Congruent triangles and the tests SSS, SAS, ASA · Mathematics 1st Year

What does congruent mean?

Two triangles are congruent if one fits exactly on top of the other. They have the same size and the same shape. You may have to turn one of them over or rotate it to make it fit.

When two triangles are congruent, the parts that match up are called corresponding parts. Corresponding sides are equal in length. Corresponding angles are equal in size. In the diagram, $A$ matches $D$, $B$ matches $E$ and $C$ matches $F$, so we write

$$\triangle ABC \equiv \triangle DEF$$

The symbol $\equiv$ means "is congruent to". The order of the letters matters: it tells you which vertices match.

Checking all six measurements (three sides and three angles) would work, but it is slow. Luckily, three well-chosen facts are enough. These are the three tests for congruence, and they are the next three sections. In geometry, an axiom is a basic fact that we accept without proof, and we build proofs on axioms. In the Junior Cycle course, SSS, SAS and ASA are accepted as axioms of congruence (Axiom 4), so you can use them as reasons in your proofs.

Practice★☆☆

State what it means for two triangles to be congruent.

Solve

Test 1: SSS (three sides)

Imagine building a triangle from three sticks of length 5 cm, 7 cm and 9 cm. However you join them, you can only make one triangle shape. The sticks lock the angles in place.

SSS stands for Side, Side, Side. Compare the three sides in matching order: shortest with shortest, middle with middle, longest with longest.

Practice★☆☆

Triangle $ABC$ has $AB = 5$ cm, $BC = 7$ cm and $CA = 9$ cm. Triangle $DEF$ has $DE = 5$ cm, $EF = 7$ cm and $FD = 9$ cm. Are the triangles congruent? Name the test.

Solve

Test 2: SAS (two sides and the angle between them)

Now use two sticks of 6 cm and 8 cm, joined at a hinge. If you fix the hinge at $50^\circ$, the third side is decided. Only one triangle is possible.

SAS stands for Side, Angle, Side. The angle must be the included angle: the one formed by the two given sides, with the angle between them in the name.

Practice★☆☆

In triangles $PQR$ and $XYZ$: $PQ = XY = 6$ cm, $\angle Q = \angle Y = 50^\circ$ and $QR = YZ = 8$ cm. Name the test showing they are congruent and list the pairs of…

In triangles $PQR$ and $XYZ$: $PQ = XY = 6$ cm, $\angle Q = \angle Y = 50^\circ$ and $QR = YZ = 8$ cm. Name the test showing they are congruent and list the pairs of corresponding vertices.

Solve

Practice★★☆

Two triangles have sides $4$ cm, $6$ cm and an angle of $30^\circ$. In the first triangle the $30^\circ$ angle is between the $4$ cm and $6$ cm sides. In the second it…

Two triangles have sides $4$ cm, $6$ cm and an angle of $30^\circ$. In the first triangle the $30^\circ$ angle is between the $4$ cm and $6$ cm sides. In the second it is opposite the $4$ cm side. Explain whether you can conclude the triangles are congruent.

Solve

Test 3: ASA (two angles and the side between them)

Now fix one side, say 7 cm, and make an angle at each end. The two arms must meet, and they meet in exactly one place. So the triangle is decided.

ASA stands for Angle, Side, Angle. The side is the one that joins the two angle vertices.

If you are given two angles, you can always work out the third, because the angles of a triangle add up to $180^\circ$. So if two angles match, all three match.

Using ASA with a missing angle

In $\triangle PQR$ the given angles are at $P$ and $R$. The given side $PQ$ is not between them, so we cannot use ASA yet.

Find the third angle: $|\angle Q| = 180^\circ - 35^\circ - 80^\circ = 65^\circ$.

Now $PQ$ is between the angles at $P$ and $Q$. In $\triangle XYZ$, $XY$ is between the angles at $X$ and $Y$.

Compare the two triangles: which angles and which side are equal? Name the test and write the congruence with the vertices in matching order.

$|\angle P| = |\angle X| = 35^\circ$, $|PQ| = |XY| = 6$ cm and $|\angle Q| = |\angle Y| = 65^\circ$. This is ASA, so $\triangle PQR \equiv \triangle XYZ$.

