# Congruent triangles

Level: GCSE Maths

Two triangles are congruent if they are the same size and the same shape. This means that all their corresponding sides and angles must be equal.

## Rules for congruent triangles

In the diagram, A, B and C are all congruent. B is the same size and shape as A, but rotated. C is the same as A but reflected.

For triangles, it is not always necessary to prove that every side and angle is the same. There are several shortcuts you can use. Two triangles are congruent if:

• If all three sides match (SSS)
• Two angles and a corresponding side match (AAS)
• Two sides and the angle between them match (SAS)
• The triangle is a right angled triangle and the hypotenuse and one other side match (RHS)

## Three sides match (SSS)

If if two triangles have three sides of matching length, then those triangles are congruent.

To understand this, look at the diagram above. With three lines of length a, b and c, there is only one way to make a triangle. You can try this, for example, using three pencils of different lengths - there is only one triangle you can make. It is impossible to create a triangle with the same lengths but different angles, so all triangles with three matching sides are congruent.

## Two angles and a corresponding side match (AAS)

If two angles match and a corresponding side match, the triangles are congruent.

In the diagram, we have started to create a triangle using a line of length a, and two angles of x and y. As you can see, once we have drawn the line and the angles, there is only one way to complete the triangle. So all triangles with a given a, x and y are congruent:

In the case above, we know a side and the two angles on that same side. The rule also applies for two angle and one of the other sides:

But the rule only applies for corresponding sides, that is the equivalent side in both triangles. In the example below, the triangles are not congruent. Both triangles have a side of length b, but it is not the equivalent side:

## Two sides and the angle between them match (SAS)

If two sides and the angle between them match, the triangles are congruent:

As the diagram shows, if you draw two lines of length a and b, with an angle x between them, there is only one possible way to complete the triangle.

Notice that is two sides and a different angle match, there will usually be two different ways to draw a triangle, so this does does not prove that they are congruent. Here is an example:

Even though both triangles have the same sides b and c, and the same angles y, they are not congruent.

## The hypotenuse and one other side match in a right angled triangle (RHS)

If two right angled triangles have a matching hypotenuse length h and a matching side length a, then they are congruent.

Looking at the diagram, given a side of length a at right angles to the base, there is only one way to place a line of length h so that it exactly meets the base. So any triangles with these lengths will be congruent.

One way to understand this is by the Pythagorean theorem. For a right angled triangle with hypotenuse h, and sides a and b the following is true:

${\displaystyle a^{2}+b^{2}=h^{2}}$

This equation tells us that is we know the length of any two sides of a right angled triangle, we can calculate the third. In other words, any two right angle triangles that have matching h and a must also have matching b. In other words, if two sides match then the third side must also match. But we know from the SSS rule that any triangles with three matching sides are congruent.