Math Study Lounge - Geometry Basics

Key Geometry Terms

Angle - either the figure formed by two rays starting from a common point or the measure of such a figure.
Complimentary Angles - a pair of angles whose sum is 90˚.
Supplementary Angles - a pair of angles whose sum is 180˚.
Congruent Angles - angles with the same angle measurement.
Vertical Angles - angles at an intersection of two lines that are not adjacent to each other.
Interior Angles - pair of angles that are in the interior of a polygon, OR angles inside or between two parallel lines.
Exterior Angles - pair of angles formed by the side of a polygon and an adjacent, extended side OR angles on the outside of two parallel lines.
Circle - a figure in which all points are the same distance from the center, that distance being known as the radius.
Pi - ratio of the circumference to the diameter of a circle. Pi is denoted by the Greek symbol “π” and is approximately 3.14159, though it is widely known to continue to an infinite number of decimals and thus, is irrational.
Pythagorean Theorem - the sum of the squares for the legs of a right triangle equal the square of the hypotenuse
Special Right Triangles - right triangles where certain relationships always hold true. These relationships, when recognized, are useful since they reduce sometimes complex work to solving a proportion or even mental math.
Regular polygon - a geometric figure with three or more sides in which all interior angles are congruent.
Area - size of a planar region or surface of a solid. Areas are measured in “square” units.
Volume - amount of space taken up by or enclosed by an object. Volumes are measured in “cubic” units.
Degree - unit of angle measurement equal to 1/360 of a circle’s circumference.
Radian - unit of angle measurement that is the ratio ofthe circle arc cut off by an angle to the central angle.
Sine - on a unit circle, the “y” coordinate for any intersection between a standard angle and the circle. In a right triangle, the ratio of the length of the opposite side to the length of the hypotenuse.
Cosine - on a unit circle, the “x” coordinate for any intersection between a standard angle and the circle. In a right triangle, the ratio of the length of the adjacent side to the length of the hypotenuse.
Tangent - the ratio of sine to cosine or, in a right triangle, the ratio of the length of the opposite side to the length of the adjacent side.
Secant - reciprocal of cosine (sec θ = 1 / cos θ).
Cosecant - reciprocal of sine (csc θ = 1 / sin θ).
Cotangent - reciprocal of tangent (cot θ = 1 / tan θ).
Vector - a directed line segment. The length of the segment indicates magnitude or amount in a measurement and the direction of the segment, indicated with an arrow, indicates the direction of the measurement

Basic Rules of Angles

Vertical angles are always congruent!
The alternate interior angles of parallel lines cut by a transversal are congruent.
The alternate exterior angles of parallel lines cut by a transversal are congruent

Shapes and Figures

Pythagorean Theorem: a² + b² = c².
3 : 4 : 5 right triangle: legs that are 3 and 4 units long resulting in a hypotenuse that is 5 units long.
5:12:13 right triangle: legs that are 5 and 12 units long resulting in a hypotenuse that is 13 units long.
30-60-90: when the angles are 30, 60 and 90 degrees, the sides are in ratios of 1 : √3 : 2 .
45-45-90: when the angles are 45, 45 and 90 degrees, the sides are in ratios of
√2 : √2 : 2 .
All regular polygons: if “n” is the number of sides, then the sum of the interior angles may be found by using: ( n – 2 ) ( 180˚ )

Area and Volume Formulas

It is possible to approximate the area of a figure by using smaller figures, such as triangles and rectangles, to approximate the figure.

We can break the figure into a number of rectangles and find the area of each:
3x2 + 3x4 + 3x6.5 + 3x11 = 70.5 square units

This may be done alternatively:
3x2 + 3x4 + 3x6.5 = 37.5 square units
We can average this and the previous result to get a better approximation than either alone: 59 square unit