WEEK 1: MATHEMATICAL SYSTEM
Mathematical System - is a set of structures
designed to provide order and procedural operation in
a certain discipline
- Generally, there are two elements that compose a
mathematical system: vocabulary and principles.
Vocabulary refers to either undefined term or
definition.
Undefined Terms
- are terms that have not been previously
categorically determined
1. Point - A point can be viewed as something
having specific position but without
dimension, magnitude, or direction
Illustrative Examples
1. Tip of a pen
2. Corner of a bond paper
3. Tip of a hair strand
4. Intersection of two strings
Recall that a dot is used to represent a point,
and a point is denoted by a capital letter. For
instance, the two points below are named
point G and point N.
2. Line - is a one- dimensional figure composed
of infinite number of points. It has
unspecified length but without width nor
thickness. In geometry, a line will always be a
straight line that extends indefinitely in two
opposite direction
Illustrative Example:
1. A straight string
2 Edge of a bond paper
3. A strand of hair
4. Intersection of a ceiling and a wall
In denoting a line two of its named points are
used. The line below is line GN. written in
symbol as GN
3. Plane - A plane is usually represented by a
flat surface where infinite number of lines
can lie. It has unspecified width and length,
but without thickness. Consequently, the
plane extends indefinitely in all directions.
Illustrative Examples
1. A white board
2 Screen of an iPad
3. Flooring of a room
4. A sheet of paper
A parallelogram is usually used to represent a
plane. In denoting a plane, a capital letter or
an uppercase Greek letter is used. The plane
below is plane P.
Defined Terms
- a term that has some sort of definition
1. Line Segment: A line segment is just part of a
line. Line segments stop somewhere in both
directions
2. Ray: A ray is like a line, but the line takes off
in one direction to infinity while the other side
is like a line segment. The end of the line is
called the endpoint.
3. Angle: Two rays that share the same
endpoint, however, the rays take off in
different directions. The area in the middle of
the two rays is the angle measure.
Postulate
- is a statement which is accepted as true without
proof. Postulates are important in studying geometry
and other mathematical systems.
Postulate 1: A line contains at least two points.
Postulate 2: A plane contains at least three
noncollinear points.
Postulate 3: Through any two points, there is exactly
one line.
Postulate 4: Through any three noncollinear points,
there is exactly one plane.
Postulate 5: If two points lie in a plane, then the line
joining them lies in that plane
Postulate 6: If two planes intersect, then their
intersection is a line.
Postulate 7: A space contains four noncoplanar
points.
Theorem
- A theorem is a statement that can be proven. Once a
theorem is proven, it can also be used as a reason in
proving other statements.
Theorem 1: If two lines intersect, then they intersect
in exactly one point.
Theorem 2: If a point lies outside a
line, then exactly one plane contains both the line and
the point. Theorem 3: If two lines intersect, then
exactly one plane contains both lines.
WEEK 2: TRIANGLE CONGRUENCE

Triangle Congruence:
• Two triangles are congruent if their vertices can be paired
so that corresponding sides are congruent and
corresponding angles are congruent.
• Two triangles are congruent if they have the same size
and shape.
• Two triangles are congruent when all corresponding sides
and interior angles are congruent.

AABC ≅ ADEF
Read as "triangle ABC is congruent to triangle DEF."
≅ symbol for congruence/congruence,
△ A symbol for triangle.
Congruent Triangle Example:
AB corresponds to RS
BC corresponds to ST
AC corresponds to RT
• As can be noted, a correspondence between two triangles
is any way of matching their vertices. The parts that
correspond are called corresponding parts of the two
triangles. Sometimes the correspondence between two
triangles is also a congruence correspondence.
• In congruent correspondence, the vertices of the 1st
triangle are paired with the vertices of the 2nd triangle in
such a way that the corresponding angles and the
corresponding sides are congruent.
WEEK 3: TRIANGLE CONGRUENCE (SAS, SSS, ASA
POSTULATES)
Congruence - two geometric figures with exactly the
same size and shape.
Congruent Triangles
Two triangles are congruent if and only if, it’s
corresponding parts are congruent (CPCTC)
CPCTC = “Corresponding Parts of Congruent
Triangles are Congruent.
