Copying and Bisecting Angles

1. Angles, arms and size

An angle is formed when two lines (called arms) meet at a point called the vertex. The size of an angle depends only on how wide the arms are opened, not on how long you draw them. A door opened a quarter turn makes a 90° angle whether the door is a school door or a small cupboard door.

In this lesson you will do two important jobs with only a pair of compasses and a ruler: copy an angle (make a new angle exactly the same size) and bisect an angle (cut it into two equal angles). Mechanics, carpenters and tailors in Kumasi, Tamale and Takoradi use these skills when they cut roofing frames, cloth patterns and metal brackets.

2. Copying an angle

Suppose you have angle AOB drawn in your notebook and you want the same angle at a point O′ on a new line. You could use a protractor, but a protractor has small errors. The compass method uses the fact that the same compass opening and the same chord make the same angle.

1. 1Draw a new line and mark the point O′ where the copied angle will be.
2. 2Put the compass point on O (the vertex of the original angle) and draw an arc that cuts both arms. Call the points X and Y.
3. 3Without changing the compass opening, put the compass point on O′ and draw an arc that cuts the new line at X′. Make the arc long and wide.
4. 4Set the compass to the distance XY by placing its point on X and the pencil on Y.
5. 5With the point on X′, draw a small arc that cuts the long arc. Call the crossing point Y′.
6. 6Use the ruler to join O′ and Y′. Angle X′O′Y′ is equal to angle XOY.

▶ Watch the compass: first the same radius from O and from O′, then the distance XY measured and transferred.

Why it works: triangle OXY and triangle O′X′Y′ have three pairs of equal sides (OX = O′X′ and OY = O′Y′ because the radius did not change, and XY = X′Y′ because we copied that distance). Triangles with all three sides equal are identical, so their angles are equal.

3. Bisecting an angle

To bisect angle AOB we want a line from O that makes two equal angles. We use two kinds of equal arcs: one centred on O, and then two centred on the places where that arc meets the arms.

1. 1With the compass point on the vertex O, draw an arc that cuts arm OA at X and arm OB at Y.
2. 2Put the compass point on X and draw an arc inside the angle. Keep the same opening, move the point to Y and draw another arc to cross it. Label the crossing point P.
3. 3Use the ruler to join O and P (and extend it if you wish). OP bisects angle AOB.

▶ A 70° angle is cut into two equal angles of 35°. Notice that the two small arcs use the same opening.

Why it works: triangles OXP and OYP have OX = OY (same radius), XP = YP (same opening) and share the side OP. All three sides are equal, so the triangles are identical and angle XOP = angle YOP.

The same method works for acute, right and obtuse angles. If the angle is very big (obtuse), draw your first arcs larger, because the crossing point P will be further away from X and Y.

4. Checking, combining and using bisectors

After you bisect, check with a protractor: measure the whole angle, then each half. The halves should be equal, and each should be half of the whole. If your measurements differ by 1° or 2° that is acceptable, but if they differ by more, check that your two small arcs used the same opening.

Which bisector is correct? Sometimes you are shown a drawing where a line has been drawn inside an angle and asked whether it is a true bisector. Measure both sides. A real bisector gives two equal angles. A line that gives 30° and 54° in an 84° angle is not a bisector, even if it looks close.

You can bisect an angle more than once. Bisect 80° and you get two 40° angles. Bisect a 40° angle and you get 20°. So bisecting twice gives a quarter of the original angle. Constructions of 45°, 30° and 15° all depend on repeated bisecting, as you will see in the next lessons.

Worked examples

Try each one yourself first, then open the solution and compare every step.