Essential Geometry Concepts: Triangles, Polygons, and Angles

Triangles

• Median: The segment connecting a vertex to the midpoint of the opposite side.
• Centroid (G): The intersection point of the three medians. It divides each median into two segments.
• Height (Altitude): The perpendicular segment from a vertex to the opposite side or its extension.
• Orthocenter: The point where the three altitudes intersect.
• Perpendicular Bisector: The line perpendicular to a segment at its midpoint.
• Angle Bisector: The ray that divides an angle into two equal angles.
• Circumcenter: The intersection point of the three perpendicular bisectors. It is equidistant from the three vertices.
• Incenter: The intersection point of the three angle bisectors. It is equidistant from the sides of the triangle.

Polygons

• Polygon: A plane region bounded by a closed path of segments.
• Regular Polygon: A polygon with all equal angles and equal sides.
• Irregular Polygon: A polygon that does not have all equal angles and sides.
• Center: A point inside the polygon equidistant from all vertices.
• Radius: A segment connecting the center to a vertex.
• Apothem: A segment perpendicular to a side, connecting the center to the midpoint of the side.
• Note: In a regular hexagon, the radius is equal to the side length.

Planar Elements

• Ray: A part of a line that starts at a point and extends infinitely in one direction.
• Segment: A part of a straight line between two points, A and B, which are the endpoints.
• Intersecting Lines: Two lines that share one common point. Perpendicular lines form four equal 90-degree angles.
• Parallel Lines: Two lines that have no points in common.
• Angle: The region between two rays that share a common origin, represented as ∠AOB.
• Degree (º): A unit of angular measure defined by dividing a right angle into 90 equal parts.
• Minute ('): An angle measure defined by dividing 1 degree into 60 equal parts.
• Second (''): An angle measure defined by dividing 1 minute into 60 equal parts.

Types of Angles

• Acute: Less than 90°.
• Right: Exactly 90°.
• Obtuse: More than 90°.
• Straight (Llano): 180°.
• Full (Completed): 360°.
• Convex: Between 0° and 180°.
• Concave: Between 180° and 360°.
• Complementary: Two angles that add up to 90°.
• Supplementary: Two angles that add up to 180°.
• Vertical Angles: Angles sharing a common vertex where the sides of one are the extensions of the sides of the other. Vertical angles are equal.