Language of Geometry

Objectives

Geometric Mathematical Structures

To define what is Geometry. To explain the nature of Geometry. To differentiate the Elements of Geometry.

Undefined Terms

To describe the undefined terms: points, lines planes and space. To define coplanar, collinear, line, line segment, rays, perpendicular and parallel

lines.

Defined Terms

To differentiate the line segment, rays, and angle.

Angle Relationships

To know the relationships between angles. To differentiate the pairs of angles such as Supplementary Angles, Complementary

Angles, Congruent Angle , Adjacent Angles, Linear Pair and Vertical Angles.

Parallel, Perpendicular and Skew Lines

To study parallel and perpendicular lines that will give better understanding about geometrical relationships.

To explain what parallel, skew and perpendicular lines are.

Postulates, Axioms and Theorems

To define what are the postulates, theorem and axioms. To know what are the differences among the three (theorems, postulates and

axioms).

Geometric Mathematical Structures

Geometry

- It came from the Greek words “geo” meaning “earth” and “metria” meaning “measurement”. Therefore Geometry is the study of Earth measurement.

- It is a mathematical subject which focuses on the properties of the undefined terms and other figures which are related to it.

- Geometry is a way of thinking about and seeing the world.- It was originated in Babylonia and Egypt.

Euclid

- The Father of Modern Geometry- The Elements

The Elements is divided into 13 books which cover plane geometry, arithmetic and number theory, irrational numbers and solid geometry. Euclid organized the known geometrical ideas starting with simple definitions, axioms; formed statements called theorems and set forth methods for logical proofs. He began with accepted mathematical truths, axioms and postulates and demonstrated logically 467 propositions in plane and solid geometry.

Thales

The first treatise of geometry. Papyrus

Elements of Geometry

1. Undefined terms - technically, they can’t be described without the aid of words which are undefined themselves. They can only be described in terms of other undefined terms and/or in terms of themselves.

Point- The most building block of geometry.- It has no size.

Line-has infinite length but has no thickness.

Plane-is a flat surface that extends indefinitely in all directions.

2. Defined terms - We used undefined terms to create a concrete definition of this defined terms.

Segment-is a part of a line that contains set if points within two endpoints.

Ray-is a part of a line that has one endpoint and extends infinitely on the other side.

Angle-is formed by two non collinear rays with a common endpoint.

3. Axioms and Postulates Axioms - are self evident or universally recognized truths. Postulates - are geometrical facts which are simple and obvious that

their validity may be assumed.

4. Theorem and Corollary Theorem - is a proven statement or proposition. Corollary - is a statement which is a direct consequence of a theorem.

Geometric Mathematical Structures

TEST I

Direction: Fill in the blanks. Write on the provided blank the correct word(s) that will complete each statement.

1. Geometry was originated in Babylonia and ____________.2. Euclid began with accepted mathematical truths, axioms and postulates and

demonstrated logically ___________ propositions in plane and __________ geometry.

3. The ___________ is divided into 13 books which cover plane geometry, arithmetic and number theory, ___________ numbers and solid geometry.

4. ____________ is a way of thinking about and seeing the world.5. ____________ is the first treatise of geometry.6. ___________ is the most building block of geometry.7. Angle is formed by 2 __________________ rays with a common _______.

Direction: Write C if the given statement is CORRECT and IC if it is INCORRECT in the space provided before each number.

______1. Euclid is the one who wrote Papyrus.

______2. The Elements is divided into 13 books.

______3.Axioms and Postulates are considered as building blocks of geometry.

______4.Thales is the father of geometry.

______5.Geometry is the study of Earth measurement.

TEST II

Direction: Answer the following questions below. Give at least 2 sentences to answer each question. Each correct answer is equivalent to 2 pts.

1. What is Geometry?2. What are the difference between Postulate and Theorem? Give at least 2

differences.3. Why it is point, line, and plane are called undefined terms?4. Who is Euclid in the field of mathematics?5. What are the contributions of Euclid in the field of mathematics?

Answer Key

Test IFill in the blanks.

1. Egypt2. 467 & Solid3. Elements4. Geometry5. Thales6. Point7. Angle & endpoint

Correct or Incorrect (C or IN)1. IN2. C3. IN4. IN5. C

Test II1. It is a mathematical subject which focuses on the properties of the undefined terms

and other figures which are related to it. It is also a way of thinking about and seeing the world.

