The idea

You can make a perfect copy of an angle without ever finding out what it measures, and you can cut any angle exactly in half the same way. Both tricks rest on one fact: if two triangles have three matching sides, they are identical in every other way as well, including their angles.

Why three equal sides is enough

Take three sticks of fixed lengths and try to build a triangle from them. You will find there is only one triangle you can build. You cannot squash it or stretch it into a different shape while keeping the same three lengths, the way you can with a four sided frame. A rectangle made of four sticks flops into a parallelogram if you push it, but a triangle made of three sticks does not move at all.

The rule this gives you

Because the shape is locked, the angles are locked too. So if you can force three sides to match, the angles match automatically and you never have to measure them. This is the whole engine behind copying an angle, behind bisecting, and behind the triangle constructions in the next unit. Mathematicians call it the side side side condition, usually shortened to SSS.

Copying an angle, step by step

Step	What you do	Which side you are matching
1	Arc from the corner B, crossing both arms at P and Q	Fixes BP and BQ
2	Draw a new ray. Same radius, arc from its end Y, crossing at M	Forces YM = BP
3	Set the compass to the straight gap PQ	Captures the third side
4	From M, cut the new arc at N	Forces MN = PQ
5	Join YN	Three sides matched, so the angles must match

Bisecting is the same trick with a symmetry

When you bisect, you draw an arc across both arms to get P and Q, then swing equal arcs from P and from Q until they cross at X. Now look at the two triangles OPX and OQX. OP equals OQ because they are the same radius. PX equals QX because you used the same width from both. And OX is shared by both triangles. Three matching sides again, so the two triangles are identical, so the two angles at O are equal. The line OX splits the angle in half, and the proof is the same one sentence as before.

The width that must be big enough

When you swing the two arcs from P and from Q, the compass width has to be more than half of the distance PQ. If it is less, the two arcs never reach each other and there is no crossing point to join to. Students often set the compass small to be neat and then wonder why nothing meets. Open it wider than half of PQ and the arcs cross cleanly. Opening it much wider is fine, and in fact makes the crossing point sharper and easier to see.

Where you will meet this again

Bisecting is not only for angles. The same two arc idea bisects a line segment, and the crossing line there turns out to be perpendicular to it. That gives you a 90 degree angle without going near the 60 and 120 rays, which is a second and often quicker route to a right angle. You will use segment bisection constantly in the next unit when you construct triangles, because it is how you find the exact middle of a side.

Worked Example 1

Construct an angle equal to a given angle ABC

Step	What you do
1	With B as centre and any radius, draw an arc cutting BA at P and BC at Q.
2	Draw a ray YZ. With Y as centre and the same radius, draw an arc cutting YZ at M.
3	Set the compass to PQ. With M as centre and this width, cut the arc at N.
4	Join YN. Then angle NYZ = angle ABC.

Reason to write downIn triangles BPQ and YMN: BP = YM (same radius), BQ = YN (same radius), PQ = MN (copied). So the triangles are congruent by SSS, and therefore angle B = angle Y.

Worked Example 2

Divide a given angle into four equal parts

Think firstYou cannot cut into four in one go. But four is two twos. Bisect once to get two halves, then bisect each half. Two rounds of bisecting, three new rays in total.

StepsBisect angle AOB with ray OX. Now bisect angle AOX with ray OY, and bisect angle XOB with ray OZ. The four angles AOY, YOX, XOZ and ZOB are all equal.

The follow up questionCan you divide an angle into eight equal parts? Yes, three rounds of bisecting. Into sixteen? Yes, four rounds. Into three? No, and no amount of bisecting will ever get you there, because halving only ever produces halves, quarters, eighths and so on.

Activity 1.2

Copy an angle a partner drew

MethodWork in pairs. One of you draws any angle at all on a slip of paper, without measuring it and without telling the other what it is. Swap slips. Copy your partner angle onto your own page using compass and ruler. Then, and only then, both of you measure both angles with a protractor and compare.

