Euclidean Geometry Grade 10 Revision Notes

Lines and Angles

• Adjacent Supplementary Angles: Sum to 180°.
• Angles Around a Point: Sum to 360°.
• Vertically Opposite Angles: Are equal.
• Corresponding Angles: Equal if lines are parallel (AB || CD).
• Alternate Angles: Equal if lines are parallel (AB || CD).
• Co-interior Angles: Sum to 180° if lines are parallel (AB || CD).

Triangles

Types of Triangles

• Scalene Triangle: No sides equal.
• Isosceles Triangle: Two sides equal, base angles equal.
• Equilateral Triangle: All sides and angles equal.
• Right-angled Triangle: One angle is 90°.

Properties

• Sum of Angles: 180°.
• Exterior Angle: Equal to sum of opposite interior angles.
• Pythagoras' Theorem: a² + b² = c² for right-angled triangles.

Congruency Conditions

• SSS: Three sides equal.
• SAS: Two sides and included angle equal.
• AAS: Two angles and one side equal.
• RHS: Hypotenuse and one side equal in right-angled triangles.

Similar Triangles

• Corresponding sides are proportional.

Solving Geometric Riders

1. Analyse Diagram: Focus on given key words.
2. Identify Useful Information: Mark equal angles/sides; look for implied info.
3. Develop Rough Proof: Link acquired information.
4. Rewrite Formal Proof.

Key Terms

• Parallel lines, triangle properties, circle center, diameter, angles on same segment, chords, cyclic quadrilaterals, tangents.

Quadrilaterals

• Polygon: 2D figure with three or more sides.
• Quadrilateral: Polygon with four sides.

Types of Quadrilaterals

• Parallelogram: Opposite sides equal and parallel.
  • Prove by showing opposite sides or angles are equal.
• Rectangle: A parallelogram with right angles.
  • Diagonals equal and bisect each other.
• Rhombus: Parallelogram with equal adjacent sides.
  • Diagonals bisect at right angles.
• Square: All properties of rectangle and rhombus.
• Trapezium: One pair of parallel sides.
• Kite: Two pairs of adjacent equal sides.

Properties of Quadrilaterals

• Opposite Sides and Angles of a Parallelogram: Are equal.

Theorems and Proofs

• Theorem 1: Opposite sides and angles of a parallelogram are equal.
  • Converse: Quadrilateral is a parallelogram if opposite sides/angles are equal.

Exercises

1. Calculate interior angles of specific quadrilaterals and triangles.
2. Prove properties of given parallelograms and kites.

Polygons

• Regular Polygon: All sides and angles equal.
• Interior Angle Sum Formula: (n-2) * 180°.
• Interior Angle of Regular Polygon: (n-2) * 180° / n where n is number of sides.

Examples

• 3 Sides: Equilateral triangle, angles sum 180°.
• 4 Sides: Square, angles sum 360°.
• 5 Sides: Pentagon, angles sum 540°.

Mixed and Challenge Exercises

• Solve geometric problems using properties and theorems of different shapes, focusing on proving congruence and similarity in various configurations.