What are complementary and supplementary angles? — Complementary: Two angles that add to 90°. Supplementary: Two angles that add to 180°. Memory trick: C comes before S in the alphabet, 90 comes before 180.
What are vertical angles? — When two lines intersect, the angles directly across from each other are vertical angles — they are always equal. The adjacent angles are supplementary.
What are alternate interior angles? — When a transversal crosses two parallel lines, alternate interior angles (on opposite sides of the transversal, between the parallel lines) are equal.
What are co-interior (same-side interior) angles? — When a transversal crosses two parallel lines, co-interior angles (on the same side, between the parallel lines) are supplementary — they add up to 180°.
What is the Pythagorean theorem? — In a right triangle: a² + b² = c² where c is the hypotenuse (longest side, opposite the right angle). Example: 3² + 4² = 5² (the 3-4-5 right triangle).
What are common Pythagorean triples? — Sets of whole numbers satisfying a²+b²=c²: 3-4-5, 6-8-10, 5-12-13, 8-15-17, 7-24-25. Multiples of these also work (e.g., 9-12-15).
What are the triangle congruence rules? — SSS (all three sides equal), SAS (two sides + included angle), ASA (two angles + included side), AAS (two angles + non-included side), RHS (right angle + hypotenuse + side).
What are the triangle similarity rules? — AA (two angles equal — most common), SSS (all sides in proportion), SAS (two sides in proportion + included angle equal). Similar triangles have equal angles and proportional sides.
What is the sum of interior angles of a polygon? — (n−2) × 180° where n = number of sides. Triangle = 180°. Quadrilateral = 360°. Pentagon = 540°. Hexagon = 720°.
What is the exterior angle of a regular polygon? — 360° ÷ n where n = number of sides. The sum of all exterior angles of any convex polygon = 360°. Example: Regular hexagon exterior angle = 60°.
What is the area of a triangle? — A = ½ × base × height. Also: A = ½ab sin C (using two sides and included angle). Also: Heron's formula A = √(s(s−a)(s−b)(s−c)) where s = (a+b+c)/2.
What is the area of a circle? — A = πr² where r is the radius. Circumference = C = 2πr = πd where d is the diameter. π ≈ 3.14159.
What is the area of a trapezoid (trapezium)? — A = ½ × (a + b) × h where a and b are the parallel sides (bases) and h is the perpendicular height between them.
What is the volume of a cylinder? — V = πr²h where r = radius of base, h = height. Surface area = 2πr² + 2πrh (two circles + curved surface).
What is the volume of a cone? — V = ⅓πr²h. Slant height l = √(r²+h²). Surface area = πr² + πrl (base + curved surface).
What is the volume of a sphere? — V = (4/3)πr³. Surface area = 4πr².
What is the volume of a pyramid? — V = ⅓ × base area × height. For a square pyramid: V = ⅓ × s² × h.
What are the circle theorems? — 1. Angle at centre = 2 × angle at circumference (same arc). 2. Angles in same segment are equal. 3. Angle in semicircle = 90°. 4. Opposite angles in cyclic quadrilateral add to 180°. 5. Tangent is perpendicular to radius at point of contact.
What is the arc length formula? — Arc length = (θ/360°) × 2πr for angle θ in degrees. In radians: Arc length = rθ. A full circle arc (θ=360°) = full circumference = 2πr.
What is the sector area formula? — Sector area = (θ/360°) × πr² for angle θ in degrees. In radians: Sector area = ½r²θ.
What is the equation of a circle? — (x−h)² + (y−k)² = r² where (h,k) is the centre and r is the radius. Standard form centred at origin: x² + y² = r².
What is the midpoint theorem for triangles? — The segment connecting the midpoints of two sides of a triangle is parallel to the third side and half its length.
What is the centroid of a triangle? — The point where all three medians intersect. It is located ⅔ of the way from each vertex to the opposite midpoint. Also the triangle's centre of mass.
What are special right triangles? — 30-60-90: Sides in ratio 1 : √3 : 2. The side opposite 30° is shortest, opposite 60° is √3 times the shortest. 45-45-90: Sides in ratio 1 : 1 : √2.
What is the surface area of a cube vs rectangular prism? — Cube: SA = 6s² (6 identical square faces). Rectangular prism (cuboid): SA = 2(lw + lh + wh) where l=length, w=width, h=height.