2. The difference between postulates and theorems is that postulates are assumed to be true, while theorems must be proven to be true based on postulates and/or already-proven theorems.

3. Because undefined terms can’t be described without the aid of words which are undefined themselves. They can only be described in terms of other undefined terms and/or in terms of themselves.

4. Euclid is the father of geometry. As we all know his work is use in geometry nowadays and it is called Euclidean Geometry.

5. Euclid's vital contribution was to gather, compile, organize, and rework the mathematical concepts of his predecessors into a consistent whole, later to become known as Euclidean geometry.

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.

Points - no dimension, usually represented by a small dot. It is named by a capital letter.

Lines - extend in one dimension, represented with straight line with two arrow heads to indicate that the line extends without end in two directions.

Named by: the letters representing the two points on the line or a lowercase script letter (AB,BC,AC,BA,BC).

Planes- extend in two dimension represented by a slanted 4 sided figure, but you must envision it extends without end, even though the representation has edges.

Named by capital script letter

Any 3 non collinear points in the plane.

Space - it is a three dimensional set of all points. It is made up of an infinite number of planes. Figures in space are called solids or surfaces. It can contain points, lines, and planes.

There are two main types of space, "Polyhedra", and "Non-Polyhedra"

Polyhedra : (they must have flat faces)

Ex: Cubes, Prisms, Pyramids

Non Polyhedra: (if any surface is not flat)

Ex: Sphere, Cylinder, Cone

Collinear points- are points that lie on the same line.

Non Collinear points- are points that do not lie on the same line.

Points A, B and C Points D, E and F

are collinear points are non collinear points

Coplanar - points that lie on the same plane.

Non coplanar – points which do not lie on the same plane.

Line segments - are one-dimensional. It has a measurable length, but has zero width.

Rays - lines which start at a point with given coordinates, and goes off in a particular direction to infinity, possibly through a second point.

Opposite rays - two rays that lie on the same line, with a common endpoint and no other points in common. Opposite rays form a straight line or a straight angle.

Parallel lines – lines in the same plane that do not intersect.

Perpendicular lines - two lines that intersect to form a right angle.

Skew lines – lines that do not intersect and not coplanar.

Undefined Terms

TEST I

Direction: Read the statement completely and determine if the statement is true or false. In the blank provided, write “True” for a true statement and “False” for a false statement.

________ 1.Points A, B and C are collinear.

________ 2.Points A, B and C are coplanar.

________ 3.Point F lies on D́E

________ 4.D́E lies on plane ¿́

________ 5. B́D and D́E intersect.

TEST II

Direction: Identify whether the following is a Plane, Point or Line. Write your answer on the space provided.

1. Desktop _________2. Tip of a Rubrics Cube _________3. Stars _________4. Paper _________5. Flag pole _________6. Wall _________7. Floor _________8. Tip of a pencil _________9. Corner of a book _________10. Spider web _________

Answer Key

TEST I

1. False

2. True

3. False

4. True

5. True

TEST II

1. Plane 6.Plane

2. Point 7. Plane

3. Point 8.Point

4. Plane 9. Line

5. Line 10. Line

Defined Terms

Defined term is a definition that explains the meaning of a term a word, phrase, or other set of symbol.

1. Line segment

A line segment is a part of a line consisting of two points, called endpoints, and the set of all points between them.

Notation:

CD or DC

2. Ray (half-line)

A ray is a part of a line consisting of a given point, called the endpoint, and the set of all points on one side of the end point.

Notation:

E⃗F and not F⃗E

3. Angle

An angle is the union of two noncollinear rays having the same endpoint. The endpoint is called the vertex of the angle and the rays are called the sides of the

angle.

Interior points: points K, and O

Exterior points: points I, and N

Vertex: point E

Sides: EA, and EL

Angle: AEL or LEA

Interior points – are points that lie on the intersection of the half-plane that contains points and has an edge of segment.

Exterior points – are points of the angle that neither or nor lie in the intersection of the half-plane that contains points and has an edge of segment.

Defined Terms

TEST I

A. Direction: Write the word TRUE if the statement is correct and FALSE if it is wrong in the blank provided.

. 1. Line segment is a part of a line consisting of two points, called the endpoints, and all points between them.

2. Angle is the union of two noncollinear rays having the same endpoint.

3. Line segment is part of a line consisting of a given point, called the endpoints.

4. Interior points are points of the angle that lie on the intersection of the half-plane that contains point and has edge of segment.

. 5. Exterior points of the angle lie on the intersection of the half-plane that contains point and has edge of segment.

