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.
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 |
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.
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.
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.
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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. |
Divide a given angle into four equal parts
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Three equal sides lock a triangle into one shape, so its angles are fixed too. This is the SSS condition.
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