courses › Geometry

Angles

level 11 course

Angles on lines, in triangles, and in polygons.

In your head, jot if needed · no calculator why?

Opens at level 11.

Builds on: Two-digit add & subtract (not open yet)

the lesson

The idea, the techniques and a tip for each skill, right here. The Learn page adds worked examples for every skill and untimed practice.

Read the lesson · about 3 minutes

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

Complements, supplements, and the third angle of a triangle.

Pick the total: 90° for a complement, 180° for a supplement or a triangle.
Subtract every angle you know from that total.

worked example

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

Angle totals, and each angle of a regular polygon (all sides and angles equal).

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

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

Tips by skill

TipComplements & supplements: Complement: take the angle away from 90. Supplement: take it away from 180.
TipThird angle of a triangle: The three angles add to 180°, so subtract both known angles from 180.
TipPolygon angles: Total inside: (n − 2) × 180°. Each interior angle: that total ÷ n. Each exterior angle: 360° ÷ n.

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°.

skills · practice stats

From rounds of this course only: box, review and test-out answers are left out. Once a skill has 40 tries, it compares your first 20 tries with your last 20.

Complements & supplements par 7 s not tried yet
worked example
What is the supplement of a 30° angle, in degrees?

Answer: 150
1. Supplementary angles add to 180°: 180 − 30 = 150°.
Third angle of a triangle par 9 s not tried yet
worked example
A triangle has angles of 42° and 58°. What is the third angle, in degrees?

Answer: 80
1. Angles in a triangle add to 180°: 180 − 42 − 58 = 80°.
Polygon angles par 17 s not tried yet
worked example
What do the interior angles of a quadrilateral add up to, in degrees?

Answer: 360
1. 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.
2. (4 − 2) × 180 = 360°.

rest ladder

Win 3 of your last 4 rounds and the course rests. A win is 90% right, within 2× the round's par. Pass the review when it comes back and the next rest is longer.

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