Math Vale

🏰 Geometry Kingdom · Geometry

Angles

Measure and classify angles, and use complementary, supplementary, vertical and parallel-line relationships to find unknown angles.

In short

  • An angle measures turn, not length — the arms can be drawn any size.
  • Complementary angles add to 90°; supplementary angles add to 180°.
  • Angles along a straight line add to 180°; angles around a point add to 360°.
  • Where two lines cross, opposite (vertical) angles are equal and neighbouring angles are supplementary.

An angle measures turning

An angle is not a distance — it is an amount of turn. Two rays share an endpoint (the vertex), and the angle records how far you would have to swing one ray to land on the other.

A full turn is 360°, so half a turn is 180° (a straight line) and a quarter turn is 90° (a square corner, called a right angle). Almost everything in this topic comes from those three facts.

Angles are named by their size:

  • acute: less than 90°
  • right: exactly 90°
  • obtuse: between 90° and 180°
  • straight: exactly 180°
  • reflex: more than 180°

The length of the arms drawn on the page has nothing to do with the size of the angle. A 40° angle drawn with very long arms is still 40°.

Complementary and supplementary

Two special partnerships come up constantly.

  • Complementary angles add to 90°. If one is 37°, the other is 90 - 37 = 53°.
  • Supplementary angles add to 180°. If one is 37°, the other is 180 - 37 = 143°.

A useful way to keep them apart: C comes before S in the alphabet, and 90 comes before 180.

Neither word says anything about the angles touching. They can sit side by side, or be in different diagrams entirely — what matters is only the total.

The two acute angles in any right triangle are always complementary, because all three angles must total 180° and one of them has already used up 90°.

Angles on a line and around a point

When several angles sit side by side without gaps or overlaps, their measures add:

  • along a straight line, they total 180°
  • all the way around a point, they total 360°

That single idea turns most angle-chasing puzzles into a one-line equation. If angles of 65°, 40° and x° lie along a straight line, then 65 + 40 + x = 180, so x = 75.

When two straight lines cross they make four angles. Angles that are opposite each other (vertical angles) are always equal, and any two angles that are next to each other form a straight line, so they are supplementary.

Why are vertical angles equal? Call the four angles a, b, a', b' in order. Then a + b = 180 (straight line) and b + a' = 180 (straight line), so a and a' are both 180 - b. They must be the same.

Writing angle relationships as equations

Once angles are described with algebra, the same facts become equations to solve.

If (3x + 10)° and (2x)° are supplementary, then

3x + 10 + 2x = 180 5x + 10 = 180 5x = 170 x = 34

Always finish by re-reading the question. It might want x, or it might want one of the angles — here the first angle is 3(34) + 10 = 112°, which is a different number from 34.

A quick check is worth the ten seconds it costs: 112 + 68 = 180. Correct.

Worked examples

Example 1

Angle A and angle B are complementary. Angle A = 26°. Find angle B.

  1. Complementary means the two angles add to 90°.
  2. So 26 + angle B = 90.
  3. Subtract 26 from both sides: angle B = 90 - 26.
  4. Angle B = 64°. Check: 26 + 64 = 90.

Example 2

Four angles meet at a point. Three of them are 95°, 120° and 80°. Find the fourth.

  1. Angles all the way around a point add to 360°.
  2. Add the three known angles: 95 + 120 + 80 = 295.
  3. The missing angle is what is left: 360 - 295.
  4. The fourth angle is 65°.

Practice problems, with solutions

Three problems of increasing difficulty, each with the full working. In the game these are generated fresh every time; these three are fixed so this page always shows the same ones.

Problem 1

Difficulty 1 of 5

Angle P and angle Q are complementary. Angle P measures 20°. How many degrees is angle Q?

Answer: 70 degrees

  1. Complementary angles sum to 90°.
  2. angle Q = 90° - 20°
  3. angle Q = 70°

Problem 2

Difficulty 3 of 5

3 angles sit side by side on a straight line. 12°, 141° and x° are the angles. Find x.

Answer: 27 degrees

  1. The angles on a straight line sum to 180°.
  2. 12 + 141 + x = 180
  3. 153 + x = 180
  4. x = 180 - 153 = 27

Problem 3

Difficulty 4 of 5

Two straight lines cross. One of the four angles formed measures 122°. How many degrees is the angle next to it (they form a straight line together)?

Answer: 58 degrees

  1. Angles side by side on a straight line add to 180°.
  2. 122° + ? = 180°
  3. ? = 180° - 122° = 58°

Common mistakes

  • Using 180° for complementary angles or 90° for supplementary ones.
  • Answering with x when the question asked for an angle (or the other way round).
  • Assuming a longer-looking arm means a bigger angle.
  • Treating two neighbouring angles at an intersection as equal — those are the opposite ones.

What you should be able to do

  • Classify angles as acute, right, obtuse or straight.
  • Use complementary and supplementary relationships to find an angle.
  • Apply vertical angle and linear pair relationships.
  • Find angles formed when a transversal crosses parallel lines.

Where this fits in the curriculum

Common Core

  • 4.MD.C.5

    Grade 4 — Recognise angles as geometric figures formed by two rays and understand angle measurement.

  • 7.G.B.5

    Grade 7 — Use supplementary, complementary, vertical and adjacent angles to write and solve equations for an unknown angle.

  • 8.G.A.5

    Grade 8 — Use informal arguments about the angles created when a transversal crosses parallel lines.

Ontario

  • G6.E2.3

    Grade 6 — Use the relationships between complementary, supplementary, opposite, interior and exterior angles to determine unknown angle measures.

  • G8.E2.2

    Grade 8 — Solve problems involving the angle properties of intersecting lines, parallel lines cut by a transversal, and polygons.

SAT

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