The Congruence Postulates
1. SSS (Side-side-side) Congruence Postulates
If three sides of one triangle are congruent to
the corresponding sides of another triangle,
then the triangles are congruent.
2. SAS (side-angle-side) Congruence Postulates
If two sides and the included angle of one
triangles are congruent to the corresponding
two sides and the included angle of another
triangle, then the triangles are congruent.
3. ASA (Angle-side-angle) Congruence Postulate
If two angles and the included side of one
triangle are congruent to the corresponding
two angles and the included side of another
triangle, then the triangles are congruent.
WEEK 4: SOLVING CORRESPONDING PARTS OF
CONGRUENT TRIANGLES
- We learnt that triangles have three sides and three
angles. We also learnt that the sum of the angles in a
triangle is 180°. If two triangles have the same size
and shape they are called congruent triangles. If we
flip, turn or rotate one of two congruent triangles they
are still congruent.
Corresponding Parts of Congruent Figures
- The word corresponding refers to parts that match
between two congruent triangles. You can
identify corresponding angles and corresponding
sides.
- In solving the measurement of corresponding parts
of congruent triangles, you must analyze the figures
shown and use the hint and information given.
Example #2
WEEK 5: CORRESPONDING PARTS OF CONGRUENT
TRIANGLES ARE CONGRUENT
CPCTC - If two triangles are congruent, then their
corresponding parts are also congruent.
- Before you use CPCTC you must prove or know that
the two triangles congruent!!!
Triangle Congruency Short-Cuts
If you can prove one of the following shortcuts, you
have two congruent triangles
1.SSS (side-side-side)
2.SAS (side-angle-side)
3.ASA (angle-side-angle)
4.AAS (angle-angle-side)
5.HL (hypotenuse-leg) right triangles only
HL (hypotenuse leg) - is used with right triangles,
BUT, not all right triangles.
WEEK 6: PROVING STATEMENTS OF CONGRUENT
TRIANGLES
Built – In Information in Triangles
- Vertical Angles
- Shared side - reflexive property
- Shared angle - reflexive property
● Reflexive Property: XY ≅ XY
Proving Statement on Congruent Triangles
1. Mark the Given and what it implies.
2. Mark … Reflexive Sides / Vertical Angles
3. Choose a Method. (SSS , SAS, ASA)
4. List the Parts …
in the order of the method.
5. Fill in the Reasons (two-column form)…
why you marked the parts.
6. Is there more?
Some Reasons for Indirect Information
• Def. of midpoint
•
Def. of a bisector
•
Vertical angles are congruent
•
Def. of perpendicular bisector
•
Reflexive property (shared side)
•
Parallel lines ….. Alt. int. angles
•
Property of Perpendicular Lines
This is called a common side. It is a side for both
triangles. We’ll use the reflexive property
HL (hypotenuse leg) is used only with right triangles,
but, not all right triangles
Two-column Form
Given △ABC, △adc right △s,
AB ≅ AD
- Prove △ABC ≅ △ADC
Using CPCTC in Proofs
• According to the definition of congruence,
if two triangles are congruent, their
corresponding parts (sides and angles) are
also congruent.
• This means that two sides or angles that are
not marked as congruent can be proven to be
congruent if they are part of two congruent
triangles.
• This reasoning, when used to prove
congruence, is abbreviated CPCTC, which
stands for Corresponding Parts of Congruent
Triangles are Congruent.
WEEK 7: BISECTORS AND PERPENDICULAR LINES
- Distance from a point to a line: the length of the
perpendicular segment from the point to the line
- The distance is also the length of the shortest
segment from the point to the line.
- In the figure the distances from A to line l
and from B to l are represented by the red segments.
Angle Bisector Theorem
- If a point is on the bisector of an angle, then the
point is equidistant from the sides of the angle
If m<PQS = m<RQS, then PS = RS.
Converse of the Angle Bisector Theorem
- If a point in the interior of an angle is equidistant
from the sides of the angle, then the point is on the
angle bisector
A point is equidistant from two objects if it is the
same distance from the objects
Perpendicular Bisector Theorem
- If a point is on the perpendicular bisector of a
segment, then it is equidistant from the endpoints of
the segment
Converse of the Perpendicular Bisector Theorem