An exterior angle of a triangle is greater than either of the two non-adjacent (remote) interior angles.

180° rotation: (x,y) → (-x,-y). Both coordinates change sign. Equivalent to reflecting over the origin.

Vertical angles are opposite angles formed when two lines intersect. They are always equal.

30-60-90 sides: 1 : √3 : 2. Short leg is x, long leg is x√3, hypotenuse is 2x.

Area of regular polygon = ½ × perimeter × apothem. The apothem is the distance from center to the middle of a side.

The centroid is the intersection of the three medians. It divides each median in a 2:1 ratio from vertex to midpoint.

Volume of rectangular prism = length × width × height.

Sector area = (θ/360) × πr². It's a fraction of the whole circle's area.

Space diagonal of a cube = s√3. Using 3D Pythagorean theorem: d = √(s²+s²+s²) = s√3.

Area of rhombus = (d₁ × d₂)/2. The diagonals of a rhombus bisect each other at right angles.

Hemisphere SA = 2πr² (curved surface) + πr² (flat circular base) = 3πr².

Lateral SA of rectangular prism = 2h(l + w). This excludes the two base (top and bottom) faces.

Triangle Inequality: the sum of any two sides of a triangle must be greater than the third side.

Area of a triangle = ½ × base × height. The ½ accounts for the triangle being half a rectangle.

Regular octagon interior angle = (8-2)×180°/8 = 1080°/8 = 135°.

In a right triangle, if altitude h is drawn to hypotenuse: h² = (segment₁)(segment₂) — the geometric mean.

A tangent line touches the circle at exactly one point. It is always perpendicular to the radius at the point of tangency.

Arc length = (θ/360) × 2πr, where θ is the central angle in degrees.

Volume of pyramid = (1/3) × base area × height. One-third of a prism with the same base and height.

Distance formula: d = √((x₂-x₁)² + (y₂-y₁)²). It comes from the Pythagorean theorem.