B. Direction: From nos. 6-10, identify the words being described in each statement.

6. It is a part of a line consisting of a given point, and the set of all points on one side of the end point.

7. It is a part of a line consisting of two points, called the endpoints, and all points between them.

8. It is the union of two noncollinear rays having the same endpoint.

9. It lies on the intersection of the half-plane that contains point and has edge of segment.

10. It neither or nor lies in the intersection of the half-plane that contains points and has an edge of segment.

TEST II

A. Circle the correct name for the lines and line segments below.

1 D BLine segment D Line BD Line segment B

2 M NLine segment M Line NM Line segment XY

3 X YLine segment X Line XY Line segment XY

4 A B Line segment AB Line AB Line segment B

B. Use the figure to name the following.

1. What is the vertex of the angle?

2-3. What are the two sides of the angle?

4-5. Name the figure in two ways.

6. Name the angle by one letter.

7-9. What are the interior points of the angle?

10-12. What are the exterior points of the angle?

Answer Key

TEST I

A.

1. True

2. True

3. False

4. True

5. False

B.

6. Ray

7. Line segment

8. Angle

9. Interior point

10. Exterior point

TEST II

A.

1. Line BD

2. Line NM

3. Line segment XY

4. Line segment AB

B.

1. M 7-9. Points R, L and H

2-3. M⃗A and M⃗K 10-12. Points E, D and N

4-5. AMK and KMA

6. M

Angle Relationships

Angle is a geometric figure formed by non-collinear rays that have a common endpoint. A common endpoint is called vertex.

A

ABC, B is the vertex

B C

An angle bisector of an angle is a ray that divides that angle into two congruent angles.

In the diagram, it shows that the B⃗D is the angle bisector of ABC.

A

D Therefore: ABD CBD

B C

“Classification of Angles”

An acute angle is an angle whose measure is less than 90°

A right angle is an angle whose measure is 90°.

An obtuse angle is an angle, whose measure is more than 90°,

A straight angle is an angle whose measure is 180°.

A reflex angle is an angle whose measure is more than 180°.

“Angle Pairs”

Supplementary angles

Angles can be supplementary if the sum of the measures of the two angles is 180.

R mQWR = 115 and mRWE = 65

Since 115 + 65 = 180, then QWR and RWE are supplementary angles.

Q W E

Complementary angles

If the sum of the measures of two angles is 90, then the angles are complementary.

If mTSR = 33.5

R U then, mUSR = 56.5.

T S

Congruent angles

Two angles are congruent if their measures are equal.

The opposite angles are congruent. This means that a parallelogram has two pairs of angles having the same measure.

Adjacent angles

Adjacent angles are angles that have the same vertex, share a common side, and have no interior points in common.

1 1 and 2 are adjacent angles.

2

Linear pairs

If two angles are supplementary, they form a linear pair. Linear pair can be defined as a ‘PAIR” of angles that form a “LINE”. The line indicates a straight angle that measures 180 . ̊�

D

C E S

CED and DES form a linear pair.

Vertical angles

Vertical angles are non-adjacent angles formed by two intersecting lines.

P F Vertical angles are:

POD and FOX

O POF and DOX

D X

Vertical angles are congruent. So, POD FOX and POF DOX.

Angle Relationships

TEST I

A. Direction: Write T if the statement is true and F if the statement is wrong in the space provided before the number.

1. Angle is formed by two collinear rays that have a common endpoint.2. Two angles whose measures have the sum of 180 is a supplementary angle.3. Angles in a linear pair are supplementary.4. Vertical angles are congruent.5. Angle bisector is a segment that divides the angle into two congruent sides.

B. Direction: For questions 6-10, refer to the figure.

N M L

P

G H I K

J

6. Name two angles that are adjacent to NPM. 7. Name two pairs of vertical angles.8. Name two angles that have H as their vertex.9. Name three angles that have segment IK as one of its sides.10. Are PIH and JIK vertical angles?

TEST II

Direction: Find the supplement and complement of the following angle measurements.

1. 20 ̊�2. 115 ̊�3. 80 ̊�4. 37 ̊�5. 70 ̊�6. 14 ̊�7. 100 ̊�8. 45 ̊�9. 69 ̊�10. 48 ̊�

Answer Key

TEST I

A.

1. False2. True3. True4. True5. True

B.

