WELCOME TO THE WORLD OF GEOMETRY

WHAT IS GEOMETRY???

Geometry (Greek γεωμετρία; geo = earth, metria = measure) arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers (arithmetic).

Classic geometry was focused in compass and straightedge constructions. Geometry was revolutionized by Euclid, who introduced mathematical rigor and the axiomatic method still in use today. His book, The Elements is widely considered the most influential textbook of all time, and was known to all educated people in the West until the middle of the 20th century.

In modern times, geometric concepts have been generalized to a high level of abstraction and complexity, and have been subjected to the methods of calculus and abstract algebra, so that many modern branches of the field are barely recognizable as the descendants of early geometry. (See areas of mathematics and algebraic geometry.)

A Brief History of Geometry

Geometry began with a practical need to measure shapes. The word geometry means to “measure the earth” and is the science of shape and size of things. It is believed that geometry first became important when an Egyptian pharaoh wanted to tax farmers who raised crops along the Nile River. To compute the correct amount of tax the pharaoh’s agents had to be able to measure the amount of land being cultivated.

Around 2900 BC the first Egyptian pyramid was constructed. Knowledge of geometry was essential for building pyramids, which consisted of a square base and triangular faces. The earliest record of a formula for calculating the area of a triangle dates back to 2000 BC. The Egyptians (5000–500 BC) and the Babylonians (4000–500 BC) developed practical geometry to solve everyday problems, but there is no evidence that they logically deduced geometric facts from basic principles.

It was the early Greeks (600 BC–400 AD) that developed the principles of modern geometry beginning with Thales of Miletus (624–547 BC). Thales is credited with bringing the science of geometry from Egypt to Greece. Thales studied similar triangles and wrote the proof that corresponding sides of similar triangles are in proportion.

The next great Greek geometer was Pythagoras (569–475 BC). Pythagoras is regarded as the first pure mathematician to logically deduce geometric facts from basic principles. Pythagoras founded a brotherhood called the Pythagoreans, who pursued knowledge in mathematics, science, and philosophy. Some people regard the Pythagorean School as the birthplace of reason and logical thought. The most famous and useful contribution of the Pythagoreans was the Pythagorean Theorem. The theory states that the sum of the squares of the legs of a right triangle equals the square of the hypotenuse.

Euclid of Alexandria (325–265 BC) was one of the greatest of all the Greek geometers and is considered by many to be the “father of modern geometry”. Euclid is best known for his 13-book treatise The Elements. The Elements is one of the most important works in history and had a profound impact on the development of Western civilization.

UNDEFINED TERMS, THEOREMS AND POSTULATES

A. UNDEFINED TERMS

An ‘undefined term’ is a term or word that doesn’t require further explanation or description.  It already exists in its most basic form.  These basic terms are used to define or explain more complicated terms or concepts.

Geometry recognizes four undefined terms. While some books only recognize three terms, all four will be included here.  Even though they are ‘undefined’, I will attempt to describe them for you below.

POINT (an undefined term) In geometry, a point has no dimension (actual size).  Even though we represent a point with a dot, the point has no length, width, or thickness.  Our dot can be very tiny or very large and it still represents a point.  A point is usually named with a capital letter.  In the coordinate plane, a point is named by an ordered pair, (x,y).
LINE (an undefined term) In geometry, a line has no thickness but its length extends in one dimension and goes on forever in both directions.  Unless otherwise stated a line is drawn as a straight line with two arrowheads indicating that the line extends without end in both directions.  A line is named by a single lowercase letter, , or by any two points on the line, .
PLANE (an undefined term) In geometry, a plane has no thickness but extends indefinitely in all directions.  Planes are usually represented by a shape that looks like a tabletop or a parallelogram.  Even though the diagram of a plane has edges, you must remember that the plane has no boundaries.  A plane is named by a single letter (plane m) or by three non-collinear points (plane ABC).

There are a few basic concepts in geometry that need to be understood, but are seldom used  as reasons in a formal proof.

 

Collinear Points	points that lie on the same line.
Coplanar points	points that lie in the same plane.
Opposite rays	2 rays that lie on the same line, with a common endpoint and no other points in common.  Opposite rays form a straight line and/or a straight angle (180°:).
Parallel lines	two coplanar lines that do not intersect
Skew lines	two non-coplanar lines that do not intersect.

