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IGCSE Maths · Geometry · Core & Extended

Angle Properties: Points, Lines, Triangles and Polygons

Angle problems are like puzzles — a handful of core facts (angles at a point, on a line, in parallel lines, in shapes) combine to let you work out almost any missing angle, as long as you can spot which rule applies where. Nearly everything here is shared between tiers; the only real split is that Core’s polygon work centres on regular polygons, while Extended explicitly covers irregular ones too.

1. Angles at a Point

Angles that meet at a single point, going all the way around, always add up to 360°.

Worked example 120° 150° 90° Three angles meet at a point: 120°, 150°, and x°. Find x.
120 + 150 + x = 360 → 270 + x = 360 → x = 90°
Try it — Angles at a Point
  1. Four angles meet at a point: 80°, 95°, 110°, and x°. Find x.
  2. Three angles meet at a point in the ratio 2:3:4. Find each angle.
Show answers
  1. 80+95+110=285. x=360−285=75°
  2. Total parts=9. One part=40°. Angles: 80°, 120°, 160°

2. Angles on a Straight Line

Angles on one side of a straight line add up to 180°.

Worked example 120° 60° Find x, given the two angles are 120° and x°, on a straight line.
120 + x = 180 → x = 60°
Try it — Angles on a Line
  1. Two angles on a straight line are 65° and x°. Find x.
  2. Three angles on a straight line are 40°, 75°, and x°. Find x.
Show answers
  1. x=180−65=115°
  2. x=180−40−75=65°

3. Vertically Opposite Angles

When two straight lines cross, the angles directly opposite each other are always equal.

Worked example 70° 70° 110° 110° Two straight lines cross. One angle is 70°. Find the other three angles.
The angle vertically opposite 70° is also 70°.
The two angles adjacent to 70° (on the straight line) are each 180 − 70 = 110°, and they’re vertically opposite each other too.
Try it — Vertically Opposite Angles
  1. Two lines cross. One angle is 55°. Find the angle vertically opposite it.
  2. Two lines cross. One angle is 55°. Find the two other (different) angles formed.
Show answers
  1. 55° (vertically opposite angles are equal)
  2. 180−55 = 125°, appearing twice (vertically opposite each other)

4. Angle Sum of Triangles and Quadrilaterals

Angles in a triangle sum to 180°
Angles in a quadrilateral sum to 360°
Worked examples A triangle has angles 55° and 72°. Find the third angle.
180 − 55 − 72 = 53°

A quadrilateral has angles 95°, 80°, and 110°. Find the fourth angle.
360 − 95 − 80 − 110 = 75°
Try it — Triangle & Quadrilateral Angles
  1. A triangle has angles 48° and 96°. Find the third angle.
  2. A quadrilateral has angles 70°, 105°, and 85°. Find the fourth angle.
  3. An isosceles triangle has a base angle of 50°. Find the other two angles.
Show answers
  1. 180−48−96 = 36°
  2. 360−70−105−85 = 100°
  3. The other base angle is also 50° (isosceles). Third angle = 180−50−50 = 80°

5. Angles Formed Within Parallel Lines

When a straight line (a transversal) crosses two parallel lines, it creates matching angle patterns at each crossing.

The three angle relationships —> —> 1 2 3 4 5 6 7 8
RelationshipRuleExample pairs
Corresponding anglesEqual1&5, 2&6, 3&7, 4&8
Alternate anglesEqual2&8, 3&5 (interior, opposite sides of the transversal)
Co-interior anglesSum to 180°2&5, 3&8 (interior, same side)
Worked examples The angle at position 1 is 65°. Find the angle at position 5 (corresponding).
Corresponding angles are equal: 65°

The angle at position 2 is 48°. Find the angle at position 8 (alternate).
Alternate angles are equal: 48°

The angle at position 3 is 110°. Find the angle at position 8 (co-interior).
Co-interior angles sum to 180°: 180 − 110 = 70°
Try it — Parallel Lines (using the diagram above)
  1. Angle 4 is 72°. Find angle 8 (corresponding).
  2. Angle 2 is 58°. Find angle 5 (co-interior).
  3. Angle 4 is 72°. Find angle 6 (alternate).
Show answers
  1. 72° (corresponding angles are equal)
  2. 180−58 = 122° (co-interior angles sum to 180°)
  3. 72° (alternate angles are equal)

6. Interior and Exterior Angles of Polygons

Sum of exterior angles = 360° (any polygon)
Sum of interior angles = (n − 2) × 180°
Regular pentagon — exterior angle 72° Each exterior angle of a regular pentagon = 360 ÷ 5 = 72°. Each interior angle = 180 − 72 = 108°.
Worked example — regular polygon, finding n A regular polygon has an interior angle of 150°. Find the number of sides.
Exterior angle = 180 − 150 = 30°
Number of sides = 360 ÷ 30 = 12 (a regular dodecagon)
Worked example — irregular polygon Extended Four of the five exterior angles of a pentagon are 50°, 70°, 80°, and 90°. Find the fifth.
Exterior angles always sum to 360°, regardless of whether the polygon is regular:
360 − 50 − 70 − 80 − 90 = 70°
Try it — Polygon Angles
  1. Find the exterior and interior angle of a regular hexagon.
  2. A regular polygon has an exterior angle of 24°. Find the number of sides.
  3. An irregular quadrilateral has three angles of 80°, 95°, and 110°. Find the fourth. Ext.
  4. Find the sum of interior angles of a regular octagon (n = 8), then find each interior angle.
Show answers
  1. Exterior = 360÷6 = 60°. Interior = 120°
  2. 360÷24 = 15 sides
  3. 360−80−95−110 = 75°
  4. (8−2)×180 = 1080°. Each interior angle = 1080÷8 = 135°
Common mistakes to watch for
  • Using 180° instead of 360° for angles at a point (or vice versa) — check whether the angles form a full circle or just a straight line.
  • Mixing up corresponding, alternate, and co-interior angles — corresponding and alternate are equal; co-interior sums to 180°.
  • Assuming lines are parallel without being told, or without a reason (matching arrow marks on a diagram) confirming it.
  • Using the exterior angle sum formula (360°) for interior angles, or vice versa.
  • Forgetting the (n−2)×180° formula counts triangles inside the polygon, not the number of sides directly — always subtract 2 first.

