Congruent Triangles

Two triangles are congruent if they have three pairs of equal side lengths and three pairs of equal angles.

Congruent

Things that are congruent look exactly the same. We say that a shape is congruent to another shape if it can be turned into the other shape by using one, two or all three of sliding, flipping and turning. Let's see what each of these moves does to a shape.

Slides, Flips and Turns

Congruent Triangles

Two triangles are congruent if they have three pairs of equal side lengths and three pairs of equal angles.

The sides and angles might appear to be in different positions if a turn or a flip has been used to turn one triangle into the other, but they will all be equal.

Triangles with All Sides Equal

If two triangles have three pairs of equal sides, then they are automatically congruent.

For example,

because they have three pairs of equal sides, but because the yellow triangle has a side of length \(12\) in place of the side of length \(8\).

Triangles with All Pairs of Angles Equal

If two triangles have three pairs of equal angles, they may or may not be congruent. It depends on whether they have the same size.

For example,

because they have the same size, but even though they have three pairs of equal angles, because the pink triangle is bigger than the green triangle.

If two triangles have three pairs of equal angles, then they are similar, but they may not be congruent.

Other Tests for Congruent Triangles

There are a number of other tests for congruent triangles. For example, two right angled triangles are congruent if their hypotenuses are equal, and one other pair of sides is equal. Another test is to show that they have two pairs of sides and the pair fo included angles equal.

You can read more about these tests in the article on proving that triangles are congruent.

How Do We Indicate That Two Triangles are Congruent?

We use markings to show which pairs of sides are equal and which pairs of angles are equal. For example,

In the above picture, we've shown the pairs of equal angles using different numbers of arcs: the angles with single arcs are equal, the angles marked with double arcs are equal, and the angles marked with triple arcs are equal. We've also marked the equal sides using different symbols: one pair of equal sides has single lines, another has double lines, and a third has a curvy symbol.

Of course there are many other ways you can show that two triangles are congruent. In symbols, we write \(ABC \equiv DEF\) to say that triangles \(ABC\) and \(DEF\) are congruent. It's usually a good idea to put the vertices in the order corresponding to the pairs of equal angles. We'll talk more about that in the article on proving that triangles are congruent.