MATH Q4

**Seed Concepts of Geometry**

The __point__, __line__, and __plane__ are the seed concepts of geometry. Although they are abstractions or mathematical ideas based on our experience, they form the starting point of the description of geometrical structures. They are often called “undefined terms.”

__Point__

A point is represented with small dots. It has no dimension. We use a __capital letter__ to name a point.

__notation:__

point A, point B, point C, point D

__figure:__

__Line__

A line has no width and it is extended endlessly in both directions. It has one dimension: ** length**.

We use a __lower-case letter__ or any two points on the line to name a line.

__notation:__

__figure:__

__Plane:__

A plane is a flat surface that extends infinitely far in all directions. It has no thickness. We use a __capital script letter__ or three non-collinear points on the plane to name a plane.

__notation:__

__figure:__

__Space__

A set of all points. It extends infinitely in all directions and has three dimensions.

__figure:__

**Basic Postulates**

**Postulates** are statements of facts or self-evident truths. It is accepted as true without proof. In Filipino, “** malinaw na katotohanan**”.

**Postulate 1:** __Points Postulate__ -

*● A line contains at least two points.*

*● A plane consists of at least three non-collinear points.*

*● A space contains at least four non-coplanar points.*

*Postulate 2: *__Line Postulate__ - Two points determine a line.

*Postulate 3: *__Plane Postulate__ - Three non-collinear points determine a plane.

*Postulate 4: *__Flat Plane Postulate__ - If two points of a line lie on the plane, then the entire line lies on the plane.

*Postulate 5: *__Plane Intersection Postulate__ - If two planes intersect, then their intersection is a line.

*Postulate 6: *__Ruler Postulate__ -

*● For every pair of points, there is only a positive real number called the distance between the two points.*

*● The length or linear measure of a segment is the distance*

*between its endpoints. This distance is denoted AB for a segment.*

*Postulate 7:*__Segment Construction Postulate__*- On any ray, there is exactly one point at a given distance from the endpoint of the ray.*

*Postulate 8: *__Segment Addition Postulate__ - If point *P** is between **A** and **B**, then the linear measures **AP̅̅̅̅ + PB̅̅̅̅ = AB̅̅̅̅.*

**Theorems**

Unlike postulates, ** theorems **are statements that must be proven true by citing undefined terms, definitions, postulates, and other proven theorems.

*Theorem 1:** Line Intersection Theorem - If two lines intersect, then*

*their intersection is exactly one point.*

*Thereom 2: *__Line-Point Theorem__ - Given a line and a point

*not on the line, there is exactly one plane that contains them.*

*Thereom 3: *__Line-Point Theorem__ - Given two intersecting

*lines, there is exactly one plane that contains the two lines.*

*Theorem 4:** Line-Plane Intersection Theorem - Given a plane and a line*

*not on the plane, their intersection is one and only one point.*

**Angle Pairs I**

*Supplementary Angles **- If the sum of the measures of two angles is 180°, then they are called*

__supplementary angles.__

** Complementary Angles** -

*If the sum of the measures of two angles is 90°, then they are called complementary angles.*

__Two angles don’t need to be adjacent to be called supplementary or complementary.__

**Angle Pairs II**

**Adjacent Pairs **- *two angles with a common vertex, a common side, and no common interior points.*

*In the figure, ∠AEB and ∠BEC share a common vertex at point E and a common side at ray EB . These pair of angles are adjacent angles. ∠BED and ∠CED are **NOT** adjacent angles since point C is in the interior of ∠BED.*

*Linear Pair of Angles** - a pair of adjacent angles whose non-common sides are collinear. The sum of the measures of the angle is 180°.*

*Example: ∠DBA and ∠DBC are a linear pair because ray BA and ray BC are collinear.*

*Vertical Angles** - pair of angles opposite each other formed by two intersecting lines in an “X”-shape. Vertical angles are equal in measure and so are congruent.*

*Example:∠EGI and ∠FGH are vertical angles which means that they are congruent.*

**Types of Triangles**

● A right triangle is a triangle with a right angle. It is composed of two legs and a

hypotenuse (the side opposite the right angle). (Figure 1)

● An acute triangle is a triangle with three acute angles. (Figure 2)

● An obtuse triangle is a triangle with an obtuse angle. (Figure 3)

● An equiangular triangle is a triangle with three equal angles. (Figure 4)

● An equilateral triangle is a triangle with three equal sides. (Figure 5)

● An isosceles triangle is a triangle with two equal sides. (Figure 6)

● A scalene triangle is a triangle with no equal sides.