Practice★★☆

Triangle $ABC$ has $\angle A = 40^\circ$, $AB = 7$ cm and $\angle B = 65^\circ$. Triangle $DEF$ has $\angle D = 40^\circ$, $DE = 7$ cm and $\angle F = 75^\circ$. a) Find…

Triangle $ABC$ has $\angle A = 40^\circ$, $AB = 7$ cm and $\angle B = 65^\circ$. Triangle $DEF$ has $\angle D = 40^\circ$, $DE = 7$ cm and $\angle F = 75^\circ$.

a) Find $\angle C$ and $\angle E$.

b) Are the triangles congruent? Name the test.

Solve

Using congruent triangles in proofs

Congruent triangles are useful because once you know two triangles are congruent, all their corresponding parts are equal, not just the three you checked. That lets you prove new facts.

A good proof has three parts:

List three equal pairs, with a reason in square brackets for each.
Name the test (SSS, SAS or ASA) and write the congruence.
Use the congruence to state what you wanted to prove.

Look out for these useful reasons: a common side (a side shared by both triangles), a midpoint (two equal halves), vertically opposite angles (equal), and given (the question tells you).

Practice★★☆

In the diagram description, $M$ is the midpoint of $AB$ and also of $CD$, where $AB$ and $CD$ are two different line segments that cross at $M$. a) Name the two…

In the diagram description, $M$ is the midpoint of $AB$ and also of $CD$, where $AB$ and $CD$ are two different line segments that cross at $M$.

a) Name the two triangles that are congruent.

b) Identify the equal pairs of sides and angles.

c) State the test used.

Solve

Practice★★★

In triangle $ABC$, $AB = AC$. $D$ is the midpoint of $BC$. Prove that triangles $ABD$ and $ACD$ are congruent, and hence prove that $\angle B = \angle C$.

Solve

Practice★★★

$ABCD$ is a parallelogram with diagonal $AC$. Using the fact that opposite sides of a parallelogram are equal, prove that triangles $ABC$ and $CDA$ are congruent. Name…

$ABCD$ is a parallelogram with diagonal $AC$. Using the fact that opposite sides of a parallelogram are equal, prove that triangles $ABC$ and $CDA$ are congruent. Name the test.

Solve

Two sides and a non-included angle: no test

It is tempting to think that any two sides and any angle will do. They do not. Look at the picture. The angle at $A$ is $30^\circ$ and $|AB| = 8$ cm. The side $BC = 5$ cm is swung round like an arm until it touches the base line. It can touch in two places, $C_1$ and $C_2$.

Triangles $ABC_1$ and $ABC_2$ have $|AB|$, $|BC|$ and $|\angle A|$ equal, but they are clearly different sizes. So "side, side, non-included angle" is not a test for congruence.

Practice★★★

Triangles $ABC$ and $DEF$ have $AB = DE$, $BC = EF$ and $\angle A = \angle D$, where these angles are not the angles between the given sides. a) Give an example, with…

Triangles $ABC$ and $DEF$ have $AB = DE$, $BC = EF$ and $\angle A = \angle D$, where these angles are not the angles between the given sides.

a) Give an example, with lengths and angle sizes, showing that the triangles need not be congruent.

b) Explain why this does not contradict the SAS test.

Solve

Back to Seán's bunting: he does not need to compare everything. If the flags match in three sides (SSS), or two sides and the angle between them (SAS), or two angles and the side between them (ASA), every flag is an exact copy.

Let's practice! 9 exercises in this section

★☆☆ State what it means for two triangles to be congruent.★☆☆ Triangle $ABC$ has $AB = 5$ cm, $BC = 7$ cm and $CA = 9$ cm. Triangle $DEF$ has $DE = 5$…★☆☆ In triangles $PQR$ and $XYZ$: $PQ = XY = 6$ cm, $\angle Q = \angle Y = 50^\circ$ and…

Summary

Two triangles are congruent ($\equiv$) if they have exactly the same size and shape. Corresponding sides and angles are equal.
SSS: three pairs of equal sides.
SAS: two pairs of equal sides and the equal angles between them.
ASA: two pairs of equal angles and the equal side between them. Find the third angle with the $180^\circ$ rule if needed.
AAA (three equal angles) and SSA (a non-included angle) are not tests.
In a proof, list three equal pairs with reasons in square brackets, name the test, then state the result you wanted.