Establishing triangle similarity through AA, SAS, and SSS similarity criteria. Covers ratio and proportion in similar figures, the Triangle Proportionality Theorem (a line parallel to one side of a triangle divides the other two sides proportionally), geometric mean, and applications to indirect measurement.
High School Advanced • 9-12
Two figures are similar if they have the same shape but not necessarily the same size. Similarity is one of the most far-reaching concepts in geometry: it underlies trigonometry, scale drawings, fractal geometry, and countless real-world applications from architecture to astronomy. In this lesson, you will learn to prove triangles similar, work with proportions in similar figures, and apply the geometric mean.
Two polygons are similar (written with the symbol ~) if and only if:
The common ratio of corresponding sides is called the scale factor, often denoted k. If triangle ABC ~ triangle DEF with scale factor k = 2, then every side of DEF is twice the corresponding side of ABC.
Unlike congruence (which requires more information), triangle similarity can be established with surprisingly little data:
| Criterion | What to Show | Why It Works |
|---|---|---|
| AA (Angle-Angle) | Two pairs of corresponding angles are congruent | The third pair must also be congruent (angle sum = 180°), which forces proportional sides |
| SAS (Side-Angle-Side) | Two pairs of corresponding sides are proportional AND the included angles are congruent | The proportional sides and fixed angle lock in the shape |
| SSS (Side-Side-Side) | All three pairs of corresponding sides are proportional | Equal ratios force equal angles |
AA is by far the most commonly used criterion. Because triangles have exactly 180 degrees, showing two angles match automatically guarantees the third matches as well.
This theorem connects parallel lines and proportional segments within a triangle:
In triangle ABC, if line DE is parallel to BC with D on AB and E on AC, then:
The converse is also true: if a line divides two sides of a triangle proportionally, then it is parallel to the third side.
The geometric mean of two positive numbers a and b is:
The geometric mean appears naturally in similar triangles. When an altitude is drawn from the right angle to the hypotenuse of a right triangle, it creates two smaller triangles, each similar to the original and to each other. The altitude is the geometric mean of the two segments of the hypotenuse:
where p and q are the two segments of the hypotenuse.
Additionally, each leg of the original right triangle is the geometric mean of the hypotenuse and the adjacent hypotenuse segment:
Problem: In triangle PQR, angle P = 50° and angle Q = 70°. In triangle XYZ, angle X = 50° and angle Z = 60°. Are the triangles similar?
Solution:
In triangle PQR: ∠R = 180° - 50° - 70° = 60°.
In triangle XYZ: ∠Y = 180° - 50° - 60° = 70°.
Comparing: ∠P = ∠X = 50°, ∠Q = ∠Y = 70°, ∠R = ∠Z = 60°.
Two pairs of angles are congruent (in fact all three are), so by AA Similarity, triangle PQR ~ triangle XYZ.
Problem: Triangle ABC ~ triangle DEF with AB = 6, BC = 8, AC = 10, and DE = 9. Find EF and DF.
Solution: The scale factor is k = DE/AB = 9/6 = 3/2.
EF = BC × k = 8 × 3/2 = 12
DF = AC × k = 10 × 3/2 = 15
We can verify: the sides 9, 12, 15 are a 3-4-5 right triangle scaled by 3, and 6, 8, 10 are a 3-4-5 right triangle scaled by 2. The ratio 3/2 checks out.
Problem: In right triangle ABC (right angle at C), the altitude from C to hypotenuse AB meets AB at point D. If AD = 4 and DB = 9, find the altitude CD and both legs AC and BC.
Solution:
The hypotenuse AB = AD + DB = 4 + 9 = 13.
The altitude is the geometric mean of the hypotenuse segments:
Each leg is the geometric mean of the hypotenuse and the adjacent segment:
Similarity lets you measure objects you cannot reach. If you know a flagpole and a person cast shadows at the same time, the triangles formed (person-shadow-sun ray and pole-shadow-sun ray) are similar by AA. Measure the person's height and both shadow lengths, then set up a proportion to find the flagpole's height. Surveyors, astronomers, and engineers use this principle constantly.
Problem 1: Determine whether the triangles with sides 5, 12, 13 and 10, 24, 26 are similar. If so, state the scale factor.
Check ratios: 10/5 = 2, 24/12 = 2, 26/13 = 2. All ratios are equal, so the triangles are similar by SSS Similarity with scale factor k = 2.
Problem 2: In triangle ABC, DE is parallel to BC with D on AB and E on AC. If AD = 6, DB = 4, and AE = 9, find EC.
By the Triangle Proportionality Theorem: AD/DB = AE/EC
6/4 = 9/EC
EC = 9 × 4/6 = 6
Problem 3: A 5-foot-tall person casts a 3-foot shadow. At the same time, a nearby tree casts a 15-foot shadow. How tall is the tree?
The sun rays create similar triangles (AA: both have a right angle at the ground and the same sun angle).
person height / person shadow = tree height / tree shadow
5/3 = h/15
h = 5 × 15/3 = 25 feet.
Problem 4: Find the geometric mean of 8 and 18. Then verify by setting up a proportion.
Geometric mean = √(8 × 18) = √144 = 12.
Verification: 8/12 = 12/18. Both simplify to 2/3. The geometric mean is the value x such that a/x = x/b.
Problem 5: Right triangle ABC has the right angle at C. The altitude from C to hypotenuse AB creates segments AD = 3 and DB = 12 on the hypotenuse. Find the altitude CD and both legs.
AB = 3 + 12 = 15
CD = √(3 × 12) = √36 = 6
AC = √(15 × 3) = √45 = 3√5
BC = √(15 × 12) = √180 = 6√5
Check: AC² + BC² = 45 + 180 = 225 = 15² = AB². Confirmed by the Pythagorean Theorem.
Similar figures share the same shape but differ in size. Triangle similarity can be established using AA, SAS, or SSS criteria, with AA being the most efficient. The Triangle Proportionality Theorem shows that a line parallel to one side of a triangle divides the other two sides proportionally. The geometric mean connects the altitude of a right triangle to the segments of its hypotenuse. Together, these tools let you solve problems from indirect measurement to complex geometric reasoning.