The similarity tests for triangles · Lập trình với Martin

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 miễn phí — 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 miễn phí”: 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. Bắt đầu 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.