Can you please send me notes from grade 10 to 12 euclidean geometry notes and theroems

Okay, Nomvelo, here are some key Euclidean Geometry notes and theorems from Grade 10 to 12. This is a summary, so make sure to refer to your textbook for full proofs and examples!

• Basic Angle Relationships (Grade 10)

  • Angles on a straight line sum to 180°.
  • Angles around a point sum to 360°.
  • Vertically opposite angles are equal.
  • Parallel Lines and Transversals: Corresponding angles are equal, alternate interior angles are equal, and co-interior angles sum to 180°.

• Triangle Properties (Grade 10-11)

  • Sum of angles in a triangle is 180°.
  • Exterior angle of a triangle equals the sum of the two opposite interior angles.
  • Congruence Conditions: SSS, SAS, AAS, RHS.
  • Similarity Conditions: AAA, SSS, SAS.
  • Midpoint Theorem: The line segment joining the midpoints of two sides of a triangle is parallel to the third side and is half the length of the third side.

• Circle Theorems (Grade 11-12)

  • The line from the center perpendicular to a chord bisects the chord.
  • The angle subtended by an arc at the center is twice the angle subtended by the same arc at any point on the circumference.
  • Angles subtended by the same arc (or in the same segment) are equal.
  • The angle in a semicircle is 90°.
  • Cyclic Quadrilateral: Opposite angles sum to 180°, and the exterior angle equals the interior opposite angle.

• Tangent Theorems (Grade 11-12)

  • A tangent is perpendicular to the radius at the point of contact.
  • Tangents drawn from an external point to a circle are equal in length.
  • Tan-Chord Theorem: The angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment.

• Proportionality and Similarity (Grade 12)

  • Proportionality Theorem (Thales' Theorem): If a line parallel to one side of a triangle intersects the other two sides, then it divides the two sides proportionally.
  • Similarity in Right-Angled Triangles: If an altitude is drawn to the hypotenuse of a right-angled triangle, then the two triangles formed are similar to the original triangle and to each other.