## What is Geometry?

Geometry is a branch of mathematics which deals with the properties, measurement, and relationships of lines, angles, points, coordinates, solids and surfaces, and solids. It is the 4th math course in high school that will help you guide through among other things inclusive of points, planes, parallel lines, lines, angles, quadrilaterals, triangles, squares, similarity, trigonometry, transformations, circles, circumference and area. Learning geometry, you will be able to study the properties of given elements that remain invariant under particular transformations.

## Major Branches of Geometry

### 1. Euclidean Geometry

In ancient cultures there developed a type of geometry apt to the relationships between lengths, areas, and volumes of physical figures. This geometry gained popularity being codified in Euclid’s elements based upon 10 axioms, or postulates, from which a hundred many theorems were proved by deductive logic.

### 2. Non-Euclidean Geometries

Several mathematicians substituted alternatives to Euclid’s parallel postulate, which, in its modern form, reads, “Given a line and a point not on the line, it is feasible to construct exactly one line along the given point parallel to the line.”

### 3. Analytic Geometry

Introduced by the French mathematician René Descartes (1596–1650), this geometry is representative of algebraic equations. This French mathematician only initiated rectangular coordinates to locate points and to allow lines and curves to be delineated with algebraic equations.

4. Projective Geometry

The French mathematician Girard Desargues (1591–1661) initiated projective geometry to enable dealing with those properties of geometric objects that are not revised by projecting their image, or “shadow,” on another surface.

### 5. Differential Geometry

In connection with practical problems of surveying and geodesy, a German mathematician introduced the domain of differential geometry. Using differential calculus, the innate properties of curves and surfaces are distinguished. For example, he characterized that the intrinsic curvature of a cylinder is just similar as that of a plane, as can be observed by cutting a cylinder through its axis and flattening, but not similar as that of a sphere, which is unable to be flattened without distortion.

### 6. Topology

Topology, the youngest and most innovative branch of geometry, emphasizes upon the properties of geometric shapes that remain unaltered upon ongoing deformation—stretching, contracting, and folding, but not tearing.

## Geometry Mathematics

Let’s get to know what you will be learning under concepts of geometry:

### Lines

Rays, Lines and line segments

Measuring Lines

Parallel, perpendicular, points, and planes

Geometry definitions

The golden ratio

### Shapes

Properties and Classification of geometric shapes

Equal parts of shapes

Polygons and Angles with polygons

Curves

Solid geometry (3D shapes)

### Angles

Introduction to Angles

Measuring and Constructing Angles

Angles between bisecting lines

Types of Angles

Angles in circle

### Triangles

Types of Triangles

Triangle angles

Triangle inequality theorem

Angle bisectors and Perpendicular bisectors

Altitudes, Medians & centroids

### Quadrilaterals

Types of Quadrilaterals

Proofs & angles of Quadrilaterals

### Coordinate Plane

Coordinate plane: Quadrants on the coordinate plane, quadrant 1 and 4 quadrants

Reflecting points on coordinate plane

Quadrilaterals and Polygons on the coordinate plane

### Area and Perimeter

Count unit squares to find area

Area of rectangles, trapezoids and Area of parallelograms

Area of triangles

Area of shapes on grids

Area of composite figures

Area and circumference of circles

Advanced area with triangles

### Volume and Surface Area

Volume of rectangular prisms

Volume with fractions

Surface area

Surface and volume density

Volume of spheres, cones, and cylinders

Volume and surface area of Solid geometry

Cross sections of 3D objects

Koch snowflake fractal

Heron's formula

### Transformations

Introduction to rigid transformations

Properties and definitions of transformations

Dilations, Translations, Rotations and Reflections of Transformations

Overview of Rigid transformations

Symmetry

### Similarity

Introduction and Definition of Similarity

triangle similarity

Solving similar triangles

Angle bisector theorem

Solving with similar and congruent triangles

Solving modeling problems

### Congruence

Transformations and congruence

Triangle congruence

Theorems in reference to triangle properties and quadrilateral properties

Working with triangles

Proofs of general theorems that apply triangle congruence

Trigonometry

Introduction to the trigonometric ratios

Special right triangles

Modeling with right triangles

Solving for a side and for an angle in a right triangle using the trigonometric ratios

Trigonometric ratios and similarity

The law of sines and cosines

Sine and cosine of complementary angles

The reciprocal trigonometric ratios

Solving problems on general triangles

### Circles

Circle basics

Introduction to radians

Arc measure, Arc length (from degrees), Arc length (from radians)

Sectors

Problem solving on Inscribed angles and Inscribed shapes

Properties of tangents

Area of inscribed triangle

Standard and Expanded equation of a circle

### Analytic Geometry

Distance and midpoints

Dividing line segments

Distance on the coordinate plane Problem solving

Parallel and perpendicular lines

Distance between a point and a line challenge

### Geometric Constructions

Constructing bisectors of lines and angles

Constructing regular polygons

Constructing inscribed in circles

Constructing incircles and circumcircles

Constructing a line tangent to a circle

### Pythagorean Theorem

Introduction and Application of Pythagorean theorem

Pythagorean theorem and distance between points

Pythagorean theorem proofs

### Miscellaneous

Worked examples pr problem solving

Q1. What is the History Behind Geometry?

Answer: The earliest known unambiguous examples begun with devising mathematical rules and methods useful for constructing buildings, surveying land areas, and measuring storage containers. The Greeks around 6th century BCE collected and delivered this practical knowledge and generalized it in the form of the abstract subject now known as geometry. It is derived from the summation of the Greek words geo (“Earth”) and metron (“measure”) for the measurement of the Earth.

Q2. What are the Various Geometric Concepts?

Answer: There are majorly 10 geometrical concepts. Geometry guides you through among other elements that include:

Points

Planes

parallel lines

Lines

Angles

Triangles

Similarity

Trigonometry

Transformations

Quadrilaterals

circles and area.

Q3. What is the Geometric Concept of Finding the Right Angle?

Answer: Builders and surveyors in ancient times required to construct right angles onto the field. The technique used by the Egyptians gained them the name “rope pullers” in Greece, might be because they used a rope for laying out their construction.

Q4. Which Ancient Method Helps Finding the Right Angle?

Answer: The simplest method to conduct the trick is to take a rope that is 12 units long, mark knot 3 units from one end and another 5 units from the other end, and then knot the ends together to form a loop. However, the Egyptian scribes generalize them to obtain the Pythagorean Theorem: the square on the line opposite the right angle is equivalent to the sum of the squares on the other two sides.