Why the secrecy mattersIf you know the angle is 50 degrees you will be tempted to construct 50 degrees rather than copy it, and the whole point is lost. Not knowing forces you to trust the method. That is exactly the situation a carpenter is in with a roof angle that is not a round number.

Worked Example

One pair did this and their two readings came out three degrees apart. Rather than shrugging, they worked out why. Here is what they wrote.

Reflection

We had assumed that if the answer was wrong then the method was wrong. It was not. The method is exact and our hands were not, and those are completely different problems with completely different fixes. Sharpening the pencil fixed two degrees of it. Our teacher said this is worth knowing for every practical subject, not just maths.

Your TurnDo the pair activity, then fill this in before you blame the method.

Check	Yes or no, and what you found
Same radius for both arcs?	-
Same PQ width when you stepped it across?	-
Pencil sharp?	-
How far apart were the two readings?	-

Three equal sides lock a triangle into one shape, so its angles are fixed too. This is the SSS condition.

A four sided frame can be squashed into a different shape; a triangle cannot. That is why braces are diagonal.

To copy an angle: arc across both arms, same arc on the new ray, then step the gap PQ across.

The straight distance PQ is what carries the size of the angle. The radius does not, because it is the same on both.

To bisect: arc across both arms at P and Q, then equal arcs from P and Q meeting at X. Join OX.

When bisecting, the compass width must be more than half of PQ or the arcs will not meet.

Bisection works because OP = OQ, PX = QX and OX is shared, giving two congruent triangles.

Neither construction ever needs the angle to be measured, which is the point of both.

Bisecting twice divides an angle into four; three times into eight. Never into three.

The same two arc idea bisects a line segment, and there the crossing line is perpendicular to it.

Copying is more accurate than measuring, because measuring adds a fresh error every single time.

A blunt pencil widens the compass by about a millimetre and is a common cause of a two degree error.

1State the SSS condition and explain in your own words why it is true.

The side side side condition says that if the three sides of one triangle are equal to the three sides of another, the two triangles are congruent, meaning identical in every way including all three angles. It is true because three fixed lengths can be assembled into only one triangle. Try it with three sticks: once they are joined, the shape cannot be pushed into any other form. A four sided frame of fixed lengths, by contrast, flops freely between many shapes, which is exactly why gates and shelves are strengthened with a diagonal that turns each half into a triangle.

2Describe how to copy an angle, and say which single measurement carries the size of the angle across.

With the corner as centre and any radius, draw an arc crossing both arms at P and Q. Draw a new ray and, with the same radius, draw an arc from its end crossing at M. Set the compass to the straight distance PQ and, from M, cut the new arc at N. Join the end of the ray to N. The measurement that carries the size across is PQ. The radius cannot carry it, because you deliberately used the same radius on both diagrams, so it holds no information about this particular angle. PQ is the only quantity that changes when the angle changes, and it is the third side that makes the two triangles congruent.

3Prove that the bisection construction really does cut the angle in half.

Let the angle be at O, with the first arc crossing the arms at P and Q, and the two later arcs crossing at X. In triangles OPX and OQX: OP equals OQ because both are radii of the first arc; PX equals QX because the same compass width was used from P and from Q; and OX is common to both triangles. So the two triangles are congruent by SSS. Congruent triangles have all corresponding angles equal, so angle POX equals angle QOX. Those two angles together make up the whole original angle, and they are equal, so each is exactly half of it.

4A student bisects an angle but the two arcs never meet. What went wrong and how is it fixed?

The compass width used for the two arcs was less than half of the distance PQ. Two circles drawn with radius r, centred at two points that are PQ apart, only overlap when r is more than half of PQ; below that they sit apart and there is no crossing point at all. The fix is simply to open the compass wider before drawing those two arcs. Opening it a good deal wider is not a problem and is in fact better, because the arcs then cross at a steeper angle and the crossing point is much easier to see and to join to accurately.

