Foundations of Geometry Definitions - Chapter 1
segment Given two points, A and B, the segment AB is the set of points whose members are
the points A and B and all the points that lie on the line AB between A and B. (p. 14)
circle Given two points O and A, the set of all points P such that segment OP is congruent to
segment OA is called a circle with center O, and each segment OP is called a radius. (p. 15)
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ray The ray AB is the following set of points lying on the line AB: those points on the segment
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AB, and those points C on AB such that B is between A and C. The ray AB is said to emanate
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from A and to be part of line AB. (p. 16)
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opposite rays Rays AB and AC are opposite rays if they are distinct, if they emanate from
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the same point A, and if they are part of the same line AB=AC. (p. 16)
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angle An angle with vertex A is a point A together with two distinct, nonopposite rays AB
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and AC (called sides of the angle) emanating from A. (p. 17)
supplementary angle If two angles 6 BAD and 6 CAD have a common side AD and the other
two sides AB and AC form opposite rays, the angles are supplements of each other, or supplementary
rays. (p. 17)
right angle An angle 6 BAD is a right angle if it has a supplementary angle to which it is
congruent. (p. 17)
parallel lines Two lines l and m are parallel if they do not intersect; i.e., if no point lies on
both of them. (p. 19)
perpendicular lines Two lines l and m are perpendicular if they intersect at a point A and
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if there is a ray AB that is part of l, and a ray AC that is part of m, such that 6 BAC is a right
angle. (p. 27)
midpoint The midpoint of a segment AB is a point M on AB such that AM is congruent to
M B.
perpendicular bisector A perpendicular bisector of a segment AB is a line that contains the
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midpoint M and a ray M C such that 6 AM C is a right angle.
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angle bisector An angle bisector of an angle 6 BAC is a ray AD emanating from A such that
6 BAD is congruent to 6 DAC.
collinear Three points A, B, and C are collinear if there exists a single line containing all three
points.
concurrent Three lines are concurrent if there exists a single point lying on all three lines.
triangle A triangle is the set of points including the three segments AB, AC, and BC for some
three distinct, noncollinear points A, B and C.
vertices, sides, angles of a triangle In the above definition, the segments are called the sides
of the triangle, the points A, B, and C are the vertices, and the angles are the angles created by
pairs of rays emanating from the vertices and including the sides.
opposite and adjacent sides to a vertex Given a particular vertex A of a triangle 4ABC,
the two sides containing A are called adjacent, whereas the side not containing A is called opposite.
median A median is a segment with one endpoint being a vertex, and the other endpoint being
the midpoint of the opposite side.
altitude An altitude of a triangle is a segment with a vertex as one endpoint, with the other
endpoint on the lines including the opposite side, and such that the line containing the altitude and
the line including that opposite side are perpendicular.
isosceles An isosceles triangle is a triangle in which (at least) two sides are congruent. The
angles opposite those sides are called the base angles, and the side included in the base angles is
called the base.
equilateral An equilateral triangle is a triangle in which all three sides are congruent.
right triangle A right triangle is a triangle in which (at least) one angle is a right angle.
quadrilateral Given four points, A, B, C, and D, no three of which are collinear, and such
that no pair of the segments AB, BC, CD, and DA have points in common aside from possibly
endpoints, a quadrilateral the the set of points consisting of those four segments, called its sides.
The four points themselves are called the vertices of the quadrilateral.
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angles of a quadrilateral For any two sides of a quadrilateral with an endpoint in common,
the rays containing those segments and emanating from the common point make up the angles of
the quadrilateral. (Note: this would have been easier if it had come after the next definition.)
adjacent sides Two sides of a quadrilateral are adjacent if they share a common point.
opposite sides Two sides of a quadrilateral are opposite if they do not share a common point.
diagonals The diagonals of a quadrilateral are segments whose endpoints are vertices of the
quadrilateral, that are not sides of the quadrilateral.
parallelogram A parallelogram is a quadrilateral in which both pairs of opposite sides are
parallel.
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vertical angles Given two sets of opposite rays all emanating from the same point – AB
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opposite AC and AD opposite AE – the angles 6 BAD and 6 CAE are called vertical angles.
rectangle A rectangle is a quadrilateral having four right angles.
square A square is a rectangle with all four sides being congruent.
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