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# Fundamentals of Geometry

Fundamentals of Geometry
• Calculus • Decision Maths • Geometry • Mechanics Maths • Probability and Statistics • Pure Maths • Statistics Though it can seem complicated at times, geometry can essentially boil down to a few fundamental concepts. These concepts have been known for thousands of years, with their origins in several different ancient cultures. The ancient Greek mathematician Euclid is often considered the 'father' of geometry, with his fundamentals of Euclidean geometry serving as a basis for much of 'modern mathematics' understanding of geometry.

## Fundamental Concepts of Euclidean Geometry

In uncovering the secrets of the fundamentals of geometry, Euclid realised that first must come the very basic fundamental concepts. Certain questions had to be answered to define these fundamental constituents of the field, such as what is a point, or what is a line?

### Points

Euclid defined a point in the following way: “a point is that which has no part." In essence, this simply means that a point is just a location in space that has no dimensions. That is to say, though it has spatial parameters to define its location, it does not actually occupy any space itself. A point P(4,3) on a set of x-y axes, John Hannah - StudySmarter Originals

### Lines

Euclid also defined something called a line. This he defined as "a length without a breadth." In essence, just a 1-dimensional segment with a finite length. He posited that a line could be extended infinitely in either direction. This is an area where modern geometry differs from Euclid's fundamentals, as now we refer to an infinitely extended line simply as a line and Euclid's line of finite length as a line segment. This is an important distinction to remember for the sake of correctness. Let's take a look at the difference below. Euclid also defined something called a ray, which similar to a line is infinitely long, however, it has a defined starting point. It is also sometimes known as a half-line.

A line is a straight, 1-dimensional figure that extends indefinitely in both directions. A line segment is a straight, 1-dimensional figure of finite length that connects two points. A ray is a straight one-dimensional figure that extends infinitely in one direction, from a defined starting point. A line on a set of x-y axis, John Hannah - StudySmarter Originals A line segment on a set of x-y axes, John Hannah - StudySmarter Original A ray on a set of x-y axes, John Hannah - StudySmarter Original

### Planes

A plane, in many ways, can be considered as similar to a line, but in two dimensions. A plane is simply a surface that extends indefinitely. A plane can exist in 2-dimensional spaces as well as 3-dimensional spaces and higher.

A plane is a 2-dimensional figure that extends indefinitely in four directions. A plane on a set of x-y-z axes, John Hannah - StudySmarter Originals

### Angles

Angles were defined by Euclid as "the inclination of two straight lines." This essentially can be described in simpler terms as the rotational distance between two lines or line segments that share a point, i.e how much would we have to rotate one line before it lines up with the other. The shared point is known as the vertex of the angle.

An angle is a measure of rotational space between two lines or line segments. Two line segments forming an acute angle, A, John Hannah - StudySmarter Originals

### Dimensions

Dimensions are an important aspect of the fundamentals of geometry, which specifically deals with spatial dimensions. Spatial dimensions in mathematics and physics can be defined as the minimum number of coordinates required to describe a point in that space. For instance, a line has 1-dimensions as only a single number is required to specify a point on that line. Equally, if you wanted to specify a point on an x-y axis you would need two coordinates an x and y coordinate and on a set of 3-dimensional axes, you would need a third coordinate - the z coordinate.

Dimensions are extensions of space in a single direction, the length along which can be used to describe the location of a point in that dimension. Multiple dimensions can be combined to describe geometric properties with increasing complexity. A point in 2-dimensional space, John Hannah - StudySmarter Originals A point in 3-dimensional space, John Hannah, StudySmarter Originals

### Area

Area is a measurement that describes the size of a certain 2-dimensional region. There are various formulas that can be used in calculating the area of certain shapes. A good way to visualise area is to divide up a 2-dimensional space into squares. The area of the shape is simply equal to the number of squares contained inside it.

Area is a measurement that describes the size of a certain 2-dimensional region of space. Depiction of the area of a triangle on a set of x-y axes, John Hannah - StudySmarter Originals

### Volume

Much like area, volume is a measure that quantifies the size of a certain region of space. Volume, however, quantifies the size of a region in 3-dimensional space. All 3-dimensional shapes have volume, and similar to area there are many useful formulas for calculating the volumes of various shapes. We can visualise volume in much the same way as area, but rather than using small squares, we count the number of small cubes inside a shape. The image below depicts a cube in 3-dimensional space. How much space does the cube take up? Well, by counting we can see that the cube takes up the space of 64 smaller cubes, each with a volume of 1 units3.

