📐 Angles

Grade 8 · Cambridge Lower Secondary Stage 8

What you'll learn

  • Angles on a straight line, around a point, and vertically opposite
  • Alternate, co-interior, and corresponding angles in parallel lines
  • Angles in triangles and quadrilaterals
  • Exterior angles of polygons
  • Bearings and directions

📖 Learn

1. Basic Angle Facts

Straight lineAngles sum to 180°
Around a pointAngles sum to 360°
Vertically oppositeEqual angles
Right angleExactly 90°
Supplementary: two angles summing to 180°
Complementary: two angles summing to 90°
💡 When two straight lines cross, 4 angles are formed. The opposite pairs (vertically opposite) are always equal.

2. Parallel Lines

When a transversal cuts two parallel lines, three types of angle pairs are formed:

Corresponding angles: same position at each intersection — EQUAL (F-shape)
Alternate angles: on opposite sides of the transversal — EQUAL (Z-shape)
Co-interior angles: same side, between parallel lines — sum to 180° (C-shape)
💡 Remember: Corresponding = F, Alternate = Z, Co-interior = C (and adds to 180)

3. Angles in Triangles

Angles in a triangle sum to 180°
Equilateral: all angles = 60°
Isosceles: base angles are equal
Right-angled: one angle = 90°, other two sum to 90°
Exterior angle: = sum of the two non-adjacent interior angles
Exterior angle = sum of two opposite interior angles

4. Angles in Quadrilaterals and Polygons

Sum of interior angles of a polygon = (n − 2) × 180°
Triangle (n=3): (3−2)×180 = 180°
Quadrilateral (n=4): (4−2)×180 = 360°
Pentagon (n=5): 540°   Hexagon (n=6): 720°
Sum of exterior angles: always 360° for any convex polygon
Regular polygon: each interior angle = (n−2)×180÷n

5. Bearings

A bearing is a direction measured clockwise from North, written as 3 digits.

North: 000°   East: 090°   South: 180°   West: 270°
Back bearing: add or subtract 180° to find the opposite direction
Example: Bearing of 065° → back bearing = 065 + 180 = 245°
⚠️ Always measure bearings clockwise from North and write 3 digits (e.g. 045°, not 45°).

✏️ Worked Examples

Example 1 – Basic Angle Facts

Find angle x: (a) angles on a straight line: 65° and x   (b) around a point: 120°, 95°, x

(a) 65 + x = 180 → x = 115°
(b) 120 + 95 + x = 360 → x = 145°

Example 2 – Parallel Lines

A transversal crosses two parallel lines. One angle is 70°. Find: alternate, corresponding, co-interior.

Alternate angle: 70° (Z-shape, equal)
Corresponding angle: 70° (F-shape, equal)
Co-interior angle: 180° − 70° = 110° (C-shape, adds to 180°)

Example 3 – Triangle Angles

A triangle has angles 48° and 67°. Find the third angle and its exterior angle.

Third angle: 180 − 48 − 67 = 65°
Exterior angle at 65°: = 180 − 65 = 115°, OR = 48 + 67 = 115° ✓

Example 4 – Interior Angles of a Polygon

Find the sum of interior angles of an octagon, and each interior angle if regular.

Sum: (8 − 2) × 180 = 6 × 180 = 1080°
Each angle (regular): 1080 ÷ 8 = 135°
Each exterior angle: 360 ÷ 8 = 45°

🎨 Visualizer

📐 Angle Explorer

Drag the slider to set angle α. See supplementary, complementary and vertically opposite.

65°

🔀 Parallel Lines Explorer

Set the transversal angle. See all 8 angles and their relationships.

70°

🔷 Polygon Angle Calculator

🧭 Bearing Calculator

Exercise 1 – Basic Angle Facts

Find the missing angle in each scenario.

Exercise 2 – Parallel Lines

Two parallel lines cut by a transversal. Find angles using alternate, corresponding or co-interior rules.

Exercise 3 – Triangle Angles

Exercise 4 – Polygon Angles

Exercise 5 – Bearings

All bearings are measured clockwise from North, 0°–360°.

📝 Practice – 20 Questions

🏆 Challenge – 8 Questions