B. THEOREMS

In mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms. The derivation of a theorem is often interpreted as a proof of the truth of the resulting expression, but different deductive systems can yield other interpretations, depending on the meanings of the derivation rules. The proof of a mathematical theorem is a logical argument demonstrating that the conclusions are a necessary consequence of the hypotheses, in the sense that if the hypotheses are true then the conclusions must also be true, without any further assumptions. The concept of a theorem is therefore fundamentally deductive, in contrast to the notion of a scientific theory, which is empirical.

Although they can be written in a completely symbolic form using, for example, propositional calculus, theorems are often expressed in a natural language such as English. The same is true of proofs, which are often expressed as logically organized and clearly worded informal arguments, intended to convince readers of the truth of the statement of the theorem beyond any doubt, and from which arguments a formal symbolic proof can in principle be constructed. Such arguments are typically easier to check than purely symbolic ones—indeed, many mathematicians would express a preference for a proof that not only demonstrates the validity of a theorem, but also explains in some way why it is obviously true. In some cases, a picture alone may be sufficient to prove a theorem. Because theorems lie at the core of mathematics, they are also central to its aesthetics. Theorems are often described as being "trivial", or "difficult", or "deep", or even "beautiful". These subjective judgments vary not only from person to person, but also with time: for example, as a proof is simplified or better understood, a theorem that was once difficult may become trivial. On the other hand, a deep theorem may be simply stated, but its proof may involve surprising and subtle connections between disparate areas of mathematics. Fermat's Last Theorem is a particularly well-known example of such a theorem. 

C. POSTULATES

Postulates are statements the validity or truth of which are assumed without proof.

DIFFERENT THEOREMS AND POSTULATES IN GEOMETRY

General:

Reflexive Property	A quantity is congruent (equal) to itself.  a = a
Symmetric Property	If a = b, then b = a.
Transitive Property	If a = b and b = c, then a = c.
Addition Postulate	If equal quantities are added to equal quantities, the sums are equal.
Subtraction Postulate	If equal quantities are subtracted from equal quantities, the differences are equal.
Multiplication Postulate	If equal quantities are multiplied by equal quantities, the products are equal.  (also Doubles of equal quantities are equal.)
Division Postulate	If equal quantities are divided by equal nonzero quantities, the quotients are equal. (also Halves of equal quantities are equal.)
Substitution Postulate	A quantity may be substituted for its equal in any expression.
Partition Postulate	The whole is equal to the sum of its parts. Also:  Betweeness of Points:  AB + BC = AC Angle Addition Postulate :  m<ABC + m<CBD = m<ABD
Construction	Two points determine a straight line.
Construction	From a given point on (or not on) a line, one and only one perpendicular can be drawn to the line.

Angles:
 

Right Angles	All right angles are congruent.
Straight Angles	All straight angles are congruent.
Congruent Supplements	Supplements of the same angle, or congruent angles, are congruent.
Congruent Complements	Complements of the same angle, or congruent angles, are congruent.
Linear Pair	If two angles form a linear pair, they are supplementary.
Vertical Angles	Vertical angles are congruent.
Triangle Sum	The sum of the interior angles of a triangle is 180º.
Exterior Angle	The measure of an exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles. The measure of an exterior angle of a triangle is greater than either non-adjacent interior angle.
Base Angle Theorem (Isosceles Triangle)	If two sides of a triangle are congruent, the angles opposite these sides are congruent.
Base Angle Converse (Isosceles Triangle)	If two angles of a triangle are congruent, the sides opposite these angles are congruent.

Triangles:
 

Side-Side-Side (SSS) Congruence	If three sides of one triangle are congruent to three sides of  another triangle, then the triangles are congruent.
Side-Angle-Side (SAS) Congruence	If two sides and the included angle of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.
Angle-Side-Angle (ASA) Congruence	If two angles and the included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.
Angle-Angle-Side (AAS) Congruence	If two angles and the non-included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.
Hypotenuse-Leg (HL) Congruence (right triangle)	If the hypotenuse and leg of one right triangle are congruent to the corresponding parts of another right triangle, the two right triangles are congruent.
CPCTC	Corresponding parts of congruent triangles are congruent.
Angle-Angle (AA) Similarity	If two angles of one triangle are congruent to two angles of another triangle, the triangles are similar.
SSS for Similarity	If the three sets of corresponding sides of two triangles are in proportion, the triangles are similar.
SAS for Similarity	If an angle of one triangle is congruent to the corresponding angle of another triangle and the lengths of the sides including these angles are in proportion, the triangles are similar.
Side Proportionality	If two triangles are similar, the corresponding sides are in proportion.
Mid-segment Theorem (also called mid-line)	The segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half as long.
Sum of Two Sides	The sum of the lengths of any two sides of a triangle must be greater than the third side
Longest Side	In a triangle, the longest side is across from the largest angle. In a triangle, the largest angle is across from the longest side.
Altitude Rule	The altitude to the hypotenuse of a right triangle is the mean proportional between the segments into which it divides the hypotenuse.
Leg Rule	Each leg of a right triangle is the mean proportional between the hypotenuse and the projection of the leg on the hypotenuse.

