Angles
Angles on lines, in triangles, and in polygons.
In your head, jot if needed · no calculator why?
Opens at level 11. You're level 1. You can read and practice here now.
the idea
Angles are measured in degrees, and a few totals do most of the work. A right angle is 90°, a straight line is 180°, and a full turn is 360°.
Two angles that add to 90° are complementary; two that add to 180° are supplementary. The three angles of any triangle add to 180°. The inside angles of a polygon (a closed shape with straight sides) with n sides add to (n − 2) × 180°. To see why, cut a convex one into n − 2 triangles from one corner.
techniques
Subtract from the total
- Pick the total: 90° for a complement, 180° for a supplement or a triangle.
- Subtract every angle you know from that total.
worked example
A triangle has angles of 52° and 71°. What is the third angle, in degrees?
- The angles of a triangle add to 180°.
- 180 − 52 − 71 = 57°.
Answer: 57
Polygon angles from triangles
- Interior angles add to (n − 2) × 180°, where n is the number of sides.
- Each interior angle of a regular polygon: divide that total by n.
- Exterior angles always add to 360°, so each one is 360° ÷ n.
- At every corner, the interior and exterior angles add to 180°.
worked example
What is each interior angle of a regular hexagon (6 sides), in degrees?
- Total: (6 − 2) × 180 = 720°.
- Each: 720 ÷ 6 = 120°.
- Check: each exterior angle is 360 ÷ 6 = 60°, and 180 − 60 = 120°.
Answer: 120
watch out for
- Mixing up the pair. Complements make 90° and supplements make 180°, so the supplement of 68° is 112°, not 22°.
- Adding the two known angles of a triangle and stopping. Subtract their sum from 180°.
- Giving the exterior angle when the question asks for the interior one. For a regular octagon, 45° is exterior; each interior angle is 135°.
- Using n × 180° for the angle total. From one corner you get only n − 2 triangles, so it is (n − 2) × 180°.
practice
Complements & supplements
worked example
What is the supplement of a 30° angle, in degrees?
Answer: 150
- Supplementary angles add to 180°: 180 − 30 = 150°.
Third angle of a triangle
worked example
A triangle has angles of 42° and 58°. What is the third angle, in degrees?
Answer: 80
- Angles in a triangle add to 180°: 180 − 42 − 58 = 80°.
Polygon angles
worked example
What do the interior angles of a quadrilateral add up to, in degrees?
Answer: 360
- Any 4-sided polygon's interior angles add up to (4 − 2) × 180°; for a convex one you can see it by splitting it into 2 triangles from one corner.
- (4 − 2) × 180 = 360°.