1. Properties of Lines and Angles

• Vertical angles are congruent.
• Supplementary angles sum to 180°.
• Complementary angles sum to 90°.
• Parallel lines and transversals:
  • Corresponding angles are congruent.
  • Alternate interior angles are congruent.
  • Alternate exterior angles are congruent.
  • Same-side interior angles are supplementary.
• Exterior Angle Theorem: An exterior angle of a triangle equals the sum of the two non-adjacent interior angles.

2. Triangle Properties and Theorems

• Triangle Sum Theorem: The interior angles of a triangle always sum to 180°.
• Isosceles Triangle: Two equal sides and two equal base angles.
• Equilateral Triangle: All sides and all angles equal (each angle = 60°).
• Triangle Similarity:
  • AA: Two matching angles imply triangle similarity.
  • SSS: Corresponding side ratios are equal.
  • SAS: Two proportional sides and the included equal angle.
• Triangle Congruence:
  • SSS: All three sides are equal.
  • SAS: Two sides and the included angle are equal.
  • ASA: Two angles and the included side are equal.
  • AAS: Two angles and a non-included side are equal.
• Right Triangle Properties:
  • Pythagorean Theorem: \( a^2 + b^2 = c^2 \)
  • Special triangles: 30-60-90 and 45-45-90
• Midsegments: Connect midpoints, parallel to the third side, and half its length.
• Medians: Line segments from a vertex to the midpoint of the opposite side.

3. Proportionality and Applications

• Triangle Similarity in Proportions: Use ratios to find missing side lengths.
• Trigonometric Applications: Use sine, cosine, and tangent with right triangles.
• Real-World Problems:
  • Find heights using shadows and triangle similarity.
  • Estimate distances using triangle properties and angle measures.