This document provides information about key concepts in geometry. It begins by defining geometry as the study of earth measurement and notes that it focuses on properties of undefined terms like points, lines, and planes. It then discusses important figures in geometry like Euclid and Thales. The document outlines several elements of geometry, including undefined terms like points, lines, and planes. It defines defined terms like line segments, rays, and angles. It also distinguishes between axioms, postulates, and theorems. Finally, it provides examples and tests to assess understanding of concepts like coplanar points, collinear points, and different geometric objects.

1. 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, Perpendicularand 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).

2. 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.

3. 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.

4. 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?

5. Answer Key
Test I
Fill in the blanks.
1. Egypt
2. 467 & Solid
3. Elements
4. Geometry
5. Thales
6. Point
7. Angle & endpoint
Correct or Incorrect (C or IN)
1. IN
2. C
3. IN
4. IN
5. C
Test II
1. 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.

6. 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
(AB̅̅̅̅, BC̅̅̅̅, AC̅̅̅̅, BA̅̅̅̅̅, BC̅̅̅̅).
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 (Plane N)
Any 3 non collinear points in the plane.

7. 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.

8. 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.

9. 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 DE⃡
________ 4.DE⃡ lies on plane DEF⃡
________ 5. BD⃡ and DE⃡ 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 _________

10. 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

11. 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:
CD̅̅̅̅ or DC̅̅̅̅
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:
EF and not FE
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.

12. Interior points: points K, and O
Exterior points: points I, and N
Vertex: point E
Sides: EA̅̅̅̅, and EL̅̅̅̅
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.

13. 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.

14. TEST II
A. Circle the correct name for the lines and line segments below.
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?
1 D B
Line segment D Line BD Line segment B
2 M N
Line segment M Line NM Line segment XY
3 X Y
Line segment X Line XY Line segment XY
4 A B
Line segment AB Line AB Line segment B

15. 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. MA and MK 10-12. Points E, D and N
4-5. AMK and KMA
6. M

16. 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 BD 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°,

17. 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 mQWR = 115 and mRWE = 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 mTSR = 33.5
R U then, mUSR = 56.5.
T S

18. 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.

19. 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.

20. 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?

21. 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 ̊

22. Answer Key
TEST I
A.
1. False
2. True
3. True
4. True
5. True
B.
6. MPI,  MPJ and NPH
7.  NPM and HIP or HPJ; NPH and MPI or MPJ; PIH and KIJ
8. PHI, PHK, PHG or MHG
9. JIK, LIK, NIK, and PIK
10. Yes
TEST II
1. COMPLEMENT: 70 SUPPLEMENT: 160
2. COMPLEMENT: NONE SUPPLEMENT: 65
3. COMPLEMENT: 10 SUPPLEMENT: 100
4. COMPLEMENT: 53 SUPPLEMENT: 143
5. COMPLEMENT: 20 SUPPLEMENT: 110
6. COMPLEMENT: 76 SUPPLEMENT: 166
7. COMPLEMENT: NONE SUPPLEMENT: 80
8. COMPLEMENT: 45 SUPPLEMENT: 135
9. COMPLEMENT: 21 SUPPLEMENT: 111
10. COMPLEMENT: 42 SUPPLEMENT: 132

23. 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

24. 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.

25. 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.

26. 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, ST̅̅̅ and UV̅̅̅̅ 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.

27. 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.

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

29. Answer Key
TEST I
1. False
2. True
3. False
4. True
5. True
6. True
7. False
8. False
9. False
10. True
TEST II
1. SSIA
2. CA
3. NOT
4. NOT
5. NOT
6. CA
7. NOT
8. AEA
9. AIA
10. AEA

30. 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.

31. 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.

32. 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 ___________________________.

33. Answer key
TEST I
1. True
2. False
3. True
4. True
5. True
6. True
7. True
8. False
9. True
10. False
TEST II
1. Line postulate
2. Acute
3. Benevolent theorem
4. Parallel Theorem
5. Plane postulate

34. 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

35. 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.

36. 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.

37. 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.