The AA, SSS and SAS criteria

Do we need all six facts?

Two triangles are similar when their three angles are equal in pairs and their three pairs of sides are in the same ratio. That is six facts. Checking all six every time would be slow.

The good news is that a few of them force the rest. This lesson gives three short tests. Each one guarantees similarity.

We write $\triangle ABC \sim \triangle PQR$ to say that A matches P, B matches Q and C matches R. The order of the letters matters.

The AA test

AA test. If two angles of one triangle are equal to two angles of another triangle, the two triangles are similar.

The angles of a triangle add up to $180^\circ$. So if two pairs of angles are equal, the third pair is equal too. All three angle pairs match.

Why do equal angles give equal ratios? Let $\angle A = \angle P$ and $\angle B = \angle Q$ in triangles ABC and PQR.

1. Cut off a copy. Mark D on AB with $AD = PQ$, and E on AC with $AE = PR$.
2. Congruent triangles. $AD = PQ$, $AE = PR$ and $\angle A = \angle P$, so $\triangle ADE \cong \triangle PQR$ by SAS. Then $\angle ADE = \angle Q = \angle B$.
3. Parallel lines. $\angle ADE$ and $\angle B$ are corresponding angles, and they are equal, so $DE \parallel BC$.
4. The basic proportionality theorem. It gives $\dfrac{AD}{DB} = \dfrac{AE}{EC}$, and so $\dfrac{AB}{AD} = \dfrac{AC}{AE}$. That is $\dfrac{AB}{PQ} = \dfrac{AC}{PR}$.
5. The third pair. In the same way, starting from the angle at B, $\dfrac{AB}{PQ} = \dfrac{BC}{QR}$. All three ratios are equal.

The SSS test

SSS test. If the three sides of one triangle are in the same ratio as the three sides of another, the triangles are similar:

Match the shortest side with the shortest, and the longest with the longest.

A triangle has sides $4$ cm, $6$ cm and $8$ cm. Another has sides $6$ cm, $9$ cm and $12$ cm.

The three ratios are equal, so the triangles are similar. Each side of the second triangle is $1.5$ times the matching side of the first.

Now take $3$ cm, $4$ cm, $6$ cm and $6$ cm, $8$ cm, $11$ cm. The first two ratios are $\dfrac{3}{6} = \dfrac{4}{8} = \dfrac{1}{2}$, but $\dfrac{6}{11}$ is not $\dfrac{1}{2}$. These triangles are not similar.

The SAS test

SAS test. If one angle of a triangle equals one angle of another, and the sides around those angles are in the same ratio, the triangles are similar:

In $\triangle ABC$, $AB = 3$ cm, $AC = 5$ cm and $\angle A = 70^\circ$. In $\triangle PQR$, $PQ = 9$ cm, $PR = 15$ cm and $\angle P = 70^\circ$.

The ratios are equal and the angle is between the two sides, so the triangles are similar by SAS.

A warning. The equal angle must sit between the two proportional sides. If it does not, the test does not work.

Which test do I use?

• Two pairs of equal angles are given: use AA.
• Only sides are given: check the three ratios, and use SSS.
• One angle and the two sides around it are given: use SAS.

When the triangles are similar, write the letters in matching order. $\triangle ABC \sim \triangle PQR$ tells you at once that $\dfrac{AB}{PQ} = \dfrac{BC}{QR}$.

Finding a length with similar triangles

Shadows. A pole $3$ m tall casts a shadow $4$ m long. At the same moment a tree casts a shadow $20$ m long. How tall is the tree?

1. Find the triangles. The sun's rays make the same angle with the ground. The pole and the tree are both upright, so both make $90^\circ$ with the ground. Two angles match, so the triangles are similar by AA.
2. Match the sides. $\dfrac{\text{height of tree}}{\text{height of pole}} = \dfrac{\text{shadow of tree}}{\text{shadow of pole}}$.
3. Solve. $\dfrac{h}{3} = \dfrac{20}{4}$, so $h = 15$ m.

Check. The shadow is $5$ times as long, so the tree is $5$ times as tall: $5 \times 3 = 15$ m.

The diagonals of a trapezium. In trapezium ABCD, $AB \parallel DC$, and the diagonals meet at O. Show that $\triangle AOB \sim \triangle COD$, and find $OC$ when $AB = 6$ cm, $DC = 9$ cm and $OA = 4$ cm.

1. $\angle AOB = \angle COD$, because they are vertically opposite angles.
2. $\angle OAB = \angle OCD$, because they are alternate angles and $AB \parallel DC$.
3. Two angles match, so $\triangle AOB \sim \triangle COD$ by AA. A matches C, O matches O and B matches D.
4. $\dfrac{OA}{OC} = \dfrac{AB}{CD}$, so $\dfrac{4}{OC} = \dfrac{6}{9}$, and $OC = 6$ cm.

Practise with new numbers

Try it in Python

The program checks the SSS test on two triangles, and then finds the angles of both with the cosine rule.

Common mistakes

• Matching the wrong sides. Always put the shortest side with the shortest, and the longest with the longest.
• Using the test without the right angle. For SAS the equal angle must lie between the two sides.
• Turning the ratio upside down. If $\dfrac{QR}{BC} = \dfrac{PQ}{AB}$, then the bigger triangle has the bigger side on top.
• Saying equal sides are enough for similar. Equal ratios are enough. Equal sides would make the triangles congruent.

Remember these

• AA: two angles equal.
• SSS: the three side ratios equal.
• SAS: one angle equal, and the two sides around it in the same ratio.
• Write similar triangles with their letters in matching order, then read off the ratios.

Check yourself

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

1. Which similarity test needs only two angles?

2. Triangles have sides $5$ cm, $12$ cm, $13$ cm and $10$ cm, $24$ cm, $26$ cm. Are they similar?

3. In $\triangle ABC$, $AB = 3$ cm, $AC = 6$ cm and $\angle A = 50^\circ$. In $\triangle PQR$, $PQ = 5$ cm, $PR = 10$ cm and $\angle P = 50^\circ$. Are they similar? Which test?

4. Triangle ABC has angles $40^\circ$ and $80^\circ$. Triangle PQR has angles $60^\circ$ and $80^\circ$. Are they similar?

5. A $2$ m stick casts a $3$ m shadow. At the same moment a flagpole casts a $12$ m shadow. How tall is the flagpole?