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7 Geometry
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Australia
Year 8

7.03 Testing for triangle congruence

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Introduction

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".

Two triangles with the same markings on the equal sides and equal angles.

Here is an example of two congruent triangles.

Three pairs of matching equal sides, and three pairs of matching equal angles.

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 the following combinations.

Test for triangle congruence

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 image shows how congruent triangles can be formed by 3 lines. Ask your teacher for more information.

Each triangle is made from the same three sides, so they are all congruent.

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 image shows that 2 sides and 1 included angle form only 1 possible triangle. Ask your teacher for more information.

After fixing the two given sides about the given angle, there is only one possible triangle.

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.

Two triangles with 2 same corresponding sides and 1 same corresponding angles but with different third corresponding side.

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

Exploration

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:

Loading interactive...

In order for the two triangles to be congruent, the third corresponding side must be equal. This means that SSS and SAS congruence hold, but not SSA.

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:

This image shows how congruent triangles can be formed by a hypotenuse, sides, and a right angle.

There is only one possible triangle that lines up the hypotenuse with the given side at a right angle.

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:

This image shows 1 side and 2 angles form 1 possible triangle. Ask your teacher for more information.

The projected lines meet at exactly one point, so we can only build one triangle from this information.

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 image shows AAS congruence. Ask your teacher for more information.

This kind of congruence is called angle-angle-side congruence, or AAS. To use AAS to show that two triangles are congruent, the matching sides must have the same position relative to the matching angles.

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.

Examples

Example 1

Consider the following:

a

Which two of the following triangles are congruent?

A
A triangle with angles of 56 and 44 degreees and sides of 5 and 9 units.
B
A triangle with a right angle and sides of 5 and 9 units. The side with 9 units is the hypotenuse.
C
A triangle with angles of 56 and 34 degrees and sides of 5 and 9 units, where 9 is between the given two angles.
D
A triangle with a right angle and sides of 5 and 9 units on either side of the right angle.
Worked Solution
Create a strategy

Determine if two of the triangles satisfy a congruence test.

Apply the idea

For option A, the triangle has two given angles of 56\degree and 44\degree. Since the angle sum of a triangle is 180\degree, the missing angle in this triangle will be 80\degree. And we do not have any other triangles that have the same angles. So there is no triangle from the other options that is congruent to the triangle in option A.

Options B and D looks the same since both have a right angle and sides with lengths 5 and 9. In option B, the side with 9 units is opposite the right angle but in option D it is not so they are not congruent.

The triangle in option C has two given angles of 56\degree and 34\degree. Since the angle sum of a triangle is 180\degree, the missing angle in this triangle will be 90\degree. The triangle also have sides with lengths 5 and 9. The side with 9 units is opposite the right angle.

The triangles in options B and C have the same angles and the same side lengths in the same positions. So the triangles in options B and C are congruent.

b

What congruence test does this pair satisfy?

A
SSS
B
SAS
C
AAS
D
RHS
Worked Solution
Create a strategy

Take note of what the triangles share in common.

Apply the idea

We have two triangles that both have right angles, equal hypotenuses, and another equal side.

So the two triangles are congruent by right angle-hypotenuse-side, or RHS.

Option D is the correct answer.

Example 2

Consider the following diagram:

Quadrilateral A B C D with diagonal B D and sides C D and A B parrallel and congruent with each other.
a

Are the triangles \triangle ABD and \triangle CDB definitely congruent?

Worked Solution
Create a strategy

Take note of the markings used for congruence and determine if any of the congruence tests are satisfied.

Apply the idea

Looking at the triangles \triangle ABD and \triangle CDB in the diagram, we have one pair of common sides, which tells that those two sides are equal, BD = DB.

Based on the given markings in the diagram, we also have sides CD = AB.

The parallel line marks on sides CD and AB also give us a pair of equal angles which are angles \angle ABD and \angle CDB, since they are alternate interior angles on parallel lines.

So we have enough information to definitely say that triangles \triangle ABD and \triangle CDB are equal.

So option A is the correct answer.

b

What congruence test does this pair satisfy?

A
SSS
B
SAS
C
AAS
D
RHS
Worked Solution
Create a strategy

Take note of what the triangles share in common.

Apply the idea

The triangles \triangle ABD and \triangle CDB have a common side DB = BD, another pair of equal sides CD = AB, and a pair of equal angles \angle CDB = \angle ABD which are between the equal sides.

This gives us side-angle-side congruence or SAS.

Option B is the correct answer.

c

Select the three statements that, when put together, establish congruence for this test.

Make sure each reason is correct as well.

A
\angle ADB = \angle CBD \, (Alternate angles on parallel lines)
B
AD = CB \, (Corresponding sides on parallel lines)
C
\angle ABD = \angle CDB \, (Corresponding angles on parallel lines)
D
AB = CD \, Given
E
BD \, is common
F
\angle ABD = \angle CDB \, (Alternate angles on parallel lines)
Worked Solution
Apply the idea

We have previously established that:

  • BD is a common side.

  • AB=CD from the same markings on the diagram.

  • \angle ABD= \angle CDB since they are alternate interior angles on parallel lines.

So the three correct statements are options D, E, and F.

Idea summary

These are the four congruence tests in proving congruence in triangles:

  • Side-side-side, or SSS: Three pairs of equal sides.

  • Side-angle-side, or SAS: Two pairs of equal sides with an equal included angle

  • Angle-angle-side, or AAS: Two pairs of equal angles and one pair of equal sides

  • Right angle-hypotenuse-side, or RHS: Both have right angles, equal hypotenuses, and another equal side

Outcomes

ACMMG200

Define congruence of plane shapes using transformations

ACMMG201

Develop the conditions for congruence of triangles

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