This geometry similar triangles lesson explains similar triangles and proportions in one printable page, with definitions, a method and a worked example.
Triangles are similar when corresponding angles are congruent and corresponding sides are proportional, which AA, SAS or SSS similarity proves. A line parallel to one side of a triangle splits the other two sides proportionally, and the altitude to the hypotenuse of a right triangle creates similar triangles and geometric-mean relationships.
Geometry Similar Triangles Lesson: key facts
- AA, SAS and SSS similarity criteria prove triangles similar (G-SRT.B.4, B.5).
- Side-splitter theorem: a line parallel to one side divides the other two sides proportionally, AD/DB = AE/EC; its converse proves lines parallel.
- In a right triangle with altitude CD to the hypotenuse, CD squared equals AD times DB.
- Each leg is the geometric mean of the hypotenuse and the segment next to that leg.
- Similar triangles justify indirect measurement with shadows, mirrors and a proof of the Pythagorean theorem.
How do you prove two triangles are similar?
Triangles are similar when their corresponding angles are congruent and their corresponding sides are proportional. You never need all six facts, because one of three criteria is enough.
| Criterion | What you must show |
|---|---|
| AA | Two pairs of congruent angles |
| SAS | Two pairs of sides in the same ratio, with congruent included angles |
| SSS | Three pairs of sides in the same ratio |
Example: the sides 4, 6, 8 and 10, 15, 20 satisfy \(\frac{10}{4}=\frac{15}{6}=\frac{20}{8}=2.5\), so the triangles are similar by SSS with scale factor 2.5.
What does the triangle proportionality theorem say?
If a line parallel to one side of a triangle meets the other two sides at D and E, it divides them proportionally: \(\frac{AD}{DB}=\frac{AE}{EC}\). Converse: if D and E divide the two sides proportionally, then DE is parallel to BC.
The reason is similarity: DE parallel to BC gives congruent corresponding angles at D and B, and angle A is shared, so triangle ADE is similar to triangle ABC by AA.
- Write the proportion \(\frac{AD}{DB}=\frac{AE}{EC}\).
- Substitute the known lengths.
- Cross-multiply.
- Solve for the unknown.
Worked example. With \(AD=6,\ DB=4,\ AE=9\): \(\frac{6}{4}=\frac{9}{EC}\), so \(6\cdot EC=36\) and \(EC=6\).
How does similarity work in right triangles?
The altitude from the right angle to the hypotenuse creates two smaller right triangles, and each one is similar to the original. The geometric mean of positive numbers \(a\) and \(b\) is \(\sqrt{ab}\), the number \(x\) with \(\frac{a}{x}=\frac{x}{b}\).
| Part of the triangle | Relationship |
|---|---|
| Altitude CD | \(CD^{2}=AD\cdot DB\) |
| Leg AC | \(AC^{2}=AD\cdot AB\) |
| Leg BC | \(BC^{2}=DB\cdot AB\) |
For \(AD=4\) and \(DB=9\), \(CD=\sqrt{36}=6\) and \(AB=13\). Adding the two leg formulas gives \(AC^{2}+BC^{2}=AB(AD+DB)=AB^{2}\), which proves the Pythagorean theorem.
How do you measure a tall object with similar triangles?
Draw the two right triangles that sunlight makes, confirm they share an angle, write a proportion of heights to shadows, and solve. A 5.5 ft person casts a 4 ft shadow while a tree casts a 28 ft shadow: \(\frac{h}{28}=\frac{5.5}{4}\), so \(h=38.5\) ft.
Similar Triangles and Proportions: frequently asked questions
What is the difference between similar and congruent triangles?
Congruent triangles have the same shape and size. Similar triangles have the same shape, so their angles match and their sides share one ratio, but their sizes can differ. Congruent triangles are the special case with ratio 1.
Why is SSA not a similarity criterion?
Two sides and an angle that is not between them can fit two different triangles, so the remaining sides are not forced into one ratio. SAS works because the included angle fixes the shape.
How do you decide which sides correspond?
Match sides opposite congruent angles, or match the shortest with the shortest and the longest with the longest. Writing the similarity statement in matching order, such as ABC ~ DEF, keeps the correspondence straight.
What is a geometric mean and why does it appear in right triangles?
The geometric mean of a and b is the square root of ab. It appears because the altitude to the hypotenuse makes similar triangles whose proportions contain the same middle length twice.
How does the side-splitter theorem differ from its converse?
The theorem starts with parallel lines and concludes that two sides are divided proportionally. The converse starts with equal ratios and concludes that the segment is parallel to the third side.
How to use this geometry similar triangles lesson
Use this geometry similar triangles lesson before the worksheet, then check what you learned with the quiz.
See every chapter on the Geometry math page.