Geometry Basics

Geometry Basics

• Geometry is the study of shapes and spaces, derived from Greek words "geo" meaning earth and "metrein" meaning to measure.
• Geometry is not attributed to a single person, with ancient Egyptian, Greek, and French mathematicians contributing to its development.

Types of Geometry

• There are three main types of geometry: Euclidean, Hyperbolic, and Elliptical.
• Other subsections of geometry include Non-Euclidean, Analytic, Differential, and Topology.

Reasoning in Geometry

• There are two types of reasoning in geometry: inductive and deductive.
• Inductive reasoning draws conclusions based on observations, but is not mathematically valid.
• Deductive reasoning bases conclusions on previously known facts and theorems, but can be incorrect if premises are not true.

Famous Mathematicians

• Thales and Pythagoras are Greek mathematicians who contributed to geometry.
• Thales is credited with teaching various geometric principles, including Thales' Theorem.
• Pythagoras is credited with teaching the Pythagorean Theorem and other geometric principles.

Axiomatic Systems

• An axiomatic system is a set of axioms used to derive theorems.
• Euclidean geometry is an example of an axiomatic system, with five axioms introduced by Euclid.
• The three properties of axiomatic systems are consistency, independence, and completeness.

Euclid's Contributions

• Euclid wrote the series of books called "Elements", which introduced five basic truths or axioms.
• These axioms are the basis for Euclid's axiomatic system.

Undefined Concepts in Geometry

• Undefined concepts in geometry, such as points, lines, and planes, are extremely important for constructing other concepts and theorems.
• These concepts are described using abstract explanations and examples.

Postulates

• Postulates are basic truths that do not require formal proof.
• Postulates are the basis for other theorems in geometry.
• Postulates deal with the basic shapes of points, lines, and planes in geometry.

Algebraic Laws

• Algebraic laws describe how things add, subtract, multiply, divide, and combine.
• Examples of algebraic laws include the commutative, associative, and distributive laws.

Logic

• Logic is the study of how to critically think about propositions or statements.
• Logic statements can involve words, words and symbols, or just symbols.
• Logic is used to make new connections based on what is known to be true.

Logical Fallacies

• Logical fallacies are failures to use logically correct reasoning.
• There are two types of logic: formal and informal.
• Logical fallacies can be used to manipulate or convince others.

Conditional Statements

• A conditional statement is an if-then statement.
• Conditional statements have a hypothesis and a conclusion.
• The hypothesis is the "if" part of the statement, and the conclusion is the "then" part.

Geometric Proofs

• Geometric proofs are a series of statements used to verify the truth of other statements.
• The main parts of geometric proofs are the given statement, statements, and their reasons.
• Figures are also important in geometric proofs.

Angles

• An angle is defined as the amount of turn between two lines, segments, or rays.
• Common types of angles include acute, obtuse, right, straight, and reflex angles.
• Angle pairs, such as alternate interior and exterior angles, are important in geometry.

Line Segments

• A line segment is a portion of a line.
• Lines continue in two directions forever, with zero endpoints.
• Line segments can be divided into equal parts using a compass and a ruler.

Geometric Construction

• Geometric construction is the drawing of lines, angles, and shapes with only a pencil, compass, and straightedge.
• No measuring or numbers are involved in geometric construction.
• Geometric construction is used to create perpendicular lines, among other things.

Shapes and Perpendicular Lines

• A rectangle is a four-sided polygon with two parallel pairs of opposite sides.
• The area of a rectangle is calculated using the formula: A = l × w
• A triangle is a three-sided polygon with straight sides that meet at three angles.
• The area of a triangle is calculated using the formula: A = (b × h) / 2### Similar Triangles
• Similar triangles have corresponding angles that are congruent or the same measure.
• Similar triangles have corresponding sides that are proportional or related to one another in a constant ratio.
• Three ways to prove that triangles are similar: Side-Angle-Side (SAS), Side-Side-Side (SSS), and Angle-Angle (AA) similarity theorems.

Triangle Properties

• A triangle is a three-sided shape with three vertices (corners) and three angles that add up to 180 degrees.
• Angles can be measured in radians or degrees.
• Base of a triangle is any side (usually the horizontal side), and height is the shortest distance from the base to the top (apex) of the triangle.

Types of Triangles

• Scalene triangle: three different sides.
• Isosceles triangle: two congruent (identical) sides.
• Equilateral triangle: three congruent sides.
• Right triangle: one right angle (90 degrees).
• Acute triangle: three angles each smaller than a right angle.
• Obtuse triangle: one angle larger than a right angle.

Trigonometry

• Trigonometry is the study of angles and side ratios of a triangle.
• SOH CAH TOA: H is the hypotenuse (longest side of a right triangle), O is the opposite side, and A is the adjacent side.

Angles and Polygons

• In triangles, the sum of the interior angles is 180 degrees.
• Exterior angle: formed by extending one side past the angle.
• Sum of exterior angles for all polygons is always 360 degrees.
• Linear pair: two adjacent angles that form a straight angle.
• Supplementary angles: add up to 180 degrees.

Medians and Centroids

• Median: a line segment drawn from the vertex to the midpoint of the opposite side.
• Each triangle has three medians that meet at a single point, the centroid.
• Centroid: the center of mass of the triangle.
• Area of a triangle: a single median gives two triangles with equal areas.

