Definition by genus and specific difference : A definition
of a mathematical
notion must contain first a broader category or concept to which the defined
notion belongs, the genus, then distinguished from other items in that category
by specific characteristics, the differentia.
Principles and Standards for schools mathematics, Appendix
2. To produce a finite straight line continuously in a
3. To describe a circle with any center and any distance
4. That all right angles are equal to each other
5. That, if a straight line falling on two straight lines
makes the interior
angles on the same side less than two right angles, the straight lines if
produced indefinitely, meet on that side on which are the angles less than
the two right angles
Basic undefined terms: point, lines, plane, real numbers ,
Parallel lines: Two distinct lines are said to be parallel
if they are contained
in the same plane and have no point in common (no point of intersection).
Skew lines: Two lines in space which are not parallel and
which do not
Euclidean postulate: If a point P is not on the line
there exist one and
only one line containing P and parallel to.”
Col linear points : Three or more points are called
collinear if they are on the
Coplanar points: Four or more points are called coplanar
if they are on the
Concurrent Lines: Three or more lines are called
concurrent if they pass
through the same point.
Axiom for points on lines: “Each line can be viewed as a
copy of the real number line . Each point P on the line has a real coordinate
Distance: The distance between two points A and B is equal
Line Segment: The line segment
is the set of all points on the line
whose coordinates are in between the coordinates
and of A
and B (i.e.
the coordinates x such that The length of
is denoted simply by AB and is the distance
between A and B. Two line
segments are said to be congruent if they have the same distance measurement.
Ray: The ray is the set
of points on the line whose coordinates x
are in the same order relation to as
is to (i.e
or if then).
The point A is called endpoint of the ray.
Polygonal line: A polygonal line is a union of segments,
called sides, denoted
here by are
distinct points in the plane which are called vertices.
Polygon: A polygon is a polygonal line in which any two
intersect only at a vertex and every non-consecutive sides do not intersect.
Regular Polygon: A regular polygon is a polygon in which
all the sides have
the same measure and all the angles have the same angular measure.
Convex Regions: A set of points in the plane (region) is
called convex if it
contains any segment de termined by two
points A and B in the region.
Angle: An angle is the union of two rays, say
joined at the
point O called the vertex of the angle The
Straight Angle: A straight angle is an angle
form the line .
Interior of an angle: An angle which is not a straight
angle divides the plane
into two regions. One is convex and one is not convex. The convex region is
called interior of the angle.
Adjacent Angles: Two angles which share the same vertex, a
ray and have
disjoint interiors are called adjacent angles.
Right Angle: A right angle is half of a straight angle.
Angle Measurement: Every angle
has associated to it a number denoted
m(), which is between 0 and 180 if “measured”
in degrees or between
0 and π if measured in radians. Conversely, every real number between 0 and
180 corresponds to an angle whose measure is that number. We say that two
angles are congruent if they have the same angular measure.
Addition Axiom : Given an angle
(which is not a straight angle) whose
interior contains the point C, then m() = m()
+ m(). If
is a straight angle and C is just another point in the plane not on its sides,
180 = m() + m().
Circle: A circle is a set of points in the plane which are
at the same distance
to a fixed point called center. The fixed distance is called the radius of the
Acute Angle: An acute angle is an angle whose measure is
less than the
measure of a right angle.
Obtuse Angle: An obtuse angle is an angle whose measure is
the measure of a right angle.
Vertical Angles: Vertical angles (which are not straight
angles) are two angles
which share the same vertex and their rays form two straight lines.
Theorem of vertical angles: Given two angles which are
vertical then these
angles are congruent.
Supplementary Angles: Two angles whose measures add up to
180 are called
Complementary Angles: Two angles whose measures add up to
90 are called
Perpendicular lines or segments: Two lines are said
perpendicular if they
contain rays which form a right angle. Two segments are perpendicular if they
are contained on two perpendicular lines.
Transversal: Transversal line is a line which intersects
Corresponding angles, alternate interior, alternate
exterior, interior of the
same side of the transversal recognize on the picture.
Triangle: A triangle is a polygon with three vertices.
Isosceles triangle: A triangle is said to be isosceles if
two of its sides are
Equilateral triangle: A triangle whose sides are all
congruent is called equilateral.
Scalene triangle: A scalene triangle is a triangle in
which all three sides have different lengths .
Midpoint of a segment A point M on a line segment
is called the midpoint
of this line segment if it is equally distant to A and B (i.e. MA = MB).
Median of a triangle: A median of a triangle is the line
segment joining a
vertex with the midpoint of the opposite side.
Centroid or center of mass: The medians in a triangle meet
at a point called
centroid or center of mass and it is denoted usually by G. The point G is
at 2/3 to the vertex and 1/3 to the corresponding midpoint on each median of
Cevian in a triangle: A Cevian is a line segment
associated with a triangle
which joins a vertex of this triangle with a point on the opposite side (or its