I.
II.
If-then statement
A. Conditional Statement
A conditional statement is a statement that can be written in the form
“If P then Q,” where P and Q are sentences. For this conditional
statement, P is called the hypothesis and Q is called the conclusion.
Intuitively, “If P then Q” means that Q must be true whenever P is true.
A conditional is a combination of two statement p and q and the words
if and then. it comes in the form " if p, then q".
Example 1: If you get good grades, then you will get into a good
college.

If you get good grades (p), then you will get into a good college
(q)
Example 2: Three non-collinear points determine a plane
p: three points are non-collinear
q: they determine a plane
Conditional:
If three points are non-collinear (p) then they determine a plane (q)
Example 3: I'll be a millionaire when i win the lottery
Hypothesis: I win the lottery
Conclusion: I will be a millionaire
In if then form, this statement would be:
-
If I win the lottery (p), then I will be a
millionaire (q)
Some examples:
Conditional: If there are three non- collinear
points, then four unique lines are formed
Solution: Given three non-collinear points A, b and C. By postulate 1,
every three points determine a line. the lines formed AC, AB and BC.
exactly three lines are formed, so the conditional is false.
Example 2: When two distinct lines intersect they intersect in onepoint
p: two distinct lines intersect
q: they intersect exactly in one point
Conditional: If two distinct lines
intersect, then they intersect exactly
in one point
Solution: Given two distinct lines C and A By postulate 1, the two
distinct lines C and A exactly intersect at point G. so the line intersect
in one point, so the conditional is true.
B. Negations, Converses, Inverses, and Contrapositive

Negations. Statements are either true or false. A negation
is the transformation of the statement such that it will be the
opposite of its true value. the negation of a true conditional
would yield a false condition negation can also be presented
symbolically.

Each type of the statement is based on the conditional
statement. referring to the preceding table for the symbolism
for the converse which is, if q then p. Based from the
conditional, the converse just reverses the order. The conclusion
becomes the hypothesis and hypotheses becomes the
conclusion. For the inverse just state the negation of the
hypothesis and the conclusion. the contrapositive first takes the
negation of the hypothesis and conclusion, then switches their
places.
III.
-
Writing Proofs
A proof is an argument to convince your audience that a
mathematical statement is true. In comparison to
computational math problems, proof writing requires greater
emphasis on mathematical rigor, organization, and
communication.
The two-column proof
-
A two-column geometric proof consists of a list of statements,
and the reasons that we know those statements are true. ... This
is the step of the proof in which you actually find out how the
proof is to be made, and whether or not you are able to prove
what is asked. Congruent sides, angles, etc.
Example 1: A line and a point not on the line are contained in one
and only one plane.
If there exist a line and a point not on the
line, then they are contained in only one
plane
Given: A line and a point not on a line
Prove: The point and the line are contained
in only one plane.
Chapter 3: Segment and Rays
I.
LINE MEASUREMENT

-
RULER POSTULATE
The points of a line can be match with set of real numbers in
way that:
1. To every point on the line there is exactly one real number.
2. To every real number there is exactly point on the line.
3. The distance between any two points is the absolute value of
the difference of their coordinates
FORMULA:
Generally, If the coordinate of P is x and the coordinate of Q is
why then the distance between them can be solved using:
◦
The measure of the length of the line segment from point A to
point B is the distance between two endpoints.
ruler A
point A = 0
point B = 9
ruler B
point A = 1
point B = 10
Ex#01
Solve: FORMULA: PQ= I y-x I
RULER A
A=0 B=9
AB = |9- 0 |
AB = |9|
AB = 9
Ex#02
RULER B
A=1
B=10
AB = |10 - 1|
AB = |9|
AB = 9
Solving:
AB = |5 - (-1) |
AB = |5 + 1|
AB = |6|
AB = 6
BA = |-1- 5|
BA = |-6|
BA = 6
DEFENITIONS:

DISTANCE: the absolute value of the difference of the
coordinates of two end points.
P= 5 Q= 11
PQ = |11-5|
PQ = |6|
PQ = 6

COORDINATES: The number that correspond to a point.

