This section is devoted to providing readers with precise definitions of geometric terms.
Axiom: A statement that cannot be proven but can be intuited as true.
Theorem: A statement that can be proven true using axioms, definitions, and other theorems.
Set: A collection of objects.
Intersection: Given a group of sets. The intersection is the set of points that are included in all of the given sets. This is denoted by $\cap$
Subset: A set of elements made from the elements a different set. (Ex: the set containing the letters a,b,c is a subset of the set that contains all the letters in the alphabet.)
Union: Given a group of sets, the union of the sets is defined as every point included in any of the given sets. This is denoted by $\cup$.
Space: The set of all points in a 3-dimensional realm.
Plane: The set of all points in a 2-dimensional realm.
Line: The set of all points in a 1-dimensional realm. Often a line is represented by $\overleftrightarrow{AB}$ where A and B are two elements of the line. Thus $\overleftrightarrow{AB}$ is read "Line AB". The degree measure of a line is equal to 180$^{\circ}$.
The distance between two points is usually denoted AB and is read as "the distance from point A to point B"
Point: An individual element of a space. Commonly denoted by capital letters, thus A is read "point A".
Collinear Points: A set of points that can be connected by a single line.
Ray: A subset of a line that starts from a specific point on the line, passes through another point on the line, and continues infinitely with the line. Given $\overleftrightarrow{AB}$ we denote a ray of this line that starts at point A and passes through B $\overrightarrow{AB}$. This is read "Ray AB
Line Segment: Given a line, a line segment is a subset of the line from one specific point to another specific point. This line segment denoted by the points AB is denoted by $\overline{AB}$ and is read "Line segment A, B."
Angle: Given three points A, B, and C. An angle is the set of points defined by the union of two rays such that $\angle{ABC}=\overrightarrow{BA}\cup\overrightarrow{BC}$ where the middle letter represents the only point in the intersection of the two rays. This notation is read "Angle ABC"
The degree measure of an angle: (denoted $m\angle{ABC}$ for a general angle ABC) The amount of rotation measured in degrees from $\overrightarrow{BA}$ to $\overrightarrow{BC}$ or vice versa.
Parallel Lines: Lines whose intersection includes no points. Lines are denoted as parallel using this notation $\parallel$
Transversal: Given two or more lines, a transversal is a line that intersects each of the lines given.
Triangle: Given three non-collinear points A, B, C. $\triangle{ABC}$ is defined as $\overline{AB}\cup\overline{BC}\cup\overline{CA}$ and is read "triangle ABC"
Hypotenuse: The side of a right triangle with the largest length.
Leg: Any side of a right triangle that is not a hypotenuse.
Supplementary angles$^{5}$: Two angles whose degree measures sum to 180$^{\circ}$.
Complementary angles: Two angles whose degree measures sum to 90$^{\circ}$.
Corresponding Angles: Two angles are corresponding given that they occupy the same relative position at each intersection.
Alternate Interior Angles: Alternate interior angles are the angles formed when a transversal intersects two coplanar lines. They lie on the inner side of the parallel lines but on the opposite sides of the transversal.
Alternate Exterior Angles: Alternate exterior angles are the angles formed when a transversal intersects two coplanar lines. They lie on the exterior of the parallel lines but on the opposite sides of the transversal.
Same-Side Interior Angles: These are the two angles that are on the inner side of the parallel lines intersected by the transversal.
Congruence: Congruent line segments are defined as line segments whose lengths are the same. Congruent angles are defined as angles whose degree measures are the same. Congruence is denoted by $\cong$.
5: Complementary and supplementary angles review (article). (n.d.). Khan Academy. Retrieved November 1, 2022, from https://www.khanacademy.org/math/cc-seventh-grade-math/cc-7th-geometry/cc-7th-angles/a/complementary-and-supplementary-angles-review