40°, 70°, 70°.

Case 2 — 40° is a base angle: the other base angle is also 40°, and the apex is \(180-80 = 100^\circ\). Angles: 40°, 40°, 100°.

Step 2 — both cases give valid triangles (all angles positive), so there are TWO possible answers.

Why this works: “one angle is 40°” does not say WHICH angle. Case-by-case analysis — and saying clearly why each case is valid — is exactly what the harder Paper 3 reasoning questions reward.
7 Exercise — in your book

📓 Write your answers in your exercise book. No peeking — reveal the answers only when finished, and mark your own work. Levels: 🟢 warm-up · 🔵 exam level · 🟠 stretch.

  1. 🟢 Two angles on a straight line: one is 65°. Find the other.
  2. 🟢 Angles round a point: 140°, 90° and x. Find x.
  3. 🟢 Write down the angle vertically opposite to 74°.
  4. 🟢 A triangle has angles 90° and 37°. Find the third.
  5. 🔵 A quadrilateral has angles 100°, 80° and 95°. Find the fourth.
  6. 🔵 Find the sum of the interior angles of a hexagon.
  7. 🔵 Find each exterior angle of a regular 10-sided polygon.
  8. 🟠 Two parallel lines are cut by a transversal. One co-interior angle is 118°. Find the other.
Show answers — mark your work
  1. 115°
  2. 130°
  3. 74° (vertically opposite angles are equal)
  4. 53°
  5. \(360-275 = 85^\circ\)
  6. \((6-2)\times180 = 720^\circ\)
  7. \(\dfrac{360}{10} = 36^\circ\)
  8. \(180-118 = 62^\circ\) (co-interior angles sum to 180°)
Marking guide: 7–8 correct — excellent, move on. 5–6 — redo the ones you missed. Below 5 — work through the lesson again, then retry this exercise. Every question you fix yourself is a mark gained in the real exam.
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