Geometry Theorems and Formulas: A Quick Reference

Posted on Feb 8, 2025 in Mathematics

Geometry Theorems and Formulas

Theorems and Postulates

  • AA (Angle-Angle): If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.
  • SAS (Side-Angle-Side): If two sides and the included angle of one triangle are proportional to two sides and the included angle of another triangle, then the triangles are similar (ab/de = ac/df).

Shapes

  • Opposite angles are always supplementary.

Parallelogram

  • Opposite sides are parallel and congruent.
  • Opposite angles are congruent.
  • Diagonals bisect each other.
  • Proving a quadrilateral is a parallelogram:
    • Show both pairs of opposite sides are parallel.
    • Show both pairs of opposite sides are congruent.
    • Show that both pairs of opposite angles are congruent.
    • Show that the diagonals bisect each other.
    • Show that one pair of opposite sides is both congruent and parallel.

(Alternate Interior Angles = Z, Same-Side Interior Angles = C, Corresponding Angles = F)

Rectangle

  • Has right angles.
  • Diagonals are congruent.

Rhombus

  • Consecutive sides are congruent.
  • Diagonals are perpendicular.
  • Each diagonal bisects two angles at the corner.

Right Triangle

  • The midpoint of the hypotenuse is equidistant from the three vertices.

Chapter 8: Geometric Mean and Special Right Triangles

Geometric Mean

Question: Find the geometric mean between 5 and 11.

Answer: 5/x = x/11, then take the square root of the answer.

When an altitude is drawn to the hypotenuse, it creates three right triangles.

Pythagorean Theorem

biggest hypotenuse = c2

Special Right Triangles

  • 45-45-90: Hypotenuse = √2 * leg
  • 30-60-90: Hypotenuse = 2 * short leg, Long leg = √3 * short leg

Trigonometry (SOH CAH TOA)

If given an angle (e.g., 56°) and one side (e.g., adjacent = 5), then to find another side (e.g., opposite), use tan(56°) = x/5.

Chapter 11: Area Formulas

  • Rectangle: base * height
  • Parallelogram: base * height (make altitude to get height, may need trig or special triangles)
  • Triangle: (base * height) / 2
  • Rhombus: (diagonal 1 * diagonal 2) / 2
  • Trapezoid: (1/2) * height * (base 1 + base 2)

Apothem

  • Triangle central angle = 120° (360°/3) makes a 30-60-90 triangle.
  • Square: 90° makes a 45-45-90 triangle.
  • Hexagon: 60° makes a 30-60-90 triangle.
  • Pentagon: 72° makes a 54-36-90 triangle (may need trig).

Circles

  • Circumference: 2πr
  • Area: πr2

Arc Length

If the radius is 5 and there are 60 degrees in arc AB, then 60/360 = 1/6. The length of AB = (1/6) * (2π * 5) = (5/3)π.

Sector Area

To find the area of sector AOB, then AOB = (1/6) * (π * 52) = 25π/6.

When the angle = 120° and the radius is 9, then it is (120/360) * (2π * 9) = (1/3) * (18π) = 6π.

To find the area of sector AOB, then (120/360) * (π * 92) = (1/3) * (81π) = 27π.

Major Arc

To find the major arc (one bigger than 180°), subtract the given angle (if it is the smaller one) from 360°. If it is 120°, then (240/360) * (2π * 9) = (2/3) * (18π) = 12π.

Chapter 9: Circle Theorems

Tangent Theorems

  1. If a line is tangent to a circle, then the line is perpendicular to the radius drawn to the point of tangency, creating right triangles.
  2. Tangents to a circle from a point are congruent.
  3. If a line in the plane of a circle is perpendicular to a radius at the outer endpoint, then the line is tangent to the circle.
  4. Congruent arcs have congruent chords, and congruent chords have congruent arcs.
  5. The measure of an inscribed angle is 1/2 of its intercepted arc; central angles are equal to the intercepted arc.
  6. The measure of an angle formed by a chord and a tangent is 1/2 of the intercepted arc.
  7. When there is a tangent segment and a secant segment from outside the circle, the product of the secant segment and its external segment equals the square of the tangent.

Chapter 12: Volume

Theorems

The lateral area of a right prism equals the perimeter of a base times the height of the prism (LA = ph). To get p, add everything on the bottom, then multiply it by the height.

The volume of a right prism equals the area of a base times the height of the prism (V = bh).

On a right trapezoid prism, find A (lateral area), B (total area), and C (volume).

  • A: Add up all the side lengths (e.g., 28), then multiply by the height (e.g., 10), which is 280 (LA).
  • B: Find the area of the base. Since it is a trapezoid, take half the height multiplied by the sum of both bases.