Volume is a measurement that describes the size of a certain 3-dimensional region of space. A cube in 3-dimensional space,
John Hannah - StudySmarter Originals

### Units

An important part of the fundamentals of geometry is the use of various units. In geometry, we use two basic types of units: units of length and units of angles.

A unit is a convention that helps us define how large something is. For instance, a unit of length can help us define how long something is, and a unit of volume can help us define how large a 3-dimensional shape is.

#### Units of Length

There are two primary systems for units of length. These systems are the metric and imperial systems. The metric system deals in units of centimetres, metres, kilometres etc. whereas the imperial system works in units inches, feet, yards, miles etc.

Length is a 1-dimensional unit, however, units of 2-dimensions (area) and 3-dimensions (volume) exist which are composed of these units of length. The convention for the naming of these dimensions is shown in the table below.

 Length Area Volume         #### Units of Angles

Degrees and radians are the two main units for the measurement of angles and it's very easy to run into problems if the distinction between the two isn't clear!

Firstly, it is important to recognise that degrees are an arbitrary unit of measurement that simply arose from the fact of the Earth's rotation. Ancient people watching the constellations in the sky move on a yearly cycle figured that since there were 360 days in a year (there are really 365 but they got pretty close!) that there should be 360 degrees in a full rotation. This has proved a simple and intuitive way to discuss angles as human beings, after all, we aren't computers and sometimes closely packed decimal numbers can be confusing.

However, since these early scholars sat looking at the stars, we have discovered another, arguably more mathematically sound way of describing angles. This unit is known as the radian.

Radians, rather than being related simply to 'amount of rotation', are related to distance travelled around an arc. Really, radians are in fact the distance travelled around an arc divided by the distance to the pivot point of that arc. If we take the equation relating a circle's circumference to its radius, we can find how many radians in a full rotation of 360o.  So in 360o there are radians. From this, we can see that

 Degrees Radians 360o 180o 90o 45o It is important when using a calculator if it is set to treat your angles as radians or degrees when dealing with trigonometric functions to obtain the correct answer. Technically all mathematical functions taking angles as inputs work in radians.

## The Fundamental Principles of Euclidean Geometry

Euclid made five fundamental postulates when delving into the field of geometry. These postulates were fundamental principles of geometry that he held to be self-evident, and informed all further principles and concepts of geometry thereafter.

 Postulates of Euclidean Geometry 1. A straight line segment can be drawn joining any two points. 2. Any straight line segment can be extended indefinitely in a straight line. 3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as centre. 4. All right angles are congruent. 5. Given a line and a point not on that line, there exist an infinite number of lines through the given point parallel to the given line.

## Fundamentals of Geometry - Key takeaways

• Many of the fundamentals of geometry that are used today were popularised by a Greek mathematician known as Euclid.
• A point is a single location space that has no size.
• A line is a 1-dimensional figure that extends indefinitely.
• A line segment is a 1-dimensional figure with finite start and end-points.
• A plane is a 2-dimensional figure that extends indefinitely.
• Angles are a measure of rotational distance.
• Dimensions can be described as the coordinates required to define a point in a certain space.
• Area is a measure of the 2-dimensional size of a shape.
• Volume is the measure of the 3-dimensional size of a shape.
• Units are a convention used to define the size of various quantities.
• Units of length are either of the imperial system or the metric system.
• Units of angles are either radians or degrees.
• Euclid defined five postulates of geometry that he held as self-evident truths.

The fundamentals of geometry are a set of rules and definitions upon which all other areas of geometry are built.

The most basic, fundamental components of geometry are points, lines, and planes.

Dimensions are an extension of space in a single direction, the length along which can be used to describe the location of a point in that dimension. To make things simpler, we can think of the number of dimensions in a space to be the number of coordinates required to fully describe the location of a point in that space.

The fundamental theorem of similarity states that a line segment splits two sides of a triangle into proportional segments if and only if the segment is parallel to the triangle's third side.

Geometry can be described largely in terms of points, lines, planes, line segments, dimensions, and angles.

