Angles opposite equal sides are equal

0:00So far, congruence has been a test. Somebody hands you two triangles, you check a few measurements, and you decide. That is useful. But it is not what congruence is really for. Here is the turn. You can use it to prove something about a single triangle. Something you were never told, and never measured. But a congruence needs two triangles, and there is only going to be one on the board.

0:29So the move is going to be audacious. If you need a second triangle, make one. Draw a line, and cut the one you have in half. Here is the triangle. A at the top, B at the bottom left, C at the bottom right. And here is everything you are told about it. The side A B and the side A C are the same length. The tick marks say so.

0:54That is what isosceles means. Two sides of equal length. And the angle at A, sitting between those two equal sides, is eighty degrees. That is all of it. Two equal sides, and one angle. Now look down at the two angles at the bottom, at B and at C. They look equal. The question is whether you can prove it, without measuring either one. There is one triangle here, and a congruence needs two. So make the second one.

1:28From the apex A, drop a line down to the side B C. Not at any old angle. Drop it so that it meets B C square on, at a right angle. Call the point where it lands D. And now count the triangles. There are two of them. A D B on the left. A D C on the right. Neither of those was given to you. You drew the line that made them.

1:54That is the whole trick, and it is one of the most useful ideas in geometry. So what did that one line actually hand you? Two right angles, one on each side of D. That is what dropping it square on means. And a shared side. The line A D belongs to the left triangle and to the right one. It is the same segment, so it is the same length in both. That costs nothing to say.

2:25Notice also that A D is new. It is not one of the sides you started with. It arrived because you drew it, and now it is doing work in both halves. So each half has a right angle, and one side that certainly matches. You need a third thing. And you already have it. Line them up. Three matching parts, one at a time. First. A B equals A C. That was given to you, right at the start.

2:55Second. The angle at D is ninety degrees on the left, and ninety on the right. That one is not a measurement either. It is exactly what the construction was for. Third. A D is the same segment in both halves, so of course it matches itself. A pair of equal sides. A pair of right angles. And one shared side. Three matching parts, in two right-angled triangles. So which condition is that?

3:28Look at where the right angle sits, and then at what A B is doing. In the left half, the right angle is down at D. And A B does not touch D at all. So A B faces the right angle. A B is that half's hypotenuse. Same on the right. A C faces the right angle at D, so A C is the other hypotenuse. And those two were equal to each other from the very start.

3:56So we have a right angle, a hypotenuse, and one more side. That is R H S. The two halves are congruent. Not because they look it. Because three named parts matched. Now write that congruence down carefully, because the order carries the answer. A pairs with A. D pairs with D. And B pairs with C. Triangle A D B is congruent to triangle A D C. Now read off the last letter of each name. B sits opposite C.

4:31So the angle at B and the angle at C are corresponding parts. And corresponding parts of congruent triangles are equal. The angle at B equals the angle at C. Nobody measured either one of them. Now pause on something we did not do, because it is easy to get wrong. Look at the two pieces the line cut the base into. B D on the left, D C on the right.

5:01It is very tempting to say those two are equal, and finish in one line. Because then all three sides of one half would match all three of the other. But stop. Where would that have come from? Nobody told you it. It is not among the things you were handed. All you were told about D is that A D meets the base square on. Assume anything more than that, and you have quietly assumed most of the answer.

5:30Now one more check, and it is the important one. Go back through everything we just did, and look for the number eighty. It is not there. Not once. We used two equal sides. We used a right angle we made ourselves. We used a shared line. The size of the angle at the top never entered the argument at all. So change it. Make it thirty. Make it a hundred and forty.

5:58The same line drops, the same two right angles appear, the same condition fires. In any triangle whatever, the angles facing two equal sides are equal to each other. Now put the eighty back, because it does have a job to do. It lets us actually find those two angles. The three angles of any triangle come to a hundred and eighty. Eighty is already spoken for at the top. That leaves a hundred to share out.

6:31And we have just proved that the two at the bottom are equal to each other. So they split that hundred evenly. Fifty each. Fifty at B, fifty at C, eighty at A. And those three do come to a hundred and eighty. One of those numbers came from the picture. The other two came from the proof. Here is the same thing wearing a disguise. A circle, with its centre at A. Two points, B and C, sitting out on the rim.

7:02Draw A B. Draw A C. And join B across to C. Now, is that triangle isosceles? Nothing is tick-marked. But A B and A C both run from the centre straight out to the rim. That is what a radius is. And every radius of a circle is the same length. So A B equals A C, and everything we proved applies. Here the angle at A is a hundred and twenty.

7:30A hundred and eighty minus a hundred and twenty leaves sixty, and half of sixty is thirty each. So what should you actually take away from this? Not the fact. The move. When there is nothing to compare, draw the thing that makes a comparison possible. Here is another figure it works on. Four corners. A at the top, B on the left, C at the bottom, D on the right.

7:58You are told that A B equals A D, and that C B equals C D. Nothing else at all. Draw the line from A across to C, and now two triangles share it. Three matching sides each, so they are congruent, so that line cuts the angle at A into two equal pieces. One drawn line, and a fact nobody gave you. That is the whole idea.