Acute angle - less than 90 degress

obtuse angle - more than 90 degrees less than 180 degrees

*Corresponding Angle*

*Corresponding angles of two triangles are the angles whose vertices are*

*matched.*

If we match or pair vertex B with vertex E, vertex A with vertex D, and vertex

C with vertex F, we get the correspondences:

∠B ↔ ∠E AB ↔ DE

∠A ↔ ∠D BC ↔ EF

∠C ↔ ∠F AC ↔ DF

*Congruent Triangle*

Two triangles are __congruent__ if and only if there is a matching of their vertices such __that corresponding parts are congruent__.

In most references, this is known as the **CPCTC** (Corresponding Parts of

Congruent Triangles are Congruent) Theorem.

**Note**: The symbol ↔ means “corresponds to”.

**SSS (Side-Side-Side) Congruence Postulate**

*If three sides of one triangle are congruent to the corresponding sides of*

__another triangle__, then the triangles are congruent.

**SAS (Side-Angle-Side) Congruence Postulate**

*If t wo sides and the included angle of one triangle are congruent to the*

__corresponding parts of another triangle__, then the triangles are __congruent.__

*Note: The “included angle” is the angle whose vertex is the common endpoint of the two sides.*

**ASA Congruence (Angle-Side-Angle) Postulate**

If t__wo angles and the included side of one triangle__ are __congruent__ to the

__corresponding parts of another triangle__, then the triangles are __congruent.__

Note: The “__included side__” is the side whose endpoints are the vertices of the two angles.

**AAS (Angle-Angle-Side) Congruence Postulate**

If __two angles and a non-included side__ in one triangle are __congruent__ to the

__corresponding parts of another triangle__, then the two triangles are __congruent__.

# MATH Q4

**Seed Concepts of Geometry**

The __point__, __line__, and __plane__ are the seed concepts of geometry. Although they are abstractions or mathematical ideas based on our experience, they form the starting point of the description of geometrical structures. They are often called “undefined terms.”

__Point__

A point is represented with small dots. It has no dimension. We use a __capital letter__ to name a point.

__notation:__

point A, point B, point C, point D

__figure:__

__Line__

A line has no width and it is extended endlessly in both directions. It has one dimension: ** length**.

We use a __lower-case letter__ or any two points on the line to name a line.

__notation:__

__figure:__

__Plane:__

A plane is a flat surface that extends infinitely far in all directions. It has no thickness. We use a __capital script letter__ or three non-collinear points on the plane to name a plane.

__notation:__

__figure:__

__Space__

A set of all points. It extends infinitely in all directions and has three dimensions.

__figure:__

**Basic Postulates**

**Postulates** are statements of facts or self-evident truths. It is accepted as true without proof. In Filipino, “** malinaw na katotohanan**”.

**Postulate 1:** __Points Postulate__ -

*● A line contains at least two points.*

*● A plane consists of at least three non-collinear points.*

*● A space contains at least four non-coplanar points.*

*Postulate 2: *__Line Postulate__ - Two points determine a line.

*Postulate 3: *__Plane Postulate__ - Three non-collinear points determine a plane.

*Postulate 4: *__Flat Plane Postulate__ - If two points of a line lie on the plane, then the entire line lies on the plane.

*Postulate 5: *__Plane Intersection Postulate__ - If two planes intersect, then their intersection is a line.

*Postulate 6: *__Ruler Postulate__ -

*● For every pair of points, there is only a positive real number called the distance between the two points.*

*● The length or linear measure of a segment is the distance*

*between its endpoints. This distance is denoted AB for a segment.*

*Postulate 7:*__Segment Construction Postulate__*- On any ray, there is exactly one point at a given distance from the endpoint of the ray.*

*Postulate 8: *__Segment Addition Postulate__ - If point *P** is between **A** and **B**, then the linear measures **AP̅̅̅̅ + PB̅̅̅̅ = AB̅̅̅̅.*

**Theorems**

Unlike postulates, ** theorems **are statements that must be proven true by citing undefined terms, definitions, postulates, and other proven theorems.