Parallels:

Corresponding Angles If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent. Corresponding Angles Converse If two lines are cut by a transversal and the corresponding angles are congruent, the lines are parallel. Alternate Interior Angles   If two parallel lines are cut by a transversal, then the alternate interior angles are congruent. Alternate Exterior Angles If two parallel lines are cut by a transversal, then the alternate exterior angles are congruent. Interiors on Same Side If two parallel lines are cut by a transversal, the interior angles on the same side of the transversal are supplementary. Alternate Interior Angles Converse If two lines are cut by a transversal and the alternate interior angles are congruent, the lines are parallel. Alternate Exterior Angles Converse If two lines are cut by a transversal and the alternate exterior angles are congruent, the lines are parallel. Interiors on Same Side Converse If two lines are cut by a transversal and the interior angles on the same side of the transversal are supplementary, the lines are parallel.

Quadrilaterals:

 

Parallelograms	About Sides	* If a quadrilateral is a parallelogram, the opposite    sides are parallel. * If a quadrilateral is a parallelogram, the opposite    sides are congruent.
	About Angles	* If a quadrilateral is a parallelogram, the opposite    angles are congruent. * If a quadrilateral is a parallelogram, the    consecutive angles are supplementary.
	About Diagonals	* If a quadrilateral is a parallelogram, the diagonals    bisect each other. * If a quadrilateral is a parallelogram, the diagonals    form two congruent triangles.
Parallelogram Converses	About Sides	* If both pairs of opposite sides of a quadrilateral    are parallel, the quadrilateral is a parallelogram. * If both pairs of opposite sides of a quadrilateral    are congruent, the quadrilateral is a    parallelogram.
	About Angles	* If both pairs of opposite angles of a quadrilateral    are congruent, the quadrilateral is a    parallelogram. * If the consecutive angles of a quadrilateral are  supplementary, the quadrilateral is a parallelogram.
	About Diagonals	* If the diagonals of a quadrilateral bisect each    other, the quadrilateral is a    parallelogram. * If the diagonals of a quadrilateral form two    congruent triangles, the quadrilateral is a    parallelogram.
Parallelogram	If one pair of sides of a quadrilateral is BOTH parallel and congruent, the quadrilateral is a parallelogram.
Rectangle	If a parallelogram has one right angle it is a rectangle
	A parallelogram is a rectangle if and only if its diagonals are congruent.
	A rectangle is a parallelogram with four right angles.
Rhombus	A rhombus is a parallelogram with four congruent sides.
	If a parallelogram has two consecutive sides congruent, it is a rhombus.
	A parallelogram is a rhombus if and only if each diagonal bisects a pair of opposite angles.
	A parallelogram is a rhombus if and only if the diagonals are perpendicular.
Square	A square is a parallelogram with four congruent sides and four right angles.
	A quadrilateral is a square if and only if it is a rhombus and a rectangle.
Trapezoid	A trapezoid is a quadrilateral with exactly one pair of parallel sides.
Isosceles Trapezoid	An isosceles trapezoid is a trapezoid with congruent legs.
	A trapezoid is isosceles if and only if the base angles are congruent
	A trapezoid is isosceles if and only if the diagonals are congruent
	If a trapezoid is isosceles, the opposite angles are supplementary.

Circles:

Radius	In a circle, a radius perpendicular to a chord bisects the chord and the arc.
	In a circle, a radius that bisects a chord is perpendicular to the chord.
	In a circle, the perpendicular bisector of a chord passes through the center of the circle.
	If a line is tangent to a circle, it is perpendicular to the radius drawn to the point of tangency.
Chords	In a circle, or congruent circles, congruent chords are equidistant from the center. (and converse)
	In a circle, or congruent circles, congruent chords have congruent arcs. (and converse0
	In a circle, parallel chords intercept congruent arcs
	In the same circle, or congruent circles, congruent central angles have congruent chords (and converse)
Tangents	Tangent segments to a circle from the same external point are congruent
Arcs	In the same circle, or congruent circles, congruent central angles have congruent arcs. (and converse)
Angles	An angle inscribed in a semi-circle is a right angle.
	In a circle, inscribed angles that intercept the same arc are congruent.
	The opposite angles in a cyclic quadrilateral are supplementary
	In a circle, or congruent circles, congruent central angles have congruent arcs.

