Geometry Cheat Sheet: Triangle Proofs & Angle Relationships

Understanding triangle similarity is crucial in geometry. Triangles can be proven similar using AA (when two angles are the same), SAS (when two sides are proportional and the included angle is congruent), or SSS (when all sides are proportional). Remember that similar triangles have the same shape but possibly different sizes.

When working with angles, keep in mind that all angles in a triangle sum to 180°. Angles can be acute, obtuse, or right angles. Pay attention to adjacent angles and special relationships like linear pairs (angles that form a straight line equal 180°) and vertical angles (which are always equal).

Parallel lines create important angle relationships including corresponding angles, alternate interior angles, alternate exterior angles, and same side angles (which sum to 180°). When proving triangles, simplify your work by taking the fastest route and noting that the smallest sides are opposite the smallest angles.

Pro Tip: When you see fractions that don't match while solving for proportions, it's a clear sign the triangles are not similar. Always check your ratios carefully!

Key symbols to remember: ⊥ means perpendicular, || means parallel, ≅ means congruent (same shape and size), and ~ means similar (same shape, different size). For coordinate transformations, reflection across the x-axis changes (x,y) to $x,-y$, while reflection across the y-axis changes (x,y) to $-x,y$.