Similar Triangles

What Does "Similar" Mean?

Two triangles are similar if they have the same shape but not necessarily the same size — one is a scaled copy of the other. Similar triangles have corresponding angles that are equal and corresponding sides that are all in the same ratio (the scale factor). We write similarity as \triangle ABC \sim \triangle DEF, and the order of the letters matters: it tells you which vertices correspond.

Similarity Shortcuts

You don't need to check every angle and every side to prove two triangles are similar. Any one of these three shortcuts is enough:

• AA (Angle-Angle) — if two angles of one triangle equal two angles of another, the triangles are similar (the third angles must also match, since all triangle angles sum to 180^{\circ}).
• SAS (Side-Angle-Side) — if two pairs of corresponding sides are in the same ratio, and the included angles (the angle between those two sides) are equal, the triangles are similar.
• SSS (Side-Side-Side) — if all three pairs of corresponding sides are in the same ratio, the triangles are similar.

Setting Up Proportions

Once you know two triangles are similar, you can find missing side lengths by setting up a proportion between corresponding sides. If \triangle ABC \sim \triangle DEF, then:

The key skill is correctly matching up which side in the small triangle corresponds to which side in the large triangle — matching letter order in the similarity statement is what tells you this.

Example

Picture two triangles, \triangle ABC and \triangle ADE, where D lies on segment AB and E lies on segment AC, with segment DE drawn parallel to segment BC. Since DE \parallel BC, angle ADE equals angle ABC (corresponding angles), and both triangles share angle A. By AA similarity, \triangle ADE \sim \triangle ABC.

Suppose AD = 4, DB = 6 (so AB = 10), and DE = 8. Find BC.

1. Set up the proportion using corresponding sides: \frac{AD}{AB} = \frac{DE}{BC}
2. Substitute known values: \frac{4}{10} = \frac{8}{BC}
3. Cross-multiply: 4 \times BC = 10 \times 8 = 80
4. Solve: BC = 20

Key Takeaways

• Similar triangles have equal corresponding angles and proportional corresponding sides.
• Three shortcuts prove similarity: AA, SAS (with included angle), and SSS.
• Set up proportions by carefully matching corresponding sides using the similarity statement's letter order.
• A line parallel to one side of a triangle, cutting the other two sides, always creates a smaller similar triangle.