ANGLES

          An angle is a figure formed by two rays with a common endpoint, and which are not on the same line. The two rays are called the sides of the angle. The common endpoints of the sides of an angle is called the vertex.

          In the figure above, rays OT and OX are the sides of the angle. Point O is the vertex of the angle.
          An angle is denoted using a number, vertex, or the vertex and two points on each side of an angle. If three letters are used, the middle letter is the vertex of the angle. The angle above is denoted by Angle TOX, or Angle O.
        
          An angle divides the plane into three parts: the interior angle, the exterior of the angle, and the angle itself.
          The color light blue portion outside the angle is called the exterior of the angle and the violet portion inside the angle is called the interior of the angle.


KINDS OF AN ANGLE

POLYGONS

A closed plane figure formed by connecting three or more segments at their endpoints is called a polygon. The segments are the sides of the polygon while the endpoints of this segments are the vertices of the polygon. Two sides of a polygon are adjacent or consecutive if they have a common endpoints. Two angles of polygon are adjacent if they have a side in common. Two vertices of a polygon are adjacent if they are the endpoints of a side.

KINDS OF POLYGONS

A. TRIANGLE

A triangle is a polygon with three sides. Every triangle has three altitudes, medians and angle bisectors. 

An altitude to a side is the segment drawn from a vertex of a triangle to the point on the line containing the opposite side such that the segment and the line intersect to form right angles.

A median to a side is a congruent drawn from a vertex of a triangle to the midpoint of the opposite side.

CLASSIFICATION OF TRIANGLE

table 1.1

Triangles can be classified according to the number of congruent sides and their angles.

Equilateral, isosceles and scalene triangles may be classified according to their congruent sides.

Acute, Obtuse and right triangles may be classified according to their angles.

In addition, equiangular triangle may be classified according to its angles. 

B. QUADRILATERALS

A quadrilateral is a polygon with four sides (or edges) and four vertices or corners. Sometimes, the term quadrangle is used, by analogy with triangle, and sometimes tetragon for consistency with pentagon (5-sided), hexagon (6-sided) and so on. The origin of the word "quadrilateral" is the two Latin words quadri, a variant of four, and latus, meaning "side."

Quadrilaterals are simple (not self-intersecting) or complex (self-intersecting), also called crossed. Simple quadrilaterals are either convex or concave.

The interior angles of a simple (and planar) quadrilateral ABCD add up to 360 degrees of arc, that is

 

This is a special case of the n-gon interior angle sum formula (n − 2) × 180°. In a crossed quadrilateral, the interior angles on either side of the crossing add up to 720°. All convex quadrilaterals tile the plane by repeated rotation around the midpoints of their edges.

KINDS OF QUADRILATERALS

C. OTHER POLYGONS

1. Pentagon                       5 sides             5 corners           5 edges
2. Hexagon                       6 sides             6 corners            6 edges
3. Heptagon                      7 sides             7 corners            7 edges 
4. Octagon                        8 sides             8 corners            8 edges
5. Nonagon                       9 sides             9 corners            9 edges 
6. Decagon                      10 sides           10 corners          10 edges
7. Undecagon                  11 sides            11 corners          11 edges
8. Dodecagon                  12 sides           12 corners          12 edges 
9. n-gon                            n sides              n corners            n edges

CIRCLES

A circle is the set of all  points in a plane that are equidistant from a fixed point in the plane. the fixed point is the center of the circle and the fixed distance is the radius.

The following are some terms related to the circle:

1. Radius - is also used to name a line segment that joints the center of the circle to the point on the circle.

2. Chord - is a segment whose end points are any two points on the circle.

3. Diameter - is a chord which passes through the center of a circle. It is twice the length of the radius. The word "diameter" can also refer to a segment or the length of a segment.

4. Secant - is a line that contains a chord.

5. Tangent - to a circle is a line, a ray or a segment in the plane of a circle that intersects the circle at exactly one point (point of  tangency).

Other basic terms needed in the study of the circle are illustrated and define as follows:

1. Central angle - is an angle formed by two radii of the circle with its vertex in the center of the circle.

2. Arc - is a connected part or portion of a circle. If it is half a circle, it is called a semicircle. if an arc is less than half a circle, it is called minor arc. If an arc is more than half a circle it is called major arc.