Congruence and similarity

Congruence

Congruent means equal. Two triangles are said to be congruent to each other if they are exactly the same shape and size.

They might look different if one is upside down or turned in some other way, though. So how can you tell whether two triangles are congruent?

Conceptually, it’s easy: trace one of the triangles with tracing paper. If you can put the traced triangle directly onto the other one, and they fit exactly, then the triangles are congruent. If you can’t, then they aren’t.

But there’s a problem with this method: what if you trace a triangle like I said, and you line them up, and it’s really close? Maybe they are congruent and your drawing is a little off… or maybe your drawing is fine, but the triangles aren’t quite congruent to each other?

It turns out that you can argue about stuff like this all day. We need a better way: a way that will stop confusion and arguments before they have a chance to get started.

Good news: we have one. The better way is something called the triangle congruence theorems.

The basic idea is that if you can show that two triangles have certain things in common, then you are allowed to say that the triangles are congruent to each other. Here’s an example:

• Look at one triangle. Now look at the lengths of its sides. Then look at a second triangle’s sides. If the lengths match up, in the same order around the triangle, then the triangles are congruent.

Let’s get a little more technical, just to make sure there’s no misunderstanding.

Imagine two triangles. One has side lengths a, b, and c (labelled clockwise around the triangle). The other has side lengths x, y, and z, also labelled clockwise.

Then, to restate the theorem above:

• The two triangles are congruent if
  • side a is congruent to side x and
  • side b is congruent to side y and
  • side c is congruent to side z.

This is known as the side-side-side congruence theorem. Or just SSS for short.

There are a few other useful related theorems: namely SAS and ASA. Let’s go through those too:

SAS says that if you have two triangles labelled as in the last example, then:

• The two triangles are congruent if
  • side a is congruent to side x and
  • side b is congruent to side y and
  • the angle between sides a and b has the same measurement as the angle between sides x and y.

ASA says that if you have two triangles labelled as in the last example, then:

• The two triangles are congruent if
  • the angle between sides a and b has the same measurement as the angle between sides x and y and
  • the angle between sides b and c has the same measurement as the angle between sides y and z and
  • side b is congruent to side y.

Okay, so, to review, you have a few different ways to prove that two triangles are congruent to each other.

• SSS: show that all three pairs of sides are congruent to each other
• SAS: show that two pairs of sides and the angles between them are all congruent
• ASA: show that two pairs of angles and the sides between them are all congruent

There’s also a theorem called HL, which stands for Hypotenuse Leg. As you might imagine, it says that if you have two right triangles (it only works on right triangles), and their hypotenuses are congruent, and one other pair of corresponding sides are congruent, then the triangles are congruent.

Okay, this is all fine, but what good is congruence?

Well, it turns out that solving problems often means figuring out that two line segments are the same length, or finding that two angles are equal to each other. In other words, figuring out that two things are congruent to each other. And usually the problem is set up in a way that makes it hard to guess whether the things are congruent to each other.

But, on the bright side, the things that might or might not be congruent to each other are either parts of triangles… or else you can add in some lines so that they become parts of triangles. And then you can often show that the triangles (and thus their parts) are congruent to each other.

And then you can solve the problem.

Similarity

Two triangles are similar if they have exactly the same shape, but different sizes.

This means that their corresponding angles are congruent, but their corresponding sides are not.

Instead, their corresponding sides are in some fixed ratio to each other. In other words, one triangle will be a certain factor bigger than the other.

For example, if you make a triangle, then make a second one with the same shape but twice as big, then the two triangles will be similar to each other. Their ratio will be 1:2, which means that each of the sides of the bigger triangle will be exactly twice as long as one of the sides of the smaller triangle. We sometimes say these triangles have a scale factor of 1:2, or that the bigger triangle has a scale factor of 2 relative to the smaller one.