5How would you divide an angle into four equal parts? Into eight? Why not into three?

Into four: bisect the angle once to get two halves, then bisect each of those halves. That is two rounds of bisecting and three new rays. Into eight: bisect once more, three rounds in total. The pattern is that each round of bisecting doubles the number of parts, so you can reach 2, 4, 8, 16 and so on. Three is not on that list and never will be, because doubling from one only ever produces powers of two. Dividing an angle into three equal parts is called trisection, and it has been proved impossible with only a compass and a straight edge.

6Why do carpenters copy an angle from a pattern instead of measuring it on each piece?

Because measuring introduces a fresh error every single time, and those errors add up. If a roof needs twenty rafters and each one is measured separately, twenty small mistakes accumulate and the last rafter may be noticeably out. If instead the first rafter is cut carefully and then used as the pattern for all the others, every piece carries the same single error rather than twenty different ones, so the pieces at least match each other. This is exactly the compass copying method carried out in wood, and the same reasoning explains why a tailor keeps a paper pattern rather than re-measuring the cloth each time.

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The video for this topic is being prepared. Check back soon!

Unit 1 · Lines and Angles

Copying and Bisecting Angles

Mathematics · Grade 7

What You Will Be Able To Do

📋

Copy any angle onto a new ray without measuring it

✂️

Cut any angle exactly in half, and into four or eight

🔺

State the SSS condition and use it to justify both constructions

🔧

Explain why a triangle is rigid and a four sided frame is not

“

Three fixed lengths can be assembled into only one triangle. Because the shape is locked, the angles are locked too. So if you can force three sides to match, the angles match by themselves and you never have to measure them.

Copying an Angle

1Arc from the corner B, crossing both arms at P and Q

2New ray from Y. Same radius, arc crossing at M

3Set the compass to the straight gap PQ. This is the third side

4From M, cut the new arc at N

5Join YN. Three sides matched, so the angles match

Two Constructions, One Reason

📋 Copying

Two radii equal, because the same width was used twice

Third side equal, because PQ was stepped across

So the two angles are equal

✂️ Bisecting

OP = OQ, because both are the first radius

PX = QX, because the same width was swung from both

OX is shared, so the two angles at O are equal

Both are SSS wearing different clothes

How Many Parts Can You Cut It Into?

2️⃣

Two. One bisection

4️⃣

Four. Bisect, then bisect each half

8️⃣

Eight. Three rounds. Then sixteen, then thirty two

3️⃣

Three. Never. Halving only ever gives powers of two

“

A carpenter cutting twenty rafters does not measure the angle twenty times. The first rafter becomes the pattern and every other one is copied from it, because measuring adds a fresh error every single time.

PAIR WORK · 20 MIN

Copy an Angle You Are Not Allowed to Know

DRAW · 3 MIN

Each student draws any angle on a slip, without measuring it and without saying what it is

SWAP · 12 MIN

Exchange slips. Copy your partner angle onto your own page with compass and ruler only

CHECK · 5 MIN

Only now bring out the protractor. Measure both and compare. If they differ, find out why before blaming the method

Bisecting, and Why It Is Exact

1Arc from O across both arms, crossing at P and Q

2Equal arcs from P and from Q, meeting at X

3Join OX. The angle is now in two equal halves

4OP = OQ, PX = QX, and OX is shared by both triangles

5Three matching sides, so the triangles are congruent and the angles equal

Presenter notes: Start by drawing an angle on the board and refusing to say what it is. Tell the class you want an exact copy of it on the far side of the board, and that nobody is allowed to measure anything. Let them suggest methods for a minute. Somebody will propose measuring, and you decline. Somebody may propose tracing, which is closer to the truth than they realise. Then say that the compass can do this, and that by the end of the lesson they will also know why it works, which is a single sentence about triangles.