Geometry and Trigonometry Fundamentals

Geometry

Points, Lines, and Planes

• A point has no size, only location
• A line extends infinitely in two directions
• A plane is a flat surface that extends infinitely in all directions
• Intersection of two lines: one point
• Intersection of two planes: a line
• Intersection of three planes: a point

Angles

• Acute angle: less than 90°
• Right angle: exactly 90°
• Obtuse angle: greater than 90°
• Straight angle: exactly 180°
• Reflex angle: greater than 180°
• Complementary angles: add up to 90°
• Supplementary angles: add up to 180°

Properties of Shapes

• Congruent shapes: same size and shape
• Similar shapes: same shape, different size
• Interior angle sum of a triangle: 180°
• Interior angle sum of a quadrilateral: 360°
• Exterior angle of a polygon: equal to the sum of the interior angles

Circle Theorems

• Angle at the center: twice the angle at the circumference
• Angles in the same segment: equal
• Cyclic quadrilateral: opposite angles add up to 180°
• Tangent and radius: perpendicular
• Tangents from the same point: equal in length

Trigonometry

Right-Angled Triangles

• Sine (sin): opposite side / hypotenuse
• Cosine (cos): adjacent side / hypotenuse
• Tangent (tan): opposite side / adjacent side
• Pythagorean identity: sin^2 + cos^2 = 1
• Trigonometric ratios: sin, cos, and tan of common angles (30°, 45°, 60°)

Graphs of Trigonometric Functions

• Sine graph: periodic, wave-like, amplitude 1
• Cosine graph: periodic, wave-like, amplitude 1
• Tangent graph: periodic, wave-like, asymptotes at 90° and 270°

Solving Triangles

• Angle-sum identity: sin(A)/a = sin(B)/b = sin(C)/c
• Sine rule: a/sin(A) = b/sin(B) = c/sin(C)
• Cosine rule: a^2 = b^2 + c^2 - 2bc cos(A)
• Area of a triangle: (1/2)ab sin(C)