Middle Years

Lesson

Two triangles are** **congruent if one can be moved (by translation, rotation, and/or reflection) to lie exactly on top of the other. It is a more precise way of saying that two triangles are "the same".

Here is an example of two congruent triangles:

If we reflected one of these triangles, rotated it, and translated it, we could place it directly on top of the other.

When deciding whether or not two triangles are congruent, we don't need to know six pieces of information (three sides and three angles), we need only three pieces in one of these combinations:

Side-side-side congruence (SSS)

If two triangles have three equal side lengths, then the triangles must be congruent. Try this yourself with three straight objects - once you put them together, you can rotate, translate, and reflect the triangle to make every other possible combination:

This kind of congruence is called side-side-side, or SSS.

Side-angle-side congruence (SAS)

If two triangles have a pair of matching sides and the angles between them are equal, then the triangles must be congruent. Try this yourself with two straight objects - if you hold them together at one end and form an angle, there is only one triangle you can form by joining the ends together:

This kind of congruence is called side-angle-side, or SAS. We write this test with the "A" in between the two "S"s, because the angle must be between the matching sides - the long name for this kind of congruence is "two sides and the included angle".

There is no SSA!

It is possible for triangles to have two pairs of equal sides and a pair of matching angles, yet not be congruent overall. Here is an example:

This can only happen when the pair of equal angles is not included between the sides.

Try using this applet to find the two different triangles that have two matching angles and a matching non-included angle, just like the picture above:

Right angle-hypotenuse-side congruence (RHS)

If two right-angled triangles have equal lengths hypotenuses and another pair of equal sides, then the triangles must be congruent:

Notice that the right angle is not included between the sides - this is the only exception to the general rule, which is why we mention it as a separate case. This congruence test is called right angle-hypotenuse-side, or RHS.

Angle-angle-side congruence (AAS)

What if we are only given one pair of equal sides? In this case we need two pairs of equal angles. Here is the construction if the two angles are made with the given side:

If one of the given angles is opposite the given side, we can always find the third one by using the angle sum of a triangle:

This kind of congruence is called angle-angle-side congruence, or AAS.

We need a matching side for AAS

To use AAS to show that two triangles are congruent, the matching sides must have the same position relative to the matching angles.

The three triangles below have two pairs of equal angles and a pair of equal sides, but because the side of length $7$7 is in a different position, none of them are congruent:

Proving congruence in triangles

These are the four congruence tests:

- SSS: Three pairs of equal sides
- SAS: Two pairs of equal sides with an equal included angle
- AAS: Two pairs of equal angles and one pair of equal sides
- RHS: Both have right angles, equal hypotenuses, and another equal side

If two triangles don't satisfy these tests, one of two things could be true:

- The two triangles are definitely not congruent, or
- We don't have enough information to know whether or not they're congruent

Two triangles are definitely not congruent if there is a pair of sides or angles that are in the same relative position but are not equal to each other.

Are these triangles congruent?

**Think:** Both have matching sides with lengths $9$9 and $5$5. We can use the angle sum of a triangle to find the missing angle in the first triangle.

**Do:** In the first triangle, we subtract the known angles to find the missing angle: $180^\circ-56^\circ-44^\circ=80^\circ$180°−56°−44°=80°. We now know that the first triangle doesn't have a right angle, so it cannot be congruent to the second triangle.

Are these triangles congruent?

**Think:** Both have matching sides with lengths $7$7 and $6$6. We don't know anything about the angles in the second triangle. They may look congruent, but that isn't enough - either we know for sure, or we don't.

**Do:** There is no way to calculate any additional side or angle information in the second triangle, and we need at least three matching pairs to make conclusions about congruence. We don't know whether or not they are congruent.

Consider the two triangles in the diagram below:

Are $\triangle PQR$△

`P``Q``R`and $\triangle STU$△`S``T``U`congruent?Yes, they satisfy SSS.

AYes, they satisfy SAS.

BYes, they satisfy AAS.

CYes, they satisfy RHS.

DNo, they are definitely not congruent.

EUnknown, there is not enough information.

FYes, they satisfy SSS.

AYes, they satisfy SAS.

BYes, they satisfy AAS.

CYes, they satisfy RHS.

DNo, they are definitely not congruent.

EUnknown, there is not enough information.

F

Consider the following:

Which two of the following triangles are congruent?

ABCDABCDWhat congruence test does this pair satisfy?

SSS

ASAS

BAAS

CRHS

DSSS

ASAS

BAAS

CRHS

D

Consider the following diagram:

Are the triangles $\triangle ABD$△

`A``B``D`and $\triangle CDB$△`C``D``B`definitely congruent?Yes

ANo

BYes

ANo

BWhat congruence test does this pair satisfy?

SSS

ASAS

BAAS

CRHS

DSSS

ASAS

BAAS

CRHS

DSelect the three correct reasons that establish congruence for this test.

$\angle ADB=\angle CBD$∠

`A``D``B`=∠`C``B``D`

Alternate angles on parallel linesA$AD=CB$

`A``D`=`C``B`

Corresponding sides on parallel linesB$\angle ABD=\angle CDB$∠

`A``B``D`=∠`C``D``B`

Corresponding angles on parallel linesC$AB=CD$

`A``B`=`C``D`

GivenD$BD$

`B``D`is commonE$\angle ABD=\angle CDB$∠

`A``B``D`=∠`C``D``B`

Alternate angles on parallel linesF$\angle ADB=\angle CBD$∠

`A``D``B`=∠`C``B``D`

Alternate angles on parallel linesA$AD=CB$

`A``D`=`C``B`

Corresponding sides on parallel linesB$\angle ABD=\angle CDB$∠

`A``B``D`=∠`C``D``B`

Corresponding angles on parallel linesC$AB=CD$

`A``B`=`C``D`

GivenD$BD$

`B``D`is commonE$\angle ABD=\angle CDB$∠

`A``B``D`=∠`C``D``B`

Alternate angles on parallel linesF