Mathematical System 2

• Definition of a Segment:

  • Segment AB, denoted by AB or BA, is the union of points A, B and all the points between them
  • Points A and B are called the endpoints of the segment
• Definition of Between:

  • Point U is between F and N if and only if F, U, and N are distinct points of the same line and FU + UN = FN
• Definition of Collinear Points and Coplanar Points:

  • Collinear points are points on the same line
  • Noncollinear points are points not on the same line
  • Coplanar points are points on the same plane
• Definition of a Ray:

  • A ray is a part of a line that has one endpoint and goes on infinitely in one direction
• Definition of an Angle:

  • An angle is the union of two noncollinear rays with a common endpoint
• Definition of Congruent Angles:

  • Two angles are congruent if and only if their measures are equal
• Definitions of Acute Angle, Right Angle, and Obtuse Angle:

  • An acute angle is an angle with a measure greater than 0° but less than 90°
  • A right angle is an angle with a measure of 90°
  • An obtuse angle is an angle with a measure greater than 90° but less than 180°
• Definition of Adjacent Angles:

  • Adjacent angles share a common vertex and a common side, but do not overlap
• Definition of Supplementary Angles:

  • Two angles are supplementary when the sum of their angles is 180°
• Definition of Linear Pairs:

  • A linear pair of angles is formed when two lines intersect
  • Two angles are linear if they are adjacent angles formed by two intersecting lines and are supplementary
• Definition of Vertical Angles:

  • Vertical angles are opposite angles formed by two intersecting lines
• Definition of Perpendicular Lines:

  • Perpendicular lines are two lines that intersect to form a right angle
• Definition of Perpendicular Bisector:

  • A perpendicular bisector of a line segment XZ is a line segment perpendicular to XZ and passing through the midpoint
• Definition of Polygon:

  • A polygon is a closed figure such that the union of three or more coplanar segments, which intersect at endpoints, with each endpoint shared by exactly two noncollinear segments
• Definition of Convex Polygon:

  • A polygon is convex if the lines containing the sides of the polygon do not contain points in its interior
• Definition of Nonconvex (Concave) Polygon:

  • A polygon is nonconvex (concave) if at least one of its sides is contained in a line, which contains also points in the interior of the polygon
• Definition of Regular Polygon:

  • A regular polygon is a polygon that is both equilateral and equiangular
• Definition of a Triangle:

  • A triangle is a three-sided polygon
• Definition of Angle Bisector of a Triangle:

  • An angle bisector of a triangle is a segment contained in the ray, which bisects the angle of the triangle, and whose endpoints are the vertex of this angle and a point on the opposite side
• Definition of an Altitude of a Triangle:

  • A segment is an altitude of a triangle if it is perpendicular from a vertex of the triangle to the line that contains the opposite side
• Definition of Median:

  • A segment is a median of a triangle if its endpoints are a vertex and the midpoint of the opposite side
• Types of triangles:

  • Scalene triangle: no congruent sides
  • Isosceles triangle: at least two congruent sides
  • Equilateral triangle: all sides congruent