6. MPI, MPJ and NPH7. NPM and HIP or HPJ; NPH and MPI or MPJ; PIH and KIJ8. PHI, PHK, PHG or MHG9. JIK, LIK, NIK, and PIK10. Yes

TEST II

1. COMPLEMENT: 70 SUPPLEMENT: 1602. COMPLEMENT: NONE SUPPLEMENT: 653. COMPLEMENT: 10 SUPPLEMENT: 1004. COMPLEMENT: 53 SUPPLEMENT: 1435. COMPLEMENT: 20 SUPPLEMENT: 1106. COMPLEMENT: 76 SUPPLEMENT: 1667. COMPLEMENT: NONE SUPPLEMENT: 808. COMPLEMENT: 45 SUPPLEMENT: 1359. COMPLEMENT: 21 SUPPLEMENT: 11110. COMPLEMENT: 42 SUPPLEMENT: 132

Parallel, Perpendicular and Skew lines

Parallel lines

l ̊║m

Parallel lines are lines in a plane that never meet.

They are always the same distance apart.

Just remember: Always the same distance apart and never touching.

How can we know that lines are parallel?

The lines are parallel if:

They are coplanar. They do not intersect. They are both perpendicular to the same line. The pairs of alternate interior/exterior angles are congruent once cut

by a transversal.

Parallel Lines Cut by a Transversal

A transversal is a line which intersects two coplanar lines in two different points.

If a set of two parallel lines, lines l ̊and m, are crossed or cut by another line, n, we say "a set of parallel lines are cut by a transversal."

Each of the parallel lines cut by the transversal has four angles surrounding the intersection. These are matched in measure and position with a counterpart at the other parallel line.

Angles formed by a Transversal

Corresponding angles – angles that are in similar positions at the intersection points of the transversal. Corresponding angles are congruent.

Alternate Interior Angles – angles in the interior sections on opposite sides and ends. Alternate interior angles are congruent.

Same Side Interior Angles/Consecutive Interior Angles – the two angles in an interior section.

Alternate Exterior Angles – opposite angles in the exterior sections. Alternate exterior angles are congruent.

Examples:

1 and 5, 3 and 7, 2 and 6, 4 and 8 are corresponding angles.

3 and 6, 4 and 5 are alternate interior angles.

1 and 8, 2 and 7 are alternate exterior angles.

3 and 5, 4 and 6 are consecutive interior angles.

Perpendicular lines

A line is said to be perpendicular to another line if the two lines intersect at a right angle. Explicitly, a first line is perpendicular to a second line if 1) the two lines meet and 2) at the point of intersection the straight angle on one side of the first line is cut by the second line into two congruent angles. Perpendicularity can be shown to be symmetric, meaning if a first line is perpendicular to a second line, then the second line is also perpendicular to the first. For this reason, we may speak of two lines as being perpendicular (to each other) without specifying an order.

They are lines that are right angles. And measure 90°.

Perpendicularity easily extends to segments and rays. For example, we say a line

segment is perpendicular to a line segment if, when each is extended in both directions to form an infinite line, these two resulting lines are perpendicular in the sense above. In symbols, we write to mean line segment AB is perpendicular to line segment CD.

Skew Lines

In solid geometry, skew lines are two lines that do not intersect and are not parallel. Equivalently, they are lines that are not coplanar. A simple example of a pair of skew lines is the pair of lines through opposite edges of a regular tetrahedron. Lines that are coplanar either intersect or are parallel, so skew lines exist only in three or more dimensions.

In the figure above, ST and UV are skew lines.

Relationship of Perpendicular and Parallel lines

If two lines (a and b) are both perpendicular to a third line (c), all of the angles formed along the third line are right angles. Therefore, in Euclidean geometry, any two lines that are both perpendicular to a third line are parallel to each other, because of the parallel postulate. Conversely, if one line is perpendicular to a second line, it is also perpendicular to any line parallel to that second line.

Parallel, Perpendicular and Skew

lines

TEST I

Direction: Read the statement completely and determine if the statement is true or false. Write “True” for a true statement and “False” for a false statement. Write your answer in the blank provided.

