Similar Triangles

Definition

Similar triangles are triangles that have the same shape but not necessarily the same size. Formally, two triangles are similar if their corresponding angles are equal and their corresponding sides are proportional.

Why It Matters

Similar triangles are the ‘indirect measurement’ tool of the ancient and modern world; they allow us to calculate heights and distances we can’t reach, serving as the foundation for both basic navigation and complex trigonometry.

Core Concepts

• Corresponding Sides: Sides that are in the same relative position in two similar triangles (e.g., the side opposite the $30^\circ$ angle in both).

• Proportionality: If triangle $ABC$ is similar to triangle $DEF$, then the ratios of their corresponding sides are equal: $\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF}$

  • How to read: “A-B over D-E equals B-C over E-F equals A-C over D-F.”
  • Meaning / when to use: All three side ratios are equal to the same scale factor $k$. Use to find unknown sides.

• Similarity Criteria:

  • AAA (Angle-Angle-Angle): If three angles of one triangle are congruent to three angles of another, the triangles are similar.
  • AA (Angle-Angle): If two angles of one triangle are equal to two angles of another, the triangles are similar.
  • SAS (Side-Angle-Side): If an angle of one triangle is equal to an angle of another and the sides including these angles are proportional.
  • SSS (Side-Side-Side): If all three pairs of corresponding sides are proportional.

• Acronyms for Proofs:

  • CSSTP: Corresponding Sides of Similar Triangles are Proportional.
  • CASTC: Corresponding Angles of Similar Triangles are Congruent.

• Similarity Theorems:

  • AA Similarity: Two congruent angles are sufficient to lock the shape of a triangle.
  • SAS~ Similarity: Requires one congruent angle and proportional sides including it.
  • SSS~ Similarity: All three sides must be in the same ratio.

• Extended Relationships:

  • Altitudes: The ratio of corresponding altitudes is equal to the ratio of corresponding sides ($k$).
  • Parallel Side Proportionality: A line parallel to one side of a triangle divides the other two sides proportionally.
  • Angle Bisector Theorem: An angle bisector divides the opposite side into segments proportional to the other two sides.

• Similarity in Right Triangles: The altitude to the hypotenuse of a right triangle divides it into two triangles similar to the given triangle and to each other.

• Ratios of Line Segments: Corresponding altitudes, medians, or angle bisectors of similar triangles have the same ratio as corresponding sides ($k$).

Connected Concepts

• Ratio
• Proportion
• Midsegment Theorem
• Trigonometric Functions: Right Triangle Approach
• Geometric Mean in Right Triangles