7. Quick-Fire Challenge Round

True or False?
  1. Angles on a straight line always sum to 180°.
  2. Vertically opposite angles sum to 180°.
  3. Co-interior angles are always equal.
  4. The exterior angles of any polygon always sum to 360°, even if it’s irregular.
Show answers
  1. True.
  2. False — vertically opposite angles are equal, not summing to 180° (that’s angles on a line).
  3. False — co-interior angles sum to 180°; they’re equal only in the special case where each is 90°.
  4. True. Ext.
Spot the Error
  1. “Three angles meet at a point: 100°, 130°, and x°. So x = 180−100−130 = −50°.”
  2. “A triangle has angles 90°, 90°, and x°, so x = 180−90−90 = 0°… that seems odd but the maths says so.”
  3. “A regular polygon has exterior angle 45°, so it has 45÷360 = 0.125 sides.”
Show answers
  1. Angles at a point sum to 360°, not 180°: x = 360−100−130 = 130°.
  2. This one’s actually a genuine mathematical impossibility — a real triangle can’t have two 90° angles (its third angle would be 0°, meaning it isn’t a real triangle at all). If a question gives these values, double-check for a typo rather than accepting an impossible result.
  3. The formula is inverted: number of sides = 360 ÷ exterior angle = 360÷45 = 8 sides, not 45÷360.

8. Mixed Practice — Bringing It All Together

  1. Three angles meet at a point: 95°, 140°, and x°. Find x.
  2. Two angles on a straight line are 78° and x°. Find x.
  3. Two lines cross. One angle is 42°. Find the other three angles.
  4. A triangle has angles 63° and 58°. Find the third.
  5. Find the interior angle of a regular decagon (n=10).
Show answers
  1. 360−95−140 = 125°
  2. 180−78 = 102°
  3. 42° (vertically opposite), and two angles of 180−42=138° each
  4. 180−63−58 = 59°
  5. Exterior = 360÷10 = 36°. Interior = 180−36 = 144°

Practice at Home

A large set again — take it steadily.

Part A — Angles at a point and on a line
  1. Four angles meet at a point: 60°, 85°, 100°, and x°. Find x.
  2. Three angles on a straight line are 35°, 60°, and x°. Find x.
  3. Three angles meet at a point in the ratio 3:4:5. Find each angle.
Part B — Vertically opposite, triangles, quadrilaterals
  1. Two lines cross. One angle is 38°. Find the other three angles.
  2. A triangle has angles 72° and 45°. Find the third.
  3. A quadrilateral has angles 88°, 92°, and 100°. Find the fourth.
  4. An isosceles triangle’s third (apex) angle is 40°. Find the two base angles.
Part C — Parallel lines (refer back to the labelled diagram in Section 5)
  1. Angle 1 is 58°. Find angle 5.
  2. Angle 3 is 115°. Find angle 8.
  3. Angle 2 is 63°. Find angle 5.
  4. Angle 4 is 80°. Find angle 7.
Part D — Polygon angles
  1. Find the exterior and interior angles of a regular nonagon (n=9).
  2. A regular polygon has an exterior angle of 18°. Find the number of sides.
  3. A regular polygon has an interior angle of 160°. Find the number of sides.
  4. An irregular pentagon has angles 100°, 110°, 95°, and 80°. Find the fifth. Ext.
Part E — Applied / challenge
  1. A regular polygon has an interior angle that is 8 times its exterior angle. Find the number of sides.
  2. Two parallel lines are cut by a transversal. One angle is 3x°, and its co-interior angle is (2x+30)°. Find x, then find both angles.
  3. Explain why the sum of exterior angles of any convex polygon is always 360°, regardless of how many sides it has, using the idea of “walking around the outside” of the shape.

Key Vocabulary Recap

Vertically opposite angles — equal angles formed by two crossing lines
Transversal — a line crossing two (usually parallel) lines
Corresponding angles — equal angles in matching positions at each crossing
Alternate angles — equal angles on opposite sides of the transversal, between the parallel lines
Co-interior angles — angles on the same side, between the parallel lines, summing to 180°
Exterior angle — the angle between a side and the extension of the next side

Syllabus Reference

This note covers Cambridge IGCSE Mathematics (0580) section C4.6 (Core subject content) and section E4.6 (Extended subject content) — Angles. Angles at a point/on a line/vertically opposite, triangle and quadrilateral angle sums, and angles in parallel lines are shared by both tiers. Core’s polygon work centres on regular polygons; Extended explicitly includes irregular polygons too, marked above.

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