___________ 1. If two lines do not lie in the same plane, they may be parallel.

___________ 2. The definition of a parallel line states that the line must remain the same distance apart.

___________ 3. If two lines are perpendicular to the same line at different points of the line, they are parallel.

___________ 4. If two lines in a plane are cut by a transversal, the alternate interior angles are congruent.

___________ 5. If a pair of alternate interior angles is congruent, then the lines are parallel.

___________ 6. Parallel lines do not intersect.

___________ 7. Two lines are parallel if they are not coplanar.

___________ 8. A transversal intersects two coplanar lines in only one point.

___________ 9. Parallel lines intersect at one point.

___________ 10. Skew lines are two lines that do not intersect and are not parallel.

TEST II

Direction: Classify each pair of angles as alternate interior angles, alternate exterior angles, same-side interior angles, corresponding angles, or none of these. Write AIA for alternate interior angles, AEA for alternate exterior angles, SSIA for same-side interior angles, CA for corresponding angles and NOT for none of these. Write your answer in the blank provided.

1. 7 and 11 ________________

2. 14 and 16 ________________

3. 4 and 10 _________________

4. 6 and 11 ________________

5. 3 and 6 ________________

6. 2 and 10 _______________

7. 2 and 3 _________________

8. 13 and 2 ________________

9. 5 and 10 _______________

10. 1 and 14 _______________

Answer Key

Line l is parallel to line m with lines t and u as transversals.

TEST I

1. False2. True3. False4. True5. True6. True7. False8. False9. False10. True

TEST II

1. SSIA2. CA3. NOT4. NOT5. NOT6. CA7. NOT8. AEA9. AIA10. AEA

Postulates, Axioms and Theorems

Postulate: our simplest and most fundamental statement will be given without proof.

Axioms: a self-evident or universally recognized truth.

Theorems: we state definitions for geometric ideas as clearly and exactly as we can and we shall establish the facts of geometry by giving logical proof.

Examples of Axioms and Postulates

A line contains of infinitely many points. Given any two points, there is exactly one line that contains both points. Plane postulate- any three points lie in at least one plane and any three non

collinear points lie in exactly one plane. Ruler postulate- the points on a line can be placed in correspondence with the real

number in such a way that ;1. To every point on their corresponds exactly one real number 2. To every real number there corresponds exactly one point of the line.

Distance postulate- to every pair of different points there correspond unique positive number.

Parallel postulate- if there is a line and a point not on a line, there is exactly one line through the point parallel to the given line.

Angle Addition Postulate- the measure of an angle created by two adjacent angles may be found by adding the measure of the two adjacent angles.

Examples of Theorem

If two angles are complementary then both are acute. Any two right angles are congruent. The Point Plotting Theorem- let can be a ray and let x be a positive number then

exactly one point P of such that AP= x If two lines are perpendicular, then they form right angles. The Benevolent Theorem- on a given plane, through a given point and the given

line there is one and only one point perpendicular to the given line.

Euclid’s Axioms

1. The things equal to the same thing are equal.

2. If equal are added to equals the results are equal.3. The whole is greater than any one of its parts.4. Things that coincide are equal.

Euclid’s Postulate

1. It shall be possible to draw a straight line joining any two points.2. A terminated straight line may be extended without line it direction.3. It shall be possible to draw a circle with given center and through a given points.4. All right angles are congruent.5. If two straights line in a plane meet another straights line in the plane so that

the sum of the interior angles on the same of the latter straight line is less than two rights angles then the two straight lines will meet on the side of the latter straight line.

Postulates, Axioms and Theorems

TEST I

Direction: Determine whether the statement is true or false. Write T if the statement is correct and F if not.

1. In theorem we can state definitions for geometric ideas clearly and exactly as we can.

2. Theorem is the simplest and fundamental statements will be given without proof.3. In theorem we shall establish the facts of geometric by giving logical proof.4. Axiom a self- evident or universally recognized truth.5. The statements that we prove will be called theorem.6. Our simplest and most fundamental statements will be given without proof.7. Postulates describe fundamental properties of space.8. Axioms agree that one point determine a line 9. A line contains of infinite points.10. A postulate is a proven statement or proposition.

TEST II

Direction: Fill in the blank with the correct word to complete the statement.

1. For every two points there is exactly one that contains both points called _____________________.

2. If two angles are complementary, then both are __________.3. In a given plane, through a given point and a given line there is one and only one

point perpendicular to the given line this postulate is called _________________________.

4. If there is a line and a point not on a line there is exactly one line through the point parallel to the given line, this postulate is called _________________.

5. Any three points lie in at least on plane and any three non collinear points lie in exactly one plane this postulate is called ___________________________.

Answer key

TEST I

1. True2. False3. True4. True5. True6. True7. True8. False9. True10. False

TEST II

1. Line postulate2. Acute3. Benevolent theorem4. Parallel Theorem5. Plane postulate

REFERENCES

BOOKS

Cruz, Carmelita. Ph. D. Geometry. Instructional Coverage System Publishing, Inc.

