Mathematics › 🔍 Grade 10 › Similarity Criteria: AA, SSS and SAS

Similarity Criteria: AA, SSS and SAS

Same shape, any size — similar triangles keep every angle equal and every side in the same fixed ratio.

Grade 10Geometry

Similarity Criteria: AA, SSS and SAS — illustration

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The big idea: Similar triangles have equal corresponding angles and proportional corresponding sides — same shape, scaled up or down by a constant factor k. Three criteria (AA, SSS-similarity, SAS-similarity) let you confirm similarity from partial information, and proportional sides let you solve for unknown lengths. Similarity is also what makes right-triangle trigonometry possible: every right triangle with a given acute angle is similar to every other, so ratios like sin θ depend only on the angle.

🎯 By the end, you'll be able to

State the three similarity criteria: AA, SSS-similarity and SAS-similarity
Distinguish similarity (proportional sides, equal angles) from congruence (equal sides)
Use the side-splitter (triangle proportionality) theorem and proportions to find missing lengths
Explain why the ratio of two sides in a right triangle depends only on its acute angle

📎 You should already know

Triangle congruence criteria (previous lesson)
Dilations and scale factor (Grade 8)

Same shape, any size

Two triangles are similar when they have the same shape but not necessarily the same size: every pair of corresponding angles is equal, and every pair of corresponding sides is in the same fixed ratio, called the scale factor k. Congruent triangles are a special case of similar triangles where k = 1.

⚠️ Similar is not congruent

Don't confuse the two: congruent triangles match in both shape and size (every side is equal); similar triangles match only in shape (sides are proportional, not necessarily equal). AA guarantees similarity — never congruence — unless the scale factor happens to be exactly 1.

AA: the fastest test

If two angles of one triangle equal two angles of another, the third angles must also be equal (angles in a triangle sum to 180°), so all three angles match — the triangles are automatically similar. This is the AA (angle-angle) criterion, and it's the one you'll reach for most often because it needs only two pieces of information.

SSS-similarity and SAS-similarity

SSS-similarity: if all three pairs of corresponding sides share the same ratio, the triangles are similar. SAS-similarity: if two pairs of corresponding sides share the same ratio and the angle between them is equal, the triangles are similar. Both are proportional cousins of the SSS and SAS congruence criteria from the last lesson — equal lengths become equal ratios.

⚠️ SAS-similarity vs. SAS-congruence: same name, different test

It's easy to conflate the two because they share a name. SAS-congruence asks whether two sides and the included angle are equal. SAS-similarity asks whether two sides are in the same ratio and the included angle is equal. One checks equal lengths; the other checks equal proportions. Read the criterion in context before assuming which one is meant.

\[ \dfrac{AB}{A'B'}=\dfrac{BC}{B'C'}=\dfrac{CA}{C'A'}=k \]

In similar triangles, every pair of corresponding sides shares the same ratio k, the scale factor.

✨ The ratio depends on the angle, not the triangle's size

Here is the fact that makes right-triangle trigonometry possible: take any two right triangles that share an acute angle θ. Because they share both the right angle and θ, they are similar by AA — and similar triangles have proportional sides. So the ratio of any two sides (say, the side opposite θ to the hypotenuse) is the same number in every right triangle with that angle, no matter how big or small the triangle is. That's exactly why a number like ‘sin 30°’ can exist at all — it depends only on the angle.

The side-splitter (triangle proportionality) theorem

If a line is drawn parallel to one side of a triangle and crosses the other two sides, it cuts those two sides into proportional segments. This follows directly from AA similarity: the small triangle cut off at the top shares two angles with the original triangle (the vertex angle, and equal corresponding angles formed by the parallel lines), so it is similar to the whole.

🔑 Area scales by k², not k

Recall from dilations: if lengths scale by a factor k, area scales by k². The same rule applies to similar triangles — and to any similar figures. Double every side (k = 2) and the area quadruples (k² = 4); triple every side (k = 3) and the area grows ninefold (k² = 9).

\[ \dfrac{\text{Area}_1}{\text{Area}_2}=k^{2} \]

For similar figures, the ratio of areas equals the square of the ratio of corresponding sides.

🎮 Similar Triangle Stretcher LIVE

Dilate a triangle by different scale factors and watch how side lengths and area respond differently.

📝 Worked example: In &triangle;ABC, segment DE is parallel to BC, with D on AB and E on AC. AD = 4, DB = 6, and EC = 9. Find AE.

Because DE ∥ BC, the side-splitter theorem gives AD⁄DB = AE⁄EC.
Substitute the known lengths: 4⁄6 = AE⁄9.
Cross-multiply: 6 × AE = 4 × 9 = 36, so AE = 36⁄6.

✓ AE = <strong>6</strong>.

📝 Worked example: A tree casts a 12 ft shadow at the same moment a 5 ft post casts a 3 ft shadow. Both stand vertically and the sun's rays hit them at the same angle. Find the height of the tree.

Each object, its shadow, and the sun's ray form a right triangle, and both triangles share the same sun angle plus their own right angle — so they're similar by AA.
Similar triangles have proportional sides: height⁄shadow is the same ratio for both objects, so h⁄12 = 5⁄3.
Solve: h = 12 × (5⁄3) = 60⁄3.

✓ The tree is about <strong>20 ft</strong> tall.

Check your understanding

1. Which criterion guarantees similarity from just two matching angles?

If two angles match, the third must too (angles sum to 180°), so all three angles agree — that's enough for similarity by AA.

2. In similar triangles, corresponding sides are ___ and corresponding angles are ___.

Similarity keeps every angle exactly equal while every side scales by the same constant ratio (the scale factor).

3. Two triangles share two sides in the ratio 2:1 and their included angles are equal. Which criterion applies, and what does it prove?

Equal ratios (not equal lengths) plus an equal included angle is SAS-similarity, which proves the triangles are similar — same shape, not necessarily same size.

4. A triangle is dilated by a scale factor of k = 3. How does its area change?

Area scales by k², and 3² = 9, so the area becomes nine times as large.

5. Why is the ratio opposite-side / hypotenuse the same number for every right triangle with a 30° angle?

Any two right triangles sharing the right angle and a 30° angle are similar by AA, so their corresponding side ratios are locked together — that ratio is exactly what we'll soon call sin 30°.

✅ Key takeaways

Similar triangles have equal corresponding angles and proportional corresponding sides, scaled by factor k.
AA, SSS-similarity and SAS-similarity are the three tests for similarity; they mirror the congruence criteria but check ratios instead of equal lengths.
The side-splitter theorem: a line parallel to one side of a triangle divides the other two sides proportionally.
Similar figures scale area by k², not k — double the sides and the area quadruples.
Because right triangles with the same acute angle are similar by AA, the ratio of their sides depends only on that angle — the fact that makes trigonometric ratios possible.