BETWEENESS: Point B is between A and C if:
a. poin A, B and c are colinear
b. AB + BC + AC
AB = |2=0|
BC = |6-2|
AB = |2|
BC = |4|
AB = 2
BC = 4
AC = |6-0|
AC = |6|
AC = 6

AB + BC = AC
2+4=6
MIDPOINT: Point B is the Midpoint of AC if:
a. AB + BC = AC
AB= |5-0|
AC= |10-0|
b. AB =BC
AB= |5|
AC= |10|
AB= 5
AC=10
BC= |10-5|
BC= |5|
BC= 5
AB+BC=AC
5 + 5 = 10

SEGMENT BISECTOR: A segment ray or a line that contains
midpoint
Using the number line, plot the following points as described:
a. Plot L if point L is 4 units to the right of T.
Y - (-2) = 4
Y+2=4
Y= 2-4
Y= 2
b.
Plot V if LV=7 and V is to the right of T.
y - (2) = 7
y = 7+2
y=9
c. Plot R if TL = RT
-2 - x = 4
-2 -4 = x
-6 = x
II.
Rays
Ray
-
Is a part of line that has one endpoint and extends endlessly to
one direction.
-
Another term we should learn in this lesson is the term opposite
rays. Two rays are opposite if they are subsets of the same line
and have a common endpoint.
Example of Not opposite Rays
III.
Convex Sets
Convex Sets
-
if a segment is drawn connecting any two points on a given
figure and the entire segment is still “inside” the figure, then
the said figure is convex. Here are the examples of convex sets
of points
Samples of Non-Convex Sets Are:
-
A set is convex, if and only if, two points in the set form a
segment with all of its points contained in the set.
Is a plane a convex set? For example, consider the plane below.
-
If we choose any segment, say TR, and both T and R on the
plane, are all the points of segment TR on the plane? Yes. So,
the plane is an example of convex set
Disjoint
-
-
sets, on the other hand are sets that do not have any common
point.
Consider a plane and a line on a plane
From the figure we can form three sets of points.
Name the sets A, B, and C.
Set A: the point contained in line g.
Set B: the points not on the line but in the plane and in the same
side as Q.
Set C: the points not on the line but in the plane and in the same
side as W.
Set A = {R, T}
Set B = {Q}
Set C = {W, K}
Chapter IV: ANGLES AND PERPENDICULAR LINES
I.
Angles
II.
Angle Measurements
Angle
-
An angle is figured formed by two
rays, called side of angle, and
sharing a common endpoint called
vertex.
Different kinds of Angles
A. Acute Angle
- is an angle which measures
less than 90°. It measures
between 0° to 90°
B. Right Angle
- IS AN ANGLE WHICH MEASURES
EXACTLY 90°
C. Obtuse Triangle
- IS AN ANGLE WHICH
MEASURES BETWEEN
90° - 180°
D. Straight Angle
- IS AN ANGLE WHICH MEASURES
EXACTLY 180°
Angle Measurement
-
the measurement of an angle is the smallest amount of rotation
about the vertex from one ray to the other.
DEGREE (°) is the unit of measure of an angle.
PROTRACTOR is the tool used to measure given angle.



The measure of <ABC as indicated in the protractor is 90
degrees. This can be written in two ways:
<ABC=90° (Angle ABC equals 90 degrees.)
m/ABC=90 (The measure of ABC 90.)
Example 1
The measure of <A is 50° and the measure of <B is 60°. Find the sum
of their measures.
m<A + m<B = 50° + 60°
= 110°
Example 2
<ABD and <CBD are two coplanar angles with a common side BD. If
m<ABD =40 and m<CBD =30. Find the measurement of Angle ABC
m<ABC + m<CBD = 40° + 30°
= 70°
Example 3
If m<ABC = 120, m<ABD =2x+10 and
m<CBD =3x. Find m<ABD
Example 4
If m<ABC = 96
m<CBD = x
m<ABD =2x
Find m<CBD