Congruence and the SAS rule

Fitting one shape on another

Meena traces her door key on a sheet of paper and cuts out the shape. She puts the cut-out on the key, and it fits exactly. The two shapes have the same size and the same form. We say they are congruent.

Here are more congruent pairs:

• two coins of the same kind,
• two circles with the same radius,
• two squares with the same side,
• two line segments of the same length.

Two figures are congruent when one can be moved on top of the other so that they cover each other exactly. You may slide the figure, turn it, or flip it over. A left glove and a right glove are congruent in this sense, because one is the mirror image of the other.

Congruent triangles and the order of letters

Suppose triangle $ABC$ fits exactly on triangle $PQR$, with $A$ on $P$, $B$ on $Q$ and $C$ on $R$. We write

The symbol $\cong$ means "is congruent to". The order of the letters matters, because it tells you which corners match: $A$ with $P$, $B$ with $Q$ and $C$ with $R$.

Matching corners give matching sides and angles. The side $AB$ matches $PQ$, the side $BC$ matches $QR$ and the side $CA$ matches $RP$. The angle $A$ matches the angle $P$, and so on. These six pairs are the corresponding parts.

Corresponding parts of congruent triangles are equal. We shorten this to CPCT. It is the reason we want to prove triangles congruent: once we know it, every matching side and every matching angle is equal.

A worked example. Triangle $DEF$ is congruent to triangle $KLM$, written $\triangle DEF \cong \triangle KLM$. Given that $DE = 6\ \text{cm}$ and $\angle E = 70^\circ$, find $KL$ and $\angle L$.

1. Match the corners. $D$ goes with $K$, $E$ with $L$ and $F$ with $M$.
2. The side. $DE$ matches $KL$, so $KL = 6\ \text{cm}$.
3. The angle. $\angle E$ matches $\angle L$, so $\angle L = 70^\circ$.

Notice how much depends on the order. If someone wrote $\triangle DEF \cong \triangle LMK$, then $D$ would match $L$, and $DE$ would match $LM$. That is a different claim.

How much do we need to check?

A triangle has six parts: three sides and three angles. To show two triangles congruent, we do not have to compare all six pairs. A few are enough. Think about building a triangle.

• One side only. Many triangles share it: a long thin one, a tall one, a wide one.
• Two sides only. Still many. Hinge two sticks at a corner and the angle between them can open or close.
• Two sides and the angle between them. Now the hinge cannot move. The third side is decided, and only one triangle is possible.

The last case is the first rule of congruence.

The SAS rule

SAS rule. If two sides and the angle between them in one triangle are equal to two sides and the angle between them in another triangle, the triangles are congruent.

The angle between two sides is called the included angle. The letters S, A, S show the order: side, angle, side, with the angle sitting between the two sides.

Why SAS works. Place triangle $PQR$ on triangle $ABC$ so that $P$ falls on $A$ and $PQ$ lies along $AB$. Since $PQ = AB$, the point $Q$ falls on $B$. Since $\angle P = \angle A$, the side $PR$ lies along $AC$. Since $PR = AC$, the point $R$ falls on $C$. Now $QR$ and $BC$ have the same two ends, so they coincide. The triangles fit exactly.

Writing a proof

A proof lists each fact with its reason, and ends with the rule. Here are two examples.

Example 1. In the figure, $AB = AD$ and the line $AC$ bisects the angle $BAD$. Show that triangle $ABC$ is congruent to triangle $ADC$. What does this tell us about $BC$ and $DC$?

1. $AB = AD$. This is given.
2. $\angle BAC = \angle DAC$. The line $AC$ bisects $\angle BAD$.
3. $AC = AC$. This side is common to both triangles.
4. So $\triangle ABC \cong \triangle ADC$, by the SAS rule.
5. Therefore $BC = DC$, by CPCT.

In step 3, a side shared by two triangles counts as one equal pair. It is equal to itself.

Example 2. In a four-sided figure $ABCD$, $AD = BC$ and $\angle DAB = \angle CBA$. Show that $BD = AC$.

1. Take the triangles $ABD$ and $BAC$. They share the side $AB$.
2. $AD = BC$. This is given.
3. $\angle DAB = \angle CBA$. This is given.
4. $AB = BA$. This is the common side.
5. So $\triangle ABD \cong \triangle BAC$, by SAS. The corners match in the order $A$ with $B$, $B$ with $A$ and $D$ with $C$.
6. Therefore $BD = AC$, by CPCT.

Practise with new numbers

Try it in Python

A computer can build a triangle from two sides and the angle between them, and then compare the three sides of two triangles. This program tests two triangles against the first one.

Common mistakes

• Writing the letters in the wrong order. If $A$ matches $P$, write $\triangle ABC \cong \triangle PQR$, not $\triangle QPR$.
• Using an angle that is not between the two sides. Two sides and an angle elsewhere do not make SAS. The next lesson shows why.
• Forgetting the common side. A side that two triangles share is equal to itself. Write it down as a step.
• Trusting the drawing. Two sides that look equal are not equal. Use only what is given or what you have proved.

Remember these

• Congruent figures have the same size and the same form.
• In $\triangle ABC \cong \triangle PQR$, the order of the letters shows which corners match.
• CPCT: corresponding parts of congruent triangles are equal.
• SAS: two sides and the angle between them. The angle must be between the sides.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Triangle $ABC$ is congruent to triangle $XYZ$, with the corners matching in the order written. Which side of triangle $XYZ$ is equal to $AC$?

2. Triangles $DEF$ and $GHI$ have $DE = GH$, $EF = HI$ and $\angle E = \angle H$. Are they congruent? Which rule?

3. In triangles $ABC$ and $PQR$, $AB = PQ$, $BC = QR$ and $\angle A = \angle P$. Can you use SAS?

4. Triangle $LMN$ is congruent to triangle $RST$ in that order, and $\angle M = 58^\circ$. What is $\angle S$?

5. In Example 1, why can we write $AC = AC$ as a step?