Triangle similarity is a way to prove that two triangles have the same shape, even if they are different sizes. Similar triangles have matching angles that are equal and matching side lengths that are proportional. This idea matters because it lets you find unknown distances, heights, and lengths without measuring them directly.
It is widely used in geometry, scale drawings, maps, shadows, and indirect measurement.
Key Facts
- Similar triangles have congruent corresponding angles and proportional corresponding sides.
- AA similarity: If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.
- SAS similarity: If two pairs of corresponding sides are proportional and the included angles are congruent, then the triangles are similar.
- SSS similarity: If all three pairs of corresponding sides are proportional, then the triangles are similar.
- Scale factor = corresponding side in image / corresponding side in original.
- If ΔABC ∼ ΔDEF, then AB/DE = BC/EF = AC/DF.
Vocabulary
- Similar triangles
- Triangles that have the same shape, with congruent corresponding angles and proportional corresponding side lengths.
- Corresponding parts
- Angles or sides in two figures that match each other based on their relative positions.
- Scale factor
- The constant multiplier that relates each side length of one similar figure to the matching side length of another.
- Proportion
- An equation showing that two ratios are equal, such as a/b = c/d.
- Included angle
- The angle formed by two given sides of a triangle.
Common Mistakes to Avoid
- Matching sides in the wrong order, which gives an incorrect proportion. Always use the triangle similarity statement to pair corresponding vertices and sides.
- Using SAS similarity without the included angle, which is not enough information. The congruent angle must be between the two proportional side pairs.
- Confusing similarity with congruence, which leads to assuming equal side lengths. Similar triangles only require proportional side lengths unless the scale factor is 1.
- Setting up ratios inconsistently, such as small/large on one side and large/small on the other. Keep the same comparison direction throughout the proportion.
Practice Questions
- 1 Triangles ABC and DEF are similar with A ↔ D, B ↔ E, and C ↔ F. If AB = 6, BC = 9, DE = 10, and EF = x, find x.
- 2 In triangles PQR and XYZ, PQ/XY = 4/10, PR/XZ = 6/15, and angle P is congruent to angle X. Which similarity shortcut proves the triangles are similar, and what is the scale factor from PQR to XYZ?
- 3 Two triangles have all three pairs of corresponding angles congruent, but one triangle has side lengths twice as large as the other. Explain why the triangles are similar but not congruent.