# 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.