Geometry Basics

• Geometry is the study of shapes and spaces, derived from Greek words "geo" meaning earth and "metrein" meaning to measure.
• Geometry is not attributed to a single person, with ancient Egyptian, Greek, and French mathematicians contributing to its development.

Types of Geometry

• There are three main types of geometry: Euclidean, Hyperbolic, and Elliptical.
• Other subsections of geometry include Non-Euclidean, Analytic, Differential, and Topology.

Reasoning in Geometry

• There are two types of reasoning in geometry: inductive and deductive.
• Inductive reasoning draws conclusions based on observations, but is not mathematically valid.
• Deductive reasoning bases conclusions on previously known facts and theorems, but can be incorrect if premises are not true.

Famous Mathematicians

• Thales and Pythagoras are Greek mathematicians who contributed to geometry.
• Thales is credited with teaching various geometric principles, including Thales' Theorem.
• Pythagoras is credited with teaching the Pythagorean Theorem and other geometric principles.

Axiomatic Systems

• An axiomatic system is a set of axioms used to derive theorems.
• Euclidean geometry is an example of an axiomatic system, with five axioms introduced by Euclid.
• The three properties of axiomatic systems are consistency, independence, and completeness.

Euclid's Contributions

• Euclid wrote the series of books called "Elements", which introduced five basic truths or axioms.
• These axioms are the basis for Euclid's axiomatic system.

Undefined Concepts in Geometry

• Undefined concepts in geometry, such as points, lines, and planes, are extremely important for constructing other concepts and theorems.
• These concepts are described using abstract explanations and examples.

Postulates

• Postulates are basic truths that do not require formal proof.
• Postulates are the basis for other theorems in geometry.
• Postulates deal with the basic shapes of points, lines, and planes in geometry.

Algebraic Laws

• Algebraic laws describe how things add, subtract, multiply, divide, and combine.
• Examples of algebraic laws include the commutative, associative, and distributive laws.

Logic

• Logic is the study of how to critically think about propositions or statements.
• Logic statements can involve words, words and symbols, or just symbols.
• Logic is used to make new connections based on what is known to be true.

Logical Fallacies

• Logical fallacies are failures to use logically correct reasoning.
• There are two types of logic: formal and informal.
• Logical fallacies can be used to manipulate or convince others.

Conditional Statements

• A conditional statement is an if-then statement.
• Conditional statements have a hypothesis and a conclusion.
• The hypothesis is the "if" part of the statement, and the conclusion is the "then" part.

Geometric Proofs

• Geometric proofs are a series of statements used to verify the truth of other statements.
• The main parts of geometric proofs are the given statement, statements, and their reasons.
• Figures are also important in geometric proofs.

Angles

• An angle is defined as the amount of turn between two lines, segments, or rays.
• Common types of angles include acute, obtuse, right, straight, and reflex angles.
• Angle pairs, such as alternate interior and exterior angles, are important in geometry.

Line Segments

• A line segment is a portion of a line.
• Lines continue in two directions forever, with zero endpoints.
• Line segments can be divided into equal parts using a compass and a ruler.

Geometric Construction

• Geometric construction is the drawing of lines, angles, and shapes with only a pencil, compass, and straightedge.
• No measuring or numbers are involved in geometric construction.
• Geometric construction is used to create perpendicular lines, among other things.

Shapes and Perpendicular Lines

• A rectangle is a four-sided polygon with two parallel pairs of opposite sides.
• The area of a rectangle is calculated using the formula: A = l × w
• A triangle is a three-sided polygon with straight sides that meet at three angles.
• The area of a triangle is calculated using the formula: A = (b × h) / 2### Similar Triangles
• Similar triangles have corresponding angles that are congruent or the same measure.
• Similar triangles have corresponding sides that are proportional or related to one another in a constant ratio.
• Three ways to prove that triangles are similar: Side-Angle-Side (SAS), Side-Side-Side (SSS), and Angle-Angle (AA) similarity theorems.

Triangle Properties

• A triangle is a three-sided shape with three vertices (corners) and three angles that add up to 180 degrees.
• Angles can be measured in radians or degrees.
• Base of a triangle is any side (usually the horizontal side), and height is the shortest distance from the base to the top (apex) of the triangle.

Types of Triangles

• Scalene triangle: three different sides.
• Isosceles triangle: two congruent (identical) sides.
• Equilateral triangle: three congruent sides.
• Right triangle: one right angle (90 degrees).
• Acute triangle: three angles each smaller than a right angle.
• Obtuse triangle: one angle larger than a right angle.

Trigonometry

• Trigonometry is the study of angles and side ratios of a triangle.
• SOH CAH TOA: H is the hypotenuse (longest side of a right triangle), O is the opposite side, and A is the adjacent side.

Angles and Polygons

• In triangles, the sum of the interior angles is 180 degrees.
• Exterior angle: formed by extending one side past the angle.
• Sum of exterior angles for all polygons is always 360 degrees.
• Linear pair: two adjacent angles that form a straight angle.
• Supplementary angles: add up to 180 degrees.

Medians and Centroids

• Median: a line segment drawn from the vertex to the midpoint of the opposite side.
• Each triangle has three medians that meet at a single point, the centroid.
• Centroid: the center of mass of the triangle.
• Area of a triangle: a single median gives two triangles with equal areas.