GEOMETRY. (Work text in Mathematics for Secondary Schools)

WEBSITES

http://en.wikipedia.org/wiki/File:Perpendicular_transversal_v3.svg

http://www.mathsisfun.com/definitions/perpendicular-lines.html

http://www.mathsisfun.com/geometry/parallel-lines.html

http://en.wikipedia.org/wiki/Parallel_(geometry)

http://regentsprep.org/regents/math/geometry/GG1/undefinedterms.htm

http://www.slideshare.net/jeffersonkaragdag/undefined-terms-in-geometry

http://www.icoachmath.com/math_dictionary/skew_lines.html

http://en.wikipedia.org/wiki/Skew_lines

http://mathworld.wolfram.com/SkewLines.html

Postulates

• A line contains of infinitely many points.

• Given any two points, there is exactly one line that contains both points.• Plane postulate- any three points lie in at least one plane and any three non

collinear points lie in exactly one plane.• Ruler postulate- the points on a line can be placed in correspondence with

the real number in such a way that ;• To every point on their corresponds exactly one real number • To every real number there corresponds exactly one point of the line.• Distance postulate- to every pair of different points there correspond unique

positive number.• Parallel postulate- if there is a line and a point not on a line, there is exactly

one line through the point parallel to the given line.• Angle Addition Postulate- the measure of an angle created by two adjacent

angles may be found by adding the measure of the two adjacent angles.• It shall be possible to draw a straight line joining any two points.• A terminated straight line may be extended without line it direction.• It shall be possible to draw a circle with given center and through a given

points.• All right angles are congruent.• If two straights line in a plane meet another straights line in the plane so that

the sum of the interior angles on the same of the latter straight line is less than two rights angles then the two straight line will meet on the side of the latter straight line

Theorems

• If two angles are complementary then both are acute.• Any two right angles are congruent.• The Point Plotting Theorem- let can be a ray and let x be a positive number then exactly one point P of such that AP= x• If two lines are perpendicular, then they form right angles.• The Benevolent Theorem- on a given plane, through a given point and the given line there is one and only one point perpendicular to the given line.

Definition of Terms

Acute angle - an angle whose measure is less than 90°.

Adjacent Angles - are angles that have the same vertex, share a common side, and have no interior points in common.

Angle - is a formed by two non collinear rays with a common endpoint.

Angle Bisector - divides that angle into two congruent angles.

Axioms - are self-evident or universally recognized truths.

Collinear points- are points that lie on the same line.

Complementary Angles - the sum of the measures of the two angles is 90 . ̊�

Congruent Angles - angles with equal measures.

Corollary - is a statement which is a direct consequence of a theorem.

Exterior point - is an exterior point of the angle if it is neither or nor lies in the intersection of the half-plane that contains points and has an edge of segment.

Geometry - came from the Greek words “geo” meaning “earth” and “metria” meaning “measurement”.

Interior point -is an interior point of the angle if it lies in the intersection of the half-plane that contains points and has an edge of segment.

Line - has infinite length but has no thickness.

Line segments - are one-dimensional. It has a measurable length, but has zero width.

Non Collinear points- are points that do not lie on the same line.

Obtuse angle - an angle, whose measure is more than 90°.

Opposite rays - two rays that lie on the same line, with a common endpoint and no other points in common. Opposite rays form a straight line or a straight angle.

Parallel lines – lines in the same plane that do not intersect.

Perpendicular lines - two lines that intersect to form a right angle.

Plane - is a flat surface that extends indefinitely in all directions.

Point - the most building block of geometry.

Postulates - are geometrical facts which are simple and obvious that their validity may be assumed.

Ray - is a part of a line that has one endpoint and extends infinitely on the other side.

Reflex angle - an angle whose measure is more than 180°.

Right angle - an angle whose measure is 90°.

Segment - is a part of a line that contains set if points within two endpoints.

Skew lines – lines that do not intersect and not coplanar.

Space - it is a three dimensional set of all points. It is made up of an infinite number of planes.

Straight angle - an angle whose measure is 180°.

Supplementary Angles - the sum of the measures of the two angles is 180 . ̊�

Theorem - is a proven statement or proposition.

Transversal - is a line which intersects two coplanar lines in two different points.

Vertical Angles - are non-adjacent angles formed by two intersecting lines.