Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
Looking for geometry similar triangles proportions? Then this Geometry lesson on Similar Triangles, Proportions and Indirect Measurement explains each idea step by step, with worked examples.
Quick answer: Triangles are similar when two angles match (AA), when two sides are proportional with equal included angles (SAS), or when all three sides are proportional (SSS). A line parallel to one side divides the other two sides proportionally. In a right triangle, the altitude to the hypotenuse creates similar triangles and geometric means.
Standards: G-SRT.B.4, G-SRT.B.5 · Level: Grade 10 Geometry
How do you prove that triangles are similar?
Similar triangles have congruent corresponding angles and proportional corresponding sides. Three shortcuts prove similarity.
| Criterion | What you need |
|---|---|
| AA | Two pairs of congruent angles |
| SAS ∼ | Two pairs of proportional sides and the congruent angles between them |
| SSS ∼ | Three pairs of proportional sides |
With the scale factor k, every side of the second triangle is k times the matching side of the first. If one triangle has sides 7, 9, 12 and another has 10.5, 13.5, 18, the ratio is 1.5 each time, so the triangles are similar by SSS ∼.
What does a line parallel to a side do?
If a line parallel to one side of a triangle meets the other two sides, it divides them proportionally. Conversely, if a line divides two sides proportionally, it is parallel to the third side.
The small triangle ADE is similar to ABC by AA, because ∠A is shared and the corresponding angles at D and B are congruent.
How is a right triangle split by its altitude?
The altitude to the hypotenuse creates two smaller right triangles. Each is similar to the original and to each other. This gives geometric mean relationships.
- Altitude: h2 = p · q, so h is the geometric mean of the two pieces of the hypotenuse.
- Each leg: a2 = p · c and b2 = q · c.
Adding the two leg equations gives a2 + b2 = c(p + q) = c2, which proves the Pythagorean Theorem using similar triangles. Check: 152 + 202 = 625 = 252.
How do you measure something you cannot reach?
Build two similar triangles from distances you can measure, write a proportion, and solve. A mirror, a shadow or a pair of stakes can create the triangles.
- Find or prove two similar triangles and name them with matching vertices.
- Write a proportion of corresponding sides.
- Cross-multiply and solve.
- Check that the answer has sensible size.
Match sides by their position in the triangle, not by how the figure looks. The proportion AD/DB = AE/EC compares parts to parts, while AD/AB = DE/BC compares part to whole.
Write the similarity statement first, such as △ADE ∼ △ABC. The order of the letters tells you which sides to put in each ratio.
- AA, SAS and SSS similarity criteria prove triangles similar.
- A line parallel to a side divides the other two sides proportionally, and the converse is true.
- In a right triangle, the altitude to the hypotenuse satisfies h² = p · q.
- Each leg is the geometric mean of the hypotenuse and the adjacent segment: a² = p · c.
- Similar triangles let you measure heights and distances indirectly.
Similar Triangles, Proportions and Indirect Measurement: frequently asked questions
What is the difference between similar and congruent triangles?
Congruent triangles have the same size and shape. Similar triangles have the same shape and corresponding sides in a constant ratio, so they can be different sizes.
How do you find the geometric mean of two numbers?
Multiply the numbers and take the square root. The geometric mean of 4 and 25 is √100 = 10.
Why does a line parallel to one side form a similar triangle?
Corresponding angles formed by the parallel line are congruent, and the angle at the top is shared. Two matching angle pairs make the triangles similar by AA.
How does similarity prove the Pythagorean Theorem?
The altitude to the hypotenuse gives a² = p·c and b² = q·c. Adding them gives a² + b² = c(p + q) = c².
What is indirect measurement?
It is finding a length you cannot measure directly by using similar triangles. A shadow, a mirror or a pair of stakes creates two triangles with a known ratio.