IGCSE Mathematics: Angle Properties Practice Questions
Angles on a straight line add to 180 degrees and angles around a point add to 360 degrees. For any polygon, the exterior angles always total 360 degrees, and the interior angles total 180 multiplied by two fewer than the number of sides.
Topic 4.6 of Cambridge IGCSE Mathematics 0580 is high frequency and low risk, provided you name the reason for every step. Polygon questions almost always run through the exterior angle, even when the interior angle is what you are given. The questions below use that route.
What you need to know for Angle Properties
- Basic angle factsAngles on a straight line total 180 degrees. Angles around a point total 360 degrees. Vertically opposite angles are equal.
- Parallel line anglesCorresponding angles are equal and form an F shape. Alternate angles are equal and form a Z shape. Co-interior angles add to 180 degrees and form a C shape.
- Triangle and quadrilateralThe angles of a triangle add to 180 degrees, and those of a quadrilateral to 360 degrees.
- Exterior angles of a polygonThe exterior angles of any polygon total 360 degrees. For a regular polygon with n sides, each exterior angle is 360 divided by n.
- Interior angles of a polygonThe sum is (n minus 2) x 180 degrees. Each interior angle and its exterior angle add to 180 degrees.
- Working backwardsGiven an interior angle, subtract from 180 to get the exterior angle, then divide 360 by that to find the number of sides.
IGCSE Mathematics Angle Properties questions and answers
4 exam-style questions written to the 0580 syllabus. Try each one on paper first, then open the worked answer to check your method against the marks.
Two parallel lines are crossed by a transversal. One of the angles formed is 62 degrees. Find the size of the co-interior angle on the same side of the transversal, and name the angle relationship you used.
Show the worked answer
- Co-interior angles, sometimes called allied angles, lie between the two parallel lines on the same side of the transversal.
- They form a C shape, and they add to 180 degrees.
- 180 minus 62 = 118 degrees.
- By contrast, the alternate angle would be 62 degrees and the corresponding angle would also be 62 degrees.
Calculate the size of each interior angle of a regular 12-sided polygon.
Show the worked answer
- The exterior angles of any polygon total 360 degrees.
- The polygon is regular, so all exterior angles are equal: 360 divided by 12 = 30 degrees.
- An interior angle and its exterior angle lie on a straight line, so they add to 180 degrees.
- Interior angle = 180 minus 30 = 150 degrees. Check using the sum formula: (12 minus 2) x 180 = 1800, and 1800 divided by 12 = 150.
A hexagon has five angles measuring 110, 125, 130, 96 and 140 degrees. Calculate the sixth angle.
Show the worked answer
- The interior angle sum of a polygon is (n minus 2) x 180. For a hexagon, n = 6.
- Sum = (6 minus 2) x 180 = 4 x 180 = 720 degrees.
- Add the five known angles: 110 + 125 + 130 + 96 + 140 = 601 degrees.
- The sixth angle = 720 minus 601 = 119 degrees.
Each interior angle of a regular polygon is 156 degrees. Calculate the number of sides.
Show the worked answer
- An interior angle and its exterior angle add to 180 degrees, so the exterior angle is 180 minus 156 = 24 degrees.
- The exterior angles of any polygon total 360 degrees.
- The polygon is regular, so the number of sides is 360 divided by the exterior angle.
- n = 360 divided by 24 = 15 sides. Check: (15 minus 2) x 180 = 2340, and 2340 divided by 15 = 156 degrees.
Common mistakes in this topic
- Treating co-interior angles as equal.
- Confusing the interior angle with the exterior angle when dividing 360.
- Using 360 degrees as the interior angle sum for any polygon.
- Omitting the reason for each step.
- Assuming a polygon is regular when the question has not said so.
Exam tips
- Route every regular polygon question through the exterior angle. It is nearly always the quicker path.
- Learn the three parallel line shapes: F for corresponding, Z for alternate, C for co-interior.
- Name the reason beside every angle. Reasoning marks are awarded separately.
- Check that the angles in your answer add to the correct total before moving on.
- Mark equal angles on the diagram as you find them. It reveals the next step.
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Angle Properties FAQs
What is the sum of interior angles of a polygon?
It is (n minus 2) multiplied by 180 degrees, where n is the number of sides. A triangle gives 180, a quadrilateral 360, a pentagon 540 and a hexagon 720. For a regular polygon, divide that total by n to find each interior angle.
How do I find the number of sides from an interior angle?
Subtract the interior angle from 180 to get the exterior angle, then divide 360 by that exterior angle. An interior angle of 156 degrees gives an exterior angle of 24 degrees, and 360 divided by 24 is 15 sides.
What is the difference between alternate and co-interior angles?
Alternate angles sit on opposite sides of the transversal between the parallel lines, forming a Z shape, and they are equal. Co-interior angles sit on the same side between the lines, forming a C shape, and they add to 180 degrees rather than being equal.
Do exterior angles always add to 360 degrees?
Yes, for any polygon regardless of the number of sides or whether it is regular. This is what makes the exterior angle route so reliable, and it is why you divide 360 rather than the interior angle sum when working with regular polygons.
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Written to the published Cambridge IGCSE Mathematics (0580) syllabus. Check your school entry code and syllabus year, because Core and Extended candidates are assessed on different content. Last reviewed 2026-08-12.