Angles
Section: Geometry | Syllabus: Cambridge IGCSE Mathematics (0580)
Angles at a Point and on a Straight Line
Several basic angle facts apply wherever lines meet at a point, and these form the foundation for solving almost every other angle problem.
- Angles at a point sum to 360°
- Angles on a straight line sum to 180°
- Vertically opposite angles (formed where two straight lines cross) are always equal
Worked Example: Finding Angles on a Straight Line
- Question: Three angles lie along a straight line: 72°, x°, and 45°. Find x.
- Step 1: Angles on a straight line sum to 180°
- Step 2: x=180-72-45=63
- Answer: x = 63°
Worked Example: Finding Angles at a Point
- Question: Four angles meet at a point: 110°, 85°, 95°, and y°. Find y.
- Step 1: Angles at a point sum to 360°
- Step 2: y=360-110-85-95=70
- Answer: y = 70°
Common Mistakes
MistakeUsing 360° for angles on a straight line, or 180° for angles at a point, mixing up the two rules
Fixa straight line always sums to 180°; a full point (all the way around) always sums to 360°
MistakeForgetting that vertically opposite angles are equal, and trying to calculate them using other angle facts unnecessarily
Fixrecognise vertically opposite angles immediately from the diagram - they are automatically equal, with no calculation needed
Angles in Parallel Lines
When a straight line (a transversal) crosses a pair of parallel lines, it creates several pairs of equal or supplementary angles with specific names.
- Corresponding angles (in matching positions at each intersection) are equal - sometimes remembered as an "F" shape
- Alternate angles (on opposite sides of the transversal, between the parallel lines) are equal - sometimes remembered as a "Z" shape
- Co-interior angles (also called allied angles, on the same side of the transversal, between the parallel lines) sum to 180° - sometimes remembered as a "C" or "U" shape
Worked Example: Finding Angles with Parallel Lines
- Question: Two parallel lines are crossed by a transversal. One angle where the transversal meets the first line is 118°. Find the corresponding angle, the alternate angle, and the co-interior angle at the second line.
- Step 1: Corresponding angle = 118° (equal, matching position)
- Step 2: Alternate angle = 118° (equal, "Z" shape)
- Step 3: Co-interior angle: 180-118=62
- Answer: corresponding = 118°, alternate = 118°, co-interior = 62°
Common Mistakes
MistakeTreating co-interior angles as equal, the same way corresponding and alternate angles are
Fixco-interior angles are the only one of the three types that are NOT equal - they are supplementary, summing to 180°
MistakeIdentifying the wrong angle type from the diagram, e.g. calling a corresponding pair "alternate"
Fixcheck the exact position carefully - alternate angles are on opposite sides of the transversal, while corresponding angles are on the same side in matching positions
Angle Sum of Triangles and Quadrilaterals
The angles inside any triangle or quadrilateral always sum to a fixed total, regardless of the specific shape.
- The angles in any triangle sum to 180°
- The angles in any quadrilateral sum to 360° (a quadrilateral can always be split into two triangles)
- The exterior angle of a triangle equals the sum of the two interior angles not adjacent to it
Worked Example: Finding an Angle in a Triangle
- Question: A triangle has angles of 54°, 76°, and z°. Find z.
- Step 1: Angles in a triangle sum to 180°
- Step 2: z=180-54-76=50
- Answer: z = 50°
Worked Example: Using the Exterior Angle of a Triangle
- Question: A triangle has interior angles of 48° and 65° at two of its vertices. Find the exterior angle at the third vertex.
- Step 1: The exterior angle equals the sum of the two interior angles not adjacent to it
- Step 2: 48+65=113
- Answer: 113°
Common Mistakes
MistakeUsing 360° for a triangle's angle sum, confusing it with a quadrilateral
Fixa triangle's angles always sum to 180°; only a quadrilateral sums to 360°
MistakeAdding all three interior angles when finding an exterior angle, instead of just the two that aren't adjacent to it
Fixthe exterior-angle rule only uses the two interior angles at the other two vertices, not the one at the same vertex as the exterior angle
Angles in Polygons
Every polygon has a predictable total for its interior angles, based on its number of sides, and its exterior angles always sum to the same fixed total no matter how many sides it has.
- The sum of the interior angles of a polygon with n sides is (n-2)180°
- The sum of the exterior angles of any polygon is always 360°, regardless of the number of sides
- For a regular polygon, each interior angle =((n-2)180°/n), and each exterior angle =(360°/n)
- An interior angle and its adjacent exterior angle always sum to 180°, since they lie on a straight line
Worked Example: Finding the Number of Sides from an Exterior Angle
- Question: A regular polygon has an exterior angle of 24°. Find the number of sides.
- Step 1: Exterior angle of a regular polygon = 360° ÷ n
- Step 2: 360÷ n=24 n=36024=15
- Answer: 15 sides
Worked Example: Finding an Interior Angle from the Number of Sides
- Question: Find the sum of the interior angles of a polygon with 9 sides, and hence the size of each interior angle if the polygon is regular.
- Step 1: Sum of interior angles: (9-2)180=7180=1260
- Step 2: Each interior angle (regular): 12609=140
- Answer: sum = 1260°, each interior angle = 140°
Common Mistakes
MistakeUsing n instead of (n-2) in the interior angle sum formula, forgetting to subtract 2
Fixthe formula is always (n-2)×180° - the subtraction of 2 comes from splitting the polygon into (n-2) triangles
MistakeAssuming the sum of exterior angles changes for polygons with more sides
Fixthe sum of exterior angles is always exactly 360°, for any polygon regardless of how many sides it has - only the sum of interior angles depends on n
Interactive revision notes, videos and practice questions load below.