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May 25th

May 25th

# Definitions

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 A, Geometry

• analyze characteristics and properties of two -and three-dimensional geometric
shapes and develop mathematical arguments about geometric relationships,

• build new mathematical knowledge through problem solving,

• solve problems that arise in mathematics and in other contexts,

• apply and adapt a variety of appropriate strategies to solve problems,

• monitor and reflect on the process of mathematical problem solving

• recognize reasoning and proof as fundamental aspects of mathematics

• make and investigate mathematical conjectures,

• develop and evaluate mathematical arguments and proofs,

• select and use various types of reasoning and methods of proof

Euclid’s Postulates

1. To draw a straight line from any point to any other

2. To produce a finite straight line continuously in a straight line

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 , rigid motions

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
intersect.

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
same line.

Coplanar points: Four or more points are called coplanar if they are on the
same plane.

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 to

where [resp. ] are the coordinates of the projections of A and
B on two perpendicular lines.

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 if
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 consecutive sides
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 and joined at the
point O called the vertex of the angle The rays and are called
sides.

Straight Angle: A straight angle is an angle whose sides and
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, then
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
circle.

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 greater than
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
supplementary angles.

Complementary Angles: Two angles whose measures add up to 90 are called
complementary angles.

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 two lines.

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
congruent.

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 located
at 2/3 to the vertex and 1/3 to the corresponding midpoint on each median of
the triangle.

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
extension).

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