Similar Triangles: Theorems & Properties
Similar Triangles: Theorems & Properties
TL;DR
Similar triangles are triangles with the same shape but not necessarily the same size: their corresponding angles are equal and their corresponding sides are in the same ratio. This article covers the definition, the AA, SAS, and SSS similarity criteria, the properties, the area-ratio rule, the difference from congruent triangles, six worked examples, and the mistakes students make most.
What Are Similar Triangles?
Similar triangles are triangles that have exactly the same shape but possibly different sizes. Two triangles are similar when both of these hold at once:
- Their corresponding angles are equal, and
- Their corresponding sides are in the same ratio (proportional).
One triangle is then a scaled copy of the other, enlarged or shrunk by a constant factor called the scale factor, and it stays similar even if it is rotated or flipped into a mirror image. We write similarity with the symbol ∼: △ABC∼△DEF means triangle ABC is similar to triangle DEF, with the vertices listed in matching order so you know which angle pairs with which.
What Are the Similarity Criteria (AA, SAS, SSS)?
A common reader question is whether you really have to check all the angles and all the sides. You do not. Three shortcut tests, the similarity criteria, each confirm similarity from partial information.
AA (Angle-Angle). If two angles of one triangle equal two angles of another, the triangles are similar. Why two is enough: the three angles of any triangle add to 180°, so once two angles match, the third must match automatically.
SAS (Side-Angle-Side). If two pairs of sides are in the same ratio and the angles between them are equal, the triangles are similar. The equal angle must be the one included between the two proportional sides.
SSS (Side-Side-Side). If all three pairs of corresponding sides are in the same ratio, the triangles are similar. No angle check is needed, because three matched side ratios lock the shape completely.
Properties of Similar Triangles
Once two triangles are known to be similar, a useful list of facts comes free:
- All corresponding angles are equal. Shape is preserved exactly.
- All corresponding sides are in the same ratio. That common ratio is the scale factor.
- The ratio of perimeters equals the scale factor. Perimeter scales the same way the sides do.
- The ratio of areas equals the square of the scale factor. If the sides are in ratio k, the areas are in ratio k².
- Corresponding medians, altitudes, and angle bisectors are all in the ratio k. Every matching length scales by the same factor as the sides.
The Area Ratio Rule
This property deserves its own heading because it is the one students most often get wrong. If two triangles are similar with sides in the ratio k, then their areas are in the ratio k², the square of the side ratio.
The reason is that area depends on two dimensions, base and height, and both scale by k. Multiplying two lengths that have each grown by k multiplies the area by k × k = k²:
Area1/Area2 = (side1/side2)² = k².
So a triangle with sides twice as long does not have twice the area, it has four times the area. A scale factor of 3 means nine times the area.
Similar Triangles vs Congruent Triangles
Similar and congruent are the two ways triangles can "match", and the difference is just one word: size. Similar triangles have the same shape, possibly different sizes. Congruent triangles have the same shape and the same size — they are similar triangles with a scale factor of exactly 1.
| Feature | Similar (∼) | Congruent (≅) |
|---|---|---|
| Shape | Same | Same |
| Size | May differ | Identical |
| Corresponding angles | Equal | Equal |
| Corresponding sides | Proportional | Equal (ratio 1:1) |
| Scale factor | Any positive number | Exactly 1 |
| Test examples | AA, SAS, SSS | SSS, SAS, ASA, RHS |
Every congruent pair is also similar (with k=1), but most similar pairs are not congruent.
Examples of Similar Triangles
Example 1 - △ABC∼△DEF with AB=4, DE=6, and BC=5. Find EF.
Final answer: EF=7.5.
Example 2 - Two similar triangles have sides in the ratio 2:3. The smaller has an area of 16 cm². Find the area of the larger.
Final answer: 36 cm².
Example 3 - Are two triangles similar if one has angles 40° and 75°, and the other has angles 75° and 65°?
Final answer: yes, they are similar (AA).
Example 4 - A 6-foot person casts a 4-foot shadow at the same time a tree casts a 30-foot shadow. How tall is the tree?
Final answer: the tree is 45 feet tall.
Example 5 - In △ABC, a line DE is drawn parallel to side BC, meeting AB at D and AC at E. If AD=3, DB=6, and AE=4, find EC.
Final answer: EC=8.
Example 6 - Two similar triangles have corresponding sides 8 cm and 12 cm. Find the ratio of their perimeters and the ratio of their areas.
Final answer: perimeters 2:3; areas 4:9.
Why Similar Triangles Matter
Similarity is the geometry of scaling, and scaling is everywhere humans build, measure, or picture the world.
- Measuring the unreachable. The shadow method is how the ancient Greek Thales is said to have measured heights, and how surveyors still estimate heights and distances they cannot reach directly.
- Maps and scale models. Every map and architectural model is a similar copy of the real thing.
- Cameras and screens. A camera lens projects a similar triangle of light onto the sensor.
- Trigonometry's foundation. The sine, cosine, and tangent ratios only make sense because all right triangles with a given acute angle are similar.
Where Students Trip Up on Similar Triangles
Mistake 1: Scaling area by the side ratio instead of its square
Where it slips in: Given the side ratio and one area, students multiply the area by the side ratio directly.
Mistake 2: Pairing the wrong corresponding sides
Where it slips in: Students match sides by their position rather than by the vertex order.
Mistake 3: Confusing similar with congruent
Where it slips in: Students conclude triangles are equal in size or insist all sides must be equal for similarity.
Key Takeaways
- Similar triangles have the same shape but possibly different sizes: equal corresponding angles and proportional corresponding sides.
- Similarity is proved by any one of three criteria — AA, SAS, or SSS.
- Corresponding sides, perimeters, medians, and altitudes all scale by the same factor; areas scale by its square.
- The most common error is scaling area by the side ratio instead of its square.