Geometry/Postulates & Definitions

A

Acute Angle

See Angle

Addition Property of Equality

For any real numbers a, b, and c, if a = b, then a + c = b + c.

Angle

Angle ABC is acute.

A figure is an angle if and only if it is composed of two rays which share a common endpoint. Each of these rays (or segments, as the case may be) is known as a side of the angle (For example, $\overline{AB}$ in the illustration at right), and the common point is known as the angle's vertex (point B in the illustration). Angles are measured by the difference of their slopes. The units for angle measure are radians and degrees. Angles may be classified by their degree measure.

• Acute Angle: an angle is an acute angle if and only if it has a measure of less than 90°
• Right Angle: an angle is an right angle if and only if it has a measure of exactly 90°
• Obtuse Angle: an angle is an obtuse angle if and only if it has a measure of greater than 90 degrees

C

Center of a circle

Point P is the center of circle C if and only if all points in circle C are equidistant from point P and point P is contained in the same plane as circle C.

Circle

A collection of points is said to be a circle with a center at point P and a radius of some distance r if and only if it is the collection of all points which are a distance of r away from point P and are contained by a plane which contain point P.

Concave

A polygon is said to be concave if and only if it contains at least one interior angle with a measure greater than 180° exclusively and less than 360° exclusively.

Corresponding angles

There are four pairs of corresponding angles: ∠1 and ∠3, ∠2 and ∠4, ∠5 and ∠7, and ∠6 and ∠8.

Two angles formed by a transversal intersecting with two lines are corresponding angles if and only if one is on the inside of the two lines, the other is on the outside of the two lines, and both are on the same side of the transversal.

Corresponding Angles Postulate

If two lines cut by a transversal are parallel, then their corresponding angles are congruent.

Corresponding Parts of Congruent Triangles are Congruent Postulate

The Corresponding Parts of Congruent Triangles are Congruent Postulate (CPCTC) states:

If ∆ABC ≅ ∆XYZ, then all parts of ∆ABC are congruent to their corresponding parts in ∆XYZ. For example:

• $\overline{AB}$ ≅ $\overline{XY}$

• $\overline{BC}$ ≅ $\overline{YZ}$

• $\overline{AC}$ ≅ $\overline{XZ}$

• ∠ABC ≅ ∠XYZ

• ∠BCA ≅ ∠YZX

• ∠CAB ≅ ∠ZXY

CPCTC also applies to all other parts of the triangles, such as a triangle's altitude, median, circumcenter, et al.

CPCTC

See Corresponding Parts of Congruent Triangles are Congruent Postulate

D

Diameter

A line segment is the diameter of a circle if and only if it is a chord of the circle which contains the circle's center.

See Circle

L

Line

A collection of points is a line if and only if the collection of points is perfectly straight (aligned), is infinitely long, and is infinitely thin. Between any two points on a line, there exists an infinite number of points which are also contained by the line.

Line segment

Line segment AB

A collection of points is a line if and only if it is perfectly straight, is infintely thin, and has a finite length. A line segment is measured by the shortest distance bewteen the two extreme points on the line segment, known as endpoints. Between any two points on a line segment, there exists an infinite number of points which are also contained by the line segment.

P

Parallel lines

Two lines or line segments are said to be parallel if and only if the lines are contained by the same plane and have no points in common if continued infinitely.

Parallel planes

Two planes are said to be parallel if and only if the planes have no points in common when continued infinitely.

Plane

An object is a plane if and only if it is a two-dimensional object which has no thickness or curvature and continues infinitely. A plane can be defined by three points. A plane may be considered to be analogous to a piece of paper[1].

Point

A point is a zero-dimensional mathematical object representing a location in one or more dimensions[2]. A point has no size; it has only location.

R

Radius

The radius of a circle is the distance between any given point on the circle and the circle's center.

See Circle

Ray

A ray is a straight collection of points which continues infinitely in one direction. The point at which the ray stops is known as the ray's endpoint. Between any two points on a ray, there exists an infinite number of points which are also contained by the ray.