Essential Geometric Definitions and Theorems

Posted on Mar 12, 2025 in Mathematics

Acute Angle: An angle that measures less than 90º.
Angles: Formed by two rays that share a common endpoint, provided that the two rays are non-collinear.
Angle Bisector: A ray that contains the vertex and divides the angle into two congruent angles.
Complementary Angle: A pair of angles that sum 90º.
Congruent Angles: Two angles are congruent if and only if they have equal measure.
Corresponding Angles: Angles that are created at the same location at each intersection where a transversal line crosses other lines.
Line: A figure that has only one direction, is infinite, and has no curvature.
Line Segment: Two points, called the endpoints of the segment, and all the points between them that are collinear with the two points.
Obtuse Angle: An angle that measures more than 90º.
Parallel Lines: Lines that never touch or are coplanar.
Parallel Line Conjecture: If two parallel lines are cut by a transversal, then the corresponding angles, the alternate interior angles, and alternate exterior angles are congruent.

Arc: Two points on the circle and a continuous part of the circle between the two points.
Circle: A set of points at a given distance from a given point.
Chord: A line segment whose endpoints lie on a circle.
Diameter: A chord that passes through the center.
Minor Arc: Two points on the circle and a continuous part of the circle between the two points that is smaller than a semicircle.

Altitude of a Triangle: A perpendicular segment from a vertex of the triangle to the opposite side or to a line containing the triangle.
Circumcenter: The point where the perpendicular bisectors of the sides of a triangle intersect; it is the center of the circle that circumscribes the triangle.
Isosceles Triangle Theorem: The angles opposite the equal sides of an isosceles triangle are equal.
Median of a Triangle: The segment connecting the vertex of a triangle to the midpoint of its opposite side.
Midpoint: The point of the segment that is the same distance from both endpoints. The midpoint bisects the segment.
Orthocenter: The point where all three altitudes of a triangle intersect.

Collinear Points: Two or more points that lie on the same line.
Congruence: When two segments have equal measures or lengths.
Convex Polygons: A polygon is convex when all the diagonals are inside of the polygon.
Diagonal of a Polygon: A line that connects two non-consecutive vertices.
Equiangular Polygon: A polygon where all angles are equal.
Equilateral Polygon: A polygon in which all the sides measure the same.
Parallelogram: A quadrilateral with two pairs of parallel sides.
Perimeter: The sum of the lengths of the sides of a shape.
Geometric Figures: Combination of different elements: points, lines, segments. There are polygons or non-polygons.

Altitude of a Cylinder: Any perpendicular segment from the plane of one base to the plane of the other.
Cone: A geometric solid that has a circle as a base; its radius is the radius of the circle, and the vertex is the greatest perpendicular distance from the base.
Cylinder: A geometric solid that has two congruent parallel bases that are interiors and circles.
Hemisphere: Half of a sphere.

Geometry: The part of Mathematics which studies the properties of the geometric figures in the space, such as points, lines, polyhedrons, planes, and polytopes (polygons, parallels, perpendiculars, surfaces, curves, polyhedra).
Area: A measure of the amount of two-dimensional space inside a boundary.