The similarity tests for triangles

You do not have to measure everything

Last time you checked similarity properly: all sides, all ratios. For triangles there is a shortcut. A triangle is such a firmly built thing that it is enough to check a piece — and the rest holds on its own.

That shortcut is called the similarity tests. There are three and they are named after what you check: AA (angle, angle), SAS (side, angle, side) and SSS (side, side, side).

The most useful is test AA: when two triangles match in two angles, they are similar. Done. No measuring sides. Why are two angles enough? Because the third is calculated for free — the sum of angles in a triangle is always 180°.

Test AA is the workhorse of all geometry: a tree’s shadow, the height of a building, ramps, maps. Wherever you find two equal angles, you have similarity — and with it a length you cannot reach with a tape measure.

Today you will feel all three tests and most of all: you will learn to see equal angles in pictures.

Adam and Tereza stand by the school and measure it — without a ladder. “The sun’s rays hit us and the school at the same angle,” Tereza explains, “and we and the school stand at right angles to the ground. Two equal angles — the triangle of me and my shadow is similar to the triangle of the school and its shadow.” Adam measures: Tereza’s shadow 2 m, Tereza 1.6 m, the school’s shadow 15 m. “So the school is 1.6 times 15 divided by 2… 12 metres!” Test AA has just measured a building.

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Look for equal angles in three places: a shared angle (a triangle inside a triangle), right angles (upright things on flat ground) and parallel lines cut by a line.

Three tests and when to use which

Test AA (angle, angle): if triangles match in two angles, they are similar. The fastest of the tests — angles often match “for free”: both triangles have a right angle, they share a common vertex, or parallels make them.

Test SSS (side, side, side): all three ratios of matching sides come out the same. That is exactly the check from the last lesson — now it has a name.

Test SAS (side, angle, side): two ratios of sides are the same and the angle BETWEEN those sides matches. Watch: the angle must be enclosed between the measured sides, otherwise the test does not hold.

How to pick: what does the question give you? Angles → AA. All sides → SSS. Two sides and the angle between them → SAS.

And then calculate. Once you have proved similarity with a test, you have the right to write the coefficient and find the unknown side:

k = known side of the large / matching side of the small, unknown side = matching side · k.

Method for problems: 1) sketch both triangles, 2) mark equal angles with arcs, 3) write the test that makes similarity hold, 4) set up the coefficient, 5) calculate. A sketch is not decoration — without it you pair the wrong sides.

Example 1: test AA with a shared angle

Question: Triangles ABC and ADE have a shared angle at vertex A and sides BC and DE are parallel. Prove similarity.

Method: First equal angle: the angle at A is shared — both triangles share it.

Second equal angle: BC and DE are parallel lines cut by line AB, so the angle at B equals the angle at D (corresponding angles).

Two equal angles → by test AA the triangles are similar.

Why it matters: this situation (a smaller triangle inside a larger one, joined by a parallel) is the most common picture in entrance-exam similarity questions.

Example 2: SSS with numbers

Question: Triangle K has sides 6, 8, 12 cm, triangle L has sides 9, 12, 18 cm. Are they similar? If yes, what is the coefficient?

Method: I sort the sides and divide matching ones: 9/6 = 1.5. Then 12/8 = 1.5. And 18/12 = 1.5.

All three ratios are 1.5 — by test SSS they are similar.

Answer: Yes, k = 1.5. Each side of triangle L is one and a half times longer.

If the third ratio came out, say, 1.4, the whole similarity falls — SSS wants all three.

Example 3: finding a side through AA

Question: The triangles are similar by AA. The smaller has sides 4 cm and 7 cm, the larger has the side matching the four-centimetre one of length 10 cm. How long is the side matching the seven?

Method: Coefficient from the paired sides: k = 10/4 = 2.5.

Unknown side = 7 · 2.5 = 17.5 cm.

Sense check: the larger triangle has everything two and a half times — 4 → 10 matches, 7 → 17.5 holds the same ratio.

Answer: 17.5 cm. First the test (the right to calculate), then the coefficient, then the side.

⚠️

For test SAS the angle must be BETWEEN the sides. Two pairs of sides in the same ratio plus a matching angle somewhere to the side does not guarantee similarity. The enclosed angle is the one both measured sides make — in a sketch you see it between them. When it is elsewhere, test SAS is silent and you have to look for another way.

Practice: now you do it 🎲

Now try it for real. You will see all the questions at once, like on paper. Start with ten. When it goes well, add more.

Similar triangles and the coefficient — find the side of the large. Write a whole number.

Practise the questions

On paper

Sketches required. Arcs on equal angles, then the test, then the calculation.

Draw a triangle with angles 40° and 60° and a second, twice as large, with the same angles. Measure matching sides and check that the ratio is the same everywhere.
Triangles with sides 5, 7, 9 and 15, 21, 27: which test proves similarity? Write all the ratios and the coefficient.
Triangles similar by AA: the smaller has sides 6 and 9 cm, the larger has 15 cm instead of the six. Find the other side.
A stick 1 m high casts a shadow 1.5 m. A tree casts a shadow 9 m. Sketch both triangles, write why they are similar (AA), and calculate the height of the tree.

Now you 💪

You can name the three tests (AA, SAS, SSS) and say what each checks.
You know why two angles are enough for AA — the third is calculated by the sum 180°.
In a shadow problem you can show where the two equal angles are.

✅

Done when: You have proved similarity with all three tests on different problems, and calculated the height of a tree through AA.

What to take from this lesson

Tests AA, SAS and SSS prove similarity without measuring everything.
AA is the fastest: two equal angles are enough.
Equal angles are made by a shared vertex, right angles and parallel lines.
For SAS the angle must be enclosed between the measured sides.
Method: sketch → test → coefficient → missing side.