MATH LESSON 2 (POSTULATE AND THEOREMS IN GEOMETRY)

A branch of mathematics that studies the properties, measurements, and relationships of points, lines, surfaces, and solids

Geometry

Cannot be defined using using simplier concepts

Undefined Terms

Ideas that cannot be defined using using simplier concepts

Undefined Teems

To build the geometry system, mathematicians begin with a few basic ideas that cannot be defined using using simplier concepts

Undefined Teems

Undefined Terms

Point

Line

Plane

It represents an exact location in space. It has no length, width, or thickness

Point

It represents an exact location in space

Point

It has no length, width, or thickness

Point

Point is usually named with a

Capital letter

Point can be named using

Capital letter

It is a straight path that extends endlessly in two opposite directions. It has no thickness but has length

Line

It is a straight path that extends endlessly in two opposite directions

Line

It has no thickness but has length

Line

Line can be name using

Any two points on it or small letter

It is a flat surface that extends infinitely in all directions. It has length and width but no thickness

Plane

It is a flat surface that extends infinitely in all directions

Plane

It has length and width but no thickness

Plane

Statements that are accepted without proof

Postulate

Statement that can be proven using postulates, definitions, and previously proven theorems

Theorem

Father of Geometry

Euclid

Euclid organized geometry into a logical system. His Five Famous Postulates include

1. A straight line may be drawn between any two points

2. A straight line may be extended indefinitely

3. A circle may be drawn with any center and radius

4. All right angles are congruent

5. If a transversal cuts two lines and the sum of the interior angles on the same side is less than two right angles, then the two lines meet on that side

Five Famous Postulates of Euclid include

1. A straight line may be drawn between any two points

2. A straight line may be extended indefinitely

3. A circle may be drawn with any center and radius

4. All right angles are congruent

5. If a transversal cuts two lines and the sum of the interior angles on the same side is less than two right angles, then the two lines meet on that side

Five Famous Postulates

1. A straight line may be drawn between any two points

2. A straight line may be extended indefinitely

3. A circle may be drawn with any center and radius

4. All right angles are congruent

5. If a transversal cuts two lines and the sum of the interior angles on the same side is less than two right angles, then the two lines meet on that side

8 Postulates

Postulate 1: The Line Postulate

Postulate 2: The Plane Postulate

Postulate 3: The Flat Plane Postulate

Postulate 4: The Plane Intersection Postulate

Postulate 5: Segment Addition Postulate

Postulate 6: Angle Addition Postulate

Postulate 7: Linear Pair Postulate

Postulate 8: Angle Measurement Postulate

8 Theorems

Theorem 1: The Line Intersection Theorem

Theorem 2: The Line-Plane Intersection Theorem

Theorem 3: The Line-Point Theorem

Theorem 4: The Lines-Plane Theorem

Theorem 5: Right Angle Congruence Theorem

Theorem 6: Vertical Angle Theorem

Theorem 7: Triangle Sum Theorem

Theorem 8: Exterior Angle Theorem

There is exactly one line through any two points

Postulate 1: The Line Postulate

Any three points lie in at least one plane and any three non-collinear points lie in exactly one plane

Postulate 2: The Plane Postulate

If two points lie in a plane, then the line containing them lies in the same plane

Postulate 3: The Flat Plane Postulate

If two planes intersect, then their intersection is a line

Postulate 4: The Plane Intersection Postulate

If point B lies between A and C, then AC=AB+BC

Postulate 5: Segment Addition Postulate

If a point lies in the interior of an angle, then the measure of the whole angle is the sum of the measures of the two adjacent angles formed

Postulate 6: Angle Addition Postulate

If two angles form a linear pair, then they are supplementary meaning their measures add up to 180°

Postulate 7: Linear Pair Postulate

It states that the measure of every angle can be matched to a real number between 0° and 180°

Postulate 8: Angle Measurement Postulate

If two distinct lines intersect, then they intersect in exactly one point

Theorem 1: The Line Intersection Theorem

If a line that does not lie on a plane intersects the plane, then their intersection is exactly one point

Theorem 2: The Line-Plane Intersection Theorem

If a line and a point not on the line are given, then there is exactly one plane that contains both the line and the point

Theorem 3: The Line-Point Theorem

If two lines intersect, then there is exactly one plane that contains both lines

Theorem 4: The Lines-Plane Theorem

All right angles are congruent. This means that any two right angles will always have the same measure, which is exactly 90°

Theorem 5: Right Angle Congruence Theorem

Vertical angles are always congruent

Theorem 6: Vertical Angle Theorem

It states that the sum of the interior angles of a triangle is always 180°

Theorem 7: Triangle Sum Theorem

If one side of a triangle is extended, the exterior angle form is equal to the sum of its two non-adjacent interior angles (also called remote interior angles)

Theorem 8: Exterior Angle Theorem

These terms are defined using undefined terms

Define terms in geometry

These terms are defined using undefined terms

Defined terms

Define terms

Line segment

Ray

Midpoint

Is a point that divides a segment into two equal parts

Midpoint

Is a line, ray, or another segment that passes through the midpoint of a segment, dividing it into two congruent parts

Segment Bisector

Is an angle measuring less than 90° but greater than 0°

Acute Angle

Is an angle whose measure is exactly 90°

Right Angle

Is an angle whose measure is greater than 90° but less than 180°

Obtuse Angle

Is a line, segment, or ray that divides an angle into two equal angles, which are called congruent angles because they have the same measure

Angle Bisector

Is a line, segment, or ray that divides an angle into two equal angles

Angle Bisector

Other term of equal angles

Congruent angles

Are two angles whose measures add up to 90°

Complementary Angles

Are two angles whose measures add up to 180°

Supplementary Angles

Is formed when two adjacent angles have their non-common sides on the same line

Linear Pair

Form by three line segments that connect three non-collinear points

Triangle

Form when two rays share a common endpoint, called the vertex

Angle

Other term of a common endpoint

Vertex

Is part of a line that has two endpoints

Line Segment

Is part of a line that starts at a single endpoint and extends infinitely in one direction, and it is named starting from the endpoint

Ray