*Theorem 1:** Line Intersection Theorem - If two lines intersect, then*

*their intersection is exactly one point.*

*Thereom 2: *__Line-Point Theorem__ - Given a line and a point

*not on the line, there is exactly one plane that contains them.*

*Thereom 3: *__Line-Point Theorem__ - Given two intersecting

*lines, there is exactly one plane that contains the two lines.*

*Theorem 4:** Line-Plane Intersection Theorem - Given a plane and a line*

*not on the plane, their intersection is one and only one point.*

**Angle Pairs I**

*Supplementary Angles **- If the sum of the measures of two angles is 180°, then they are called*

__supplementary angles.__

** Complementary Angles** -

*If the sum of the measures of two angles is 90°, then they are called complementary angles.*

__Two angles don’t need to be adjacent to be called supplementary or complementary.__

**Angle Pairs II**

**Adjacent Pairs **- *two angles with a common vertex, a common side, and no common interior points.*

*In the figure, ∠AEB and ∠BEC share a common vertex at point E and a common side at ray EB . These pair of angles are adjacent angles. ∠BED and ∠CED are **NOT** adjacent angles since point C is in the interior of ∠BED.*

*Linear Pair of Angles** - a pair of adjacent angles whose non-common sides are collinear. The sum of the measures of the angle is 180°.*

*Example: ∠DBA and ∠DBC are a linear pair because ray BA and ray BC are collinear.*

*Vertical Angles** - pair of angles opposite each other formed by two intersecting lines in an “X”-shape. Vertical angles are equal in measure and so are congruent.*

*Example:∠EGI and ∠FGH are vertical angles which means that they are congruent.*

**Types of Triangles**

● A right triangle is a triangle with a right angle. It is composed of two legs and a

hypotenuse (the side opposite the right angle). (Figure 1)

● An acute triangle is a triangle with three acute angles. (Figure 2)

● An obtuse triangle is a triangle with an obtuse angle. (Figure 3)

● An equiangular triangle is a triangle with three equal angles. (Figure 4)

● An equilateral triangle is a triangle with three equal sides. (Figure 5)

● An isosceles triangle is a triangle with two equal sides. (Figure 6)

● A scalene triangle is a triangle with no equal sides.

Acute angle - less than 90 degress

obtuse angle - more than 90 degrees less than 180 degrees

*Corresponding Angle*

*Corresponding angles of two triangles are the angles whose vertices are*

*matched.*

If we match or pair vertex B with vertex E, vertex A with vertex D, and vertex

C with vertex F, we get the correspondences:

∠B ↔ ∠E AB ↔ DE

∠A ↔ ∠D BC ↔ EF

∠C ↔ ∠F AC ↔ DF

*Congruent Triangle*

Two triangles are __congruent__ if and only if there is a matching of their vertices such __that corresponding parts are congruent__.

In most references, this is known as the **CPCTC** (Corresponding Parts of

Congruent Triangles are Congruent) Theorem.

**Note**: The symbol ↔ means “corresponds to”.

**SSS (Side-Side-Side) Congruence Postulate**

*If three sides of one triangle are congruent to the corresponding sides of*

__another triangle__, then the triangles are congruent.

**SAS (Side-Angle-Side) Congruence Postulate**

*If t wo sides and the included angle of one triangle are congruent to the*

__corresponding parts of another triangle__, then the triangles are __congruent.__

*Note: The “included angle” is the angle whose vertex is the common endpoint of the two sides.*

**ASA Congruence (Angle-Side-Angle) Postulate**

If t__wo angles and the included side of one triangle__ are __congruent__ to the

__corresponding parts of another triangle__, then the triangles are __congruent.__

Note: The “__included side__” is the side whose endpoints are the vertices of the two angles.

**AAS (Angle-Angle-Side) Congruence Postulate**

If __two angles and a non-included side__ in one triangle are __congruent__ to the

__corresponding parts of another triangle__, then the two triangles are __congruent__.