Similar Triangles and Proportionality Theorems
Two triangles are similar if they have the same shape: their corresponding angles are equal and their corresponding sides are proportional. In brief: the criteria AA, SAS (proportional sides with an equal included angle) and SSS (all three side ratios equal) prove similarity. Similar triangles lead to the intercept theorem: a line parallel to one side of a triangle divides the other two sides in the same ratio.
Similarity lets you find lengths that cannot be measured directly, such as the height of a tree from its shadow, and is the foundation of trigonometry and scale drawings. This guide explains the definition, the three criteria, the theorems about proportional segments, how areas change and how to solve typical problems, with a table of corresponding ratios.
What does similar mean?
Triangles ABC and DEF are similar, written △ABC ~ △DEF, if ∠A = ∠D, ∠B = ∠E, ∠C = ∠F and AB/DE = BC/EF = CA/FD = k. The number k is the scale factor. The order of the vertices shows the correspondence. If k = 1 the triangles are congruent. Similarity is an equivalence: it is reflexive, symmetric and transitive.
What are the similarity criteria?
- AA: two pairs of equal angles. Then the third pair is equal as well, since the angles sum to 180°.
- SAS: two pairs of proportional sides and equal included angles.
- SSS: three pairs of proportional sides.
Example 1
Triangle 1 has angles 50° and 60°, triangle 2 has angles 60° and 70°. The third angle of triangle 1 is 70°, and of triangle 2 is 50°. They have the same set of angles, so they are similar by AA, with 50° ↔ 50°, 60° ↔ 60° and 70° ↔ 70°.
Example 2
Triangle with sides 4, 6, 8 and triangle with sides 6, 9, 12: ratios 6/4 = 9/6 = 12/8 = 1.5. SSS similarity with k = 1.5. A triangle with sides 4, 6, 9 is not similar to the one with sides 4, 6, 8: the first two ratios are 4/4 = 1 and 6/6 = 1, but 9/8 ≠ 1.
What is the intercept theorem?
If a line parallel to side BC of triangle ABC meets AB at D and AC at E, then triangle ADE is similar to triangle ABC (AA, by corresponding angles), and so AD/AB = AE/AC = DE/BC. Equivalent forms: AD/DB = AE/EC. The converse also holds: if a line divides two sides in the same ratio, it is parallel to the third side.
A related result: the line segment connecting the midpoints of two sides of a triangle is parallel to the third side and half as long, the midsegment theorem (the case k = 1/2).
Example 3
In triangle ABC, DE ∥ BC with AD = 4, DB = 6 and DE = 5. Then AB = 10, and DE/BC = AD/AB = 4/10, so BC = 12.5. If AE = 3, then EC = 3 · 6/4 = 4.5 (from AE/EC = AD/DB = 4/6, EC = 4.5).
How do perimeters and areas change?
If the scale factor is k, the perimeters have ratio k, the areas have ratio k2 and corresponding heights, medians and radii of the circumcircle or incircle also have ratio k. For solids with scale factor k, volumes scale as k3.
Example: similar triangles with k = 3 and an area of 8 cm2 for the small one: the larger has area 8 · 9 = 72 cm2. If the larger triangle has perimeter 45, the smaller has 15.
How do you solve problems with similar triangles?
- Find two triangles and show they are similar (AA is usually enough).
- Write the correspondence and the equal ratios.
- Substitute the known lengths and solve the proportion.
- Check that the result is sensible.
Example 4: the height of a tree
A person 1.8 m tall casts a shadow 2.4 m long. At the same time, a tree casts a shadow 14 m long. The sun’s rays are parallel, so the triangles are similar by AA. Then h/14 = 1.8/2.4 = 0.75, so h = 10.5 m.
Example 5: crossing ladders
Two poles of heights 6 m and 10 m stand upright. A rope joins the top of each pole to the foot of the other. How high above the ground is the crossing point? Let the height be h. The triangles formed above the ground give 1/h = 1/6 + 1/10, so h = 1/(0.1667 + 0.1) = 3.75 m. The result does not depend on the distance between the poles.
Example 6: a right triangle
In a right triangle ABC with the right angle at C, the altitude CD divides it into similar triangles ACD, CBD and ABC. With AD = 4 and DB = 9, the altitude CD satisfies CD/4 = 9/CD, so CD = 6.
Similarity ratios at a glance
| Quantity | Ratio of the larger to the smaller | Example for k = 2 | Example for k = 3 |
|---|---|---|---|
| Side or height | k | ×2 | ×3 |
| Perimeter | k | ×2 | ×3 |
| Angles | 1 | unchanged | unchanged |
| Area | k2 | ×4 | ×9 |
| Volume (similar solids) | k3 | ×8 | ×27 |
Where do these ideas come from?
The theory of similar figures is developed in Book VI of Euclid’s Elements. According to tradition, Thales used similar triangles to estimate the height of the Egyptian pyramids from their shadows, and to determine the distance of ships at sea. The details of these stories are known from later authors, so they should be treated with caution. Similar triangles are the foundation of trigonometry, since the ratios of sides in a right triangle depend only on the angles.
Where are similar triangles used today?
Surveyors measure heights and widths indirectly. Maps and models rely on similarity, and photographers use it for the relationship between the size of an object and its image. Architects scale plans, and in computing, resizing images preserves similarity. The ratio of sides determines sine, cosine and tangent.
Which mistakes are common?
Matching the wrong sides
Corresponding sides are opposite equal angles. Label the vertices carefully and write the ratios in the same order.
Using the scale factor for areas
For areas use k2, for volumes k3.
Assuming similarity from equal sides only
Two sides in proportion are not enough. The included angle must be equal as well (SAS).
Confusing similar and congruent
Similar triangles have the same shape. Congruent triangles have the same shape and size.