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Angles

An angle is the amount of turn between two lines that meet. It has nothing to do with how long the lines are — stretch both arms out twice as far and the angle is exactly the same. The corner of a page is the one worth memorising: that is a right angle, and every other angle is named by whether it is smaller than that corner, bigger than it, or opened out flat into a straight line.

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Naming an angle by eye

Is this angle acute, right, obtuse, or straight? (an angle measuring 130 degrees)

  1. Find the corner of something square — a page, a card, a book.
  2. Put that corner against the angle, with the two edges lined up.
  3. The angle opens wider than the card's corner, so it is more than a right angle.
  4. It has not opened all the way into a straight line, so it is not straight.
  5. Wider than a right angle but not flat means obtuse.

Worth knowing. The arms being long or short changes nothing. An angle is the turn between the lines, not the lines themselves.

Finding a complement

Two angles are complementary. One measures 35 degrees. What does the other measure?

  1. Complementary means the two angles make a right angle together.
  2. A right angle is 90 degrees.
  3. So the two must add to 90: 35 plus something is 90.
  4. 90 take away 35 is 55.
  5. The other angle is 55 degrees.

Worth knowing. Complementary goes with corner — both begin with a c, and a corner is 90 degrees. Supplementary is the bigger word for the bigger angle, 180.

The third angle of a triangle

Two angles of a triangle measure 40 and 75 degrees. What does the third measure?

  1. The three angles of any triangle add to 180 degrees — always, whatever shape it is.
  2. Add the two you have been given: 40 plus 75 is 115.
  3. Take that from 180: 180 take away 115 is 65.
  4. The third angle is 65 degrees.
  5. Check it: 40 plus 75 plus 65 is 180.

Worth knowing. The check at the end costs one addition and catches almost every slip. It is worth doing every time.

Where it usually goes wrong

  • Thinking a longer pair of arms means a bigger angle. Only the turn between them counts.
  • Mixing up complementary and supplementary. Complementary makes a Corner, 90 degrees; supplementary makes a Straight line, 180.
  • Calling an angle of 85 degrees a right angle because it looks close. Close is not equal — a right angle is exactly 90.
  • Forgetting that a triangle's angles always add to 180, and adding the two given angles instead of taking them away from 180.

The Professor holds the corner of a card against the angle. If the angle hides behind the card it is acute; if it pokes out past it, obtuse; if it fits exactly, a right angle. One piece of card and no protractor.

Practise it

All level paths

Questions teachers and parents ask

How can I tell an acute angle from an obtuse one without a protractor?

Compare it with the corner of a page. Anything that fits inside that corner is acute, and anything wider than it is obtuse. These worksheets deliberately draw angles that are clearly one or the other, and only close in on the boundary at the harder levels — so if it genuinely looks borderline, that is the level being difficult on purpose.

Why do the worksheets never ask me to measure an angle in degrees?

Because reading a drawn angle to the degree needs a protractor lined up on the page, and a printed sheet cannot hand a child one. Asking for the number would be asking for a guess. The sheets ask for the kind of angle instead, which can be answered by eye, and ask for degrees only where the number is given in the wording.

What is a reflex angle, and why is it not here?

A reflex angle is one bigger than a straight line — more than 180 degrees. It is a real thing and it is left out on purpose, because the same two rays can show a 60-degree angle or a 300-degree one and only a marked arc says which is meant. That is a question about the drawing rather than about the angle.

Do the angles of a triangle really always add to 180?

On a flat page, yes, always — tall thin ones, wide flat ones, all of them. Tear the three corners off a paper triangle and lay them side by side and they make a straight line every time. That is the fact these levels are built on.

Where this shows up

Standards

  • 4.MD · Measurement and data
  • 4.G · Geometry
  • 7.G · Geometry
  • 8.G · Geometry
  • CCSS.MATH.CONTENT.4.G.A.1 — Draw and identify points, lines, line segments, rays, angles and perpendicular and parallel lines in two-dimensional figures.
  • CCSS.MATH.CONTENT.4.MD.C.5 — Recognise angles as geometric shapes formed wherever two rays share an endpoint, and understand concepts of angle measurement.
  • CCSS.MATH.CONTENT.7.G.B.5 — Use facts about supplementary, complementary, vertical and adjacent angles to write and solve simple equations for an unknown angle.
  • CCSS.MATH.CONTENT.8.G.A.5 — Use informal arguments to establish facts about the angle sum of a triangle and the angles created when parallel lines are cut by a transversal.

Formats