Angles and Triangles

1An angle is an amount of turn

Open your scissors a little. Now open them wider. The two blades still start from the same little screw, but the gap between them has grown.

Whenever two straight lines start from the same point, they make an angle.

• The two straight lines are the arms.
• The point where they meet is the vertex (say it: VER-teks). It is just the corner.

The angle is not about how long the arms are. A short pair of scissors and a long pair of scissors can be opened by exactly the same amount. The angle measures the opening, nothing else.

2Degrees: the units of turning

Stand up and spin all the way round until you face the front again. That whole spin is called a full turn, and we say it is 360°. The little circle mark ° means degrees.

From that one fact, everything else follows:

• A full turn = 360°
• Half a turn (you end up facing backwards) = 180°
• A quarter turn (you end up facing sideways) = 90°

A quarter turn, 90°, is the most useful angle of all. It is the corner of your exercise book, the corner of a window, the corner of a brick. We call it a right angle. Every corner of this page is a right angle.

3Naming angles by their size

Compare the angle to a right angle and it names itself.

• Acute angle — smaller than 90°. A sharp, narrow angle, like a slice of pineapple. (25°, 40°, 89°)
• Right angle — exactly 90°. The book-corner angle.
• Obtuse angle — bigger than 90° but smaller than 180°. A wide, lazy angle. (91°, 130°, 179°)
• Straight angle — exactly 180°. The two arms lie flat in a straight line.
• Reflex angle — bigger than 180° but smaller than 360°. The huge one, going the long way round. (200°, 300°)

4Measuring with a protractor

A protractor is the half-circle of clear plastic with numbers curving round its edge. Look closely and you will see two rows of numbers. One row counts up from the left, the other counts up from the right. That double row is where nearly every mistake happens, so read the next part slowly.

To measure an angle:

1. Find the small hole or cross in the middle of the protractor's flat edge. That is the centre. Put it exactly on the vertex of your angle.
2. Turn the protractor until the flat edge lies exactly along one arm of the angle.
3. Look at where that arm passes through the numbers. It should sit on a 0. Two rows, but only one of them shows 0 on your arm — that is the row you will read from.
4. Now follow the other arm out to the edge of the protractor and read the number, counting from your 0.

Worked example 1: reading the right row

Keza has drawn an angle. She lines up the protractor properly. The first arm sits on 0 on the outer row.

She follows the second arm to the edge. It lands where two numbers are printed together: 130 on the outer row and 50 on the inner row. Which one is her answer?

Her arm started at 0 on the outer row, so she must stay on the outer row. The angle is 130°.

And she can check it with her eyes: the angle is clearly wider than the corner of her book, so it must be more than 90°. 130° is more than 90°. 50° is not. The answer 130° is right.

Worked example 2: drawing an angle of 65°

Mugisha must draw an angle of exactly 65°.

1. He draws one straight arm with his ruler and marks the vertex at one end.
2. He puts the protractor's centre on the vertex, flat edge along the arm, so that the arm reads 0.
3. Starting from that 0, he counts round to 65 and makes a small neat dot at 65.
4. He takes the protractor away and rules a straight line from the vertex through the dot.

Done. His angle is 65° — acute, because 65 is less than 90.

5The three angles of a triangle

Here is something you can prove with your own hands right now. Cut any triangle out of paper — any shape at all, it does not matter. Tear off its three corners. Now lay the three torn corners side by side so their points all touch.

Worked example 3: finding the missing angle

In a triangle, two angles measure 55° and 65°. What is the third angle?

Step 1 Add the two you know: 55 + 65 = 120°.

Step 2 The three must total 180°, so take away what you have used: 180 − 120 = 60°.

Check: 55 + 65 + 60 = 180. It works.

6Naming triangles by their sides

Get out your ruler and measure the three sides. What you find tells you the name.

• Equilateral triangle — all three sides the same length. Every angle is 60°, because 180 ÷ 3 = 60. It looks perfectly even, whichever way you turn it.
• Isosceles triangle — exactly two sides the same length. The two angles at the bottom of those equal sides are also equal to each other.
• Scalene triangle — no two sides the same. All three sides different, so all three angles are different too.

Worked example 4: an isosceles triangle

Umutoni draws an isosceles triangle. The angle at the top (between the two equal sides) is 40°. What is each of the other two angles?

Step 1 The three angles add to 180°, and 40° is used up: 180 − 40 = 140° is left for the other two.

Step 2 Because the triangle is isosceles, those two remaining angles are equal. So share the 140° between them: 140 ÷ 2 = 70°.

Each of the other two angles is 70°. Check: 40 + 70 + 70 = 180. It works.

7Naming triangles by their angles

You can also name a triangle by looking at its biggest angle.

• Acute-angled triangle — all three angles are less than 90°. (Example: 60°, 70°, 50°)
• Right-angled triangle — one angle is exactly 90°. (Example: 90°, 60°, 30°)
• Obtuse-angled triangle — one angle is more than 90°. (Example: 110°, 40°, 30°)

A triangle can only ever have one angle of 90° or more. Think about why: two right angles would already use up 90 + 90 = 180°, and there would be nothing left for the third angle.

Worked example 5: giving a triangle both its names

The full name is an obtuse-angled isosceles triangle. Notice how the two equal sides sit opposite the two equal angles — that is always true.

Try it yourself

Work each one out on paper first, then check. The answers are right underneath, so you can never get stuck.

1. Name this angle by its size: 37°.
   Answer: acute. 37 is less than 90, so the angle is narrower than a book corner.
2. Name this angle by its size: 145°.
   Answer: obtuse. 145 is more than 90 but less than 180, so it is a wide, lazy angle.
3. Two angles of a triangle are 45° and 85°. Find the third angle.
   Answer: 45 + 85 = 130. Then 180 − 130 = 50°. Check: 45 + 85 + 50 = 180.
4. A triangle has sides of 7 cm, 7 cm and 7 cm. Name it, and give the size of each angle.
   Answer: all three sides are equal, so it is equilateral. Its three angles are equal too, so each one is 180 ÷ 3 = 60°.
5. An isosceles triangle has two base angles of 50° each. What is the third angle?
   Answer: the two base angles use up 50 + 50 = 100°. So the third angle is 180 − 100 = 80°.
6. Byiringiro says his triangle has angles of 30°, 60° and 90°. Name it by its angles, and name it by its sides.
   Answer: one angle is exactly 90°, so it is right-angled. All three angles are different, which means all three sides are different too, so it is also scalene.
7. Mutesi says she has drawn a triangle with two angles of 90°. Is she right?
   Answer: no, that is impossible. 90 + 90 = 180, and that is the whole 180° used up already. There would be 0° left for the third angle, so the shape could never close.

Now find three right angles in the room you are sitting in. Check each one with the corner of your exercise book.