Geometry Essentials

Cheatsheet Content

A. Basic Definitions & Concepts Point: A location in space, no size or dimension. Line: A straight path extending infinitely in two directions. Defined by two points. Notation: $\overleftrightarrow{AB}$ or $l$. Plane: A flat surface extending infinitely in all directions. Defined by three non-collinear points. Notation: Plane $ABC$ or $P$. Line Segment: Part of a line with two endpoints. Notation: $\overline{AB}$. Length denoted as $AB$. Ray: Part of a line with one endpoint, extending infinitely in one direction. Notation: $\overrightarrow{AB}$ (starts at A, goes through B). Angle: Formed by two rays sharing a common endpoint (vertex). Types: Acute ($ 90^\circ, Complementary angles sum to $90^\circ$. Supplementary angles sum to $180^\circ$. Perpendicular Lines: Lines intersecting at a $90^\circ$ angle. Notation: $l \perp m$. Parallel Lines: Lines in the same plane that never intersect. Notation: $l \parallel m$. Transversal: A line intersecting two or more other lines. Corresponding Angles: Equal (e.g., $\angle1 = \angle5$). Alternate Interior Angles: Equal (e.g., $\angle3 = \angle6$). Alternate Exterior Angles: Equal (e.g., $\angle1 = \angle8$). Consecutive Interior Angles: Supplementary (e.g., $\angle3 + \angle5 = 180^\circ$). B. Triangles Sum of Angles: The sum of interior angles in any triangle is $180^\circ$. Types by Sides: Equilateral: All 3 sides equal, all 3 angles $60^\circ$. Isosceles: 2 sides equal, angles opposite those sides are equal. Scalene: No sides equal, no angles equal. Types by Angles: Acute: All angles acute. Right: One angle is $90^\circ$. Obtuse: One angle is obtuse. Pythagorean Theorem (Right Triangles): $a^2 + b^2 = c^2$, where $a, b$ are legs and $c$ is the hypotenuse. Area: $A = \frac{1}{2} \cdot \text{base} \cdot \text{height} = \frac{1}{2}bh$. Perimeter: $P = a+b+c$. Congruence Postulates: SSS: Side-Side-Side SAS: Side-Angle-Side ASA: Angle-Side-Angle AAS: Angle-Angle-Side HL (Right $\triangle$s only): Hypotenuse-Leg Similarity Postulates: AA: Angle-Angle SSS: Side-Side-Side (proportional sides) SAS: Side-Angle-Side (proportional sides, congruent angle) Special Right Triangles: $45^\circ-45^\circ-90^\circ$: Sides in ratio $x:x:x\sqrt{2}$. $30^\circ-60^\circ-90^\circ$: Sides in ratio $x:x\sqrt{3}:2x$. C. Quadrilaterals Sum of Interior Angles: $360^\circ$. Parallelogram: Opposite sides parallel & equal, opposite angles equal, diagonals bisect each other. Area: $A = \text{base} \cdot \text{height} = bh$. Rectangle: Parallelogram with four right angles. Diagonals are equal. Area: $A = \text{length} \cdot \text{width} = lw$. Perimeter: $P = 2l + 2w$. Square: Rectangle with four equal sides. Diagonals are perpendicular bisectors. Area: $A = s^2$. Perimeter: $P = 4s$. Rhombus: Parallelogram with four equal sides. Diagonals are perpendicular bisectors of each other. Area: $A = \frac{1}{2} d_1 d_2$ (where $d_1, d_2$ are diagonals). Trapezoid: One pair of parallel sides (bases). Area: $A = \frac{1}{2} (b_1 + b_2)h$. Kite: Two pairs of adjacent sides are equal. Diagonals are perpendicular. Area: $A = \frac{1}{2} d_1 d_2$. D. Polygons Sum of Interior Angles: $S = (n-2) \cdot 180^\circ$, where $n$ is the number of sides. Each Interior Angle (Regular Polygon): $A = \frac{(n-2) \cdot 180^\circ}{n}$. Sum of Exterior Angles: $360^\circ$ (for any convex polygon). Each Exterior Angle (Regular Polygon): $E = \frac{360^\circ}{n}$. Number of Diagonals: $D = \frac{n(n-3)}{2}$. E. Circles Circumference: $C = 2\pi r$ or $C = \pi d$. Area: $A = \pi r^2$. Radius: Distance from center to any point on the circle ($r$). Diameter: Distance across the circle through the center ($d=2r$). Chord: A segment connecting two points on the circle. Secant: A line intersecting a circle at two points. Tangent: A line intersecting a circle at exactly one point. Tangent is perpendicular to the radius at the point of tangency. Arc Length: $L = \frac{\theta}{360^\circ} \cdot 2\pi r$ (where $\theta$ is central angle in degrees). Sector Area: $A = \frac{\theta}{360^\circ} \cdot \pi r^2$ (where $\theta$ is central angle in degrees). Inscribed Angle Theorem: An inscribed angle is half the measure of its intercepted arc. F. Solids (3D Shapes) Prism: Volume: $V = \text{Area of Base} \cdot \text{height} = Bh$. Surface Area: Sum of areas of all faces. (e.g., Rectangular Prism: $SA = 2(lw + lh + wh)$). Cylinder: Volume: $V = \pi r^2 h$. Surface Area: $SA = 2\pi r^2 + 2\pi r h$. Pyramid: Volume: $V = \frac{1}{3} \cdot \text{Area of Base} \cdot \text{height} = \frac{1}{3}Bh$. Cone: Volume: $V = \frac{1}{3} \pi r^2 h$. Surface Area: $SA = \pi r^2 + \pi r l$ (where $l$ is slant height). Sphere: Volume: $V = \frac{4}{3} \pi r^3$. Surface Area: $SA = 4\pi r^2$. G. Coordinate Geometry Distance Formula: Between $(x_1, y_1)$ and $(x_2, y_2)$ is $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$. Midpoint Formula: Midpoint of a segment between $(x_1, y_1)$ and $(x_2, y_2)$ is $M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$. Slope Formula: Slope $m = \frac{y_2-y_1}{x_2-x_1}$. Equation of a Line: Slope-intercept form: $y = mx + b$. Point-slope form: $y - y_1 = m(x - x_1)$. Parallel Lines: Have the same slope ($m_1 = m_2$). Perpendicular Lines: Have negative reciprocal slopes ($m_1 m_2 = -1$). Equation of a Circle: $(x-h)^2 + (y-k)^2 = r^2$, where $(h,k)$ is the center and $r$ is the radius.