## Final Fundamentals of Geometry Quiz

Question

What are vertical angles?

Vertical angles are two nonadjacent angles that are generated by when two lines intersect.

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What is a linear pair?

If the non-common sides of two adjacent angles are opposing rays, they form a linear pair.

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What are supplementary angles?

If the total of two angles' measurements is 180o , they are called supplementary angles. Also, angles in a linear pair are supplements to each other.

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What are the four types of angles?

Acute, right, obtuse, and straight

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What is the name of the tool to measure angles?

Protractor

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Adjacent angles are two angles that are in the same plane, share a vertex and a side, but have no interior points in common.

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What are congruent angles?

Two angles are congruent angles if they have the same measure.

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What are complementary angles?

If the total of two angles' measurements is 90o , they are called complementary angles.

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∠A and ∠B are complementary. If m∠A= x+10 and m∠B=4x-20, then find m∠A and m∠B.

m∠A=30o , m∠B=60o.

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What are parallel lines?

Parallel lines are lines which never intersect and are equidistant.

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How are parallel lines shown on a diagram?

Parallel lines are shown by drawing an arrowhead on each of the lines.

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What is the slope of a parallel line?

Parallel lines have the same slope.

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A line has a slope of 4. What is the slope of a line parallel to it?

A parallel line would also have a slope of 4

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For a line with the equation y = mx + b, what is the equation of a parallel line?

A parallel line will have the same slope (m) and a different y-intercept (b)

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A line is given by the equation y = 3x + 2. What is the equation of a line parallel to it?

y = 3x + b, where b is any number other than 2

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What is the difference between parallel and perpendicular lines?

Parallel lines will never intersect. Perpendicular lines intersect at right angles.

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What is the difference between the equations of parallel and perpendicular lines?

Parallel lines have the same slope. Perpendicular lines have slopes that multiply to -1.

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What is the name of a line that cuts through two lines?

Intersecting transversal

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What are alternate angles?

Alternate angles are angles in parallel lines that create a Z shape in the lines. Alternate angles are equal.

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What are interior angles?

Interior angles are angles in parallel lines that create a C shape in the lines. Interior angles add to 180°.

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What are corresponding angles?

Corresponding angles are angles in parallel lines that create an F shape in the lines. Corresponding angles add to 180°.

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How can you prove that two lines are parallel?

Two lines are parallel if, when a transversal line is cut through them, the angles made between the lines follow the alternate, interior and corresponding angle rules.

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Are two lines parallel if the alternate angles are not equal?

No, as alternate angles are equal in parallel lines.

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What is a figure?

A figure is a geometric shape that is a combination of lines, points and planes that form a closed boundary.

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What are the types of figures in geometry?

1. Two dimensional figures.

2. Three dimensional figures.

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What is a composite figure?

Composite figure is a figure that can be separated into different geometric shapes

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What is the difference between congruent and similar figures?

Congruent figures are of the same shape and size but Similar figures are of the same shape but not the same size.

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What is a 2 dimensional figure?

A two dimensional figure is a flat or plane figure that has just two dimensions which are the length and breadth

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What are 3 dimensional figures?

A three dimensional figure is a solid and tangible figure that has its three dimensions as its length, breadth and depth

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What are Similar Figures?

Similar figures are figures that look alike. They have the same shape but not the same size.

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What are Congruent figures?

Congruent figures are figures that are of the same size and shape. They have the same length, width, height and even angles.

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What is the difference between a closed figure and an open figure?

A closed figure is a figure that is connected on both ends, that is, its start and end point meet but an open figure is a figure that is not connected on both ends. The start and the end point of an open figure do not meet.

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What dimensions do 2-dimensional figures have?

Length and width

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Which is the name of the lines that form the 2-dimensional figures?

Sides

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Are open figures a type of 2-dimensional figures?

Yes, although we cannot determine its perimeter and area

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How much do the angles of a triangle sum?

180º

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Which one of these shapes have always all of their sides equal?

Squares

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Which is the name of the distance from the center of a circle to any of its points?

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What is the perimeter of a 2d figure?

The sum of all of its sides

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How much is the perimeter of a rectangle with two sides a and b equal to 3 meters (a,b = 3m ) and the other two c and d equal to 6 meters (c,d = 6m)?

18 m

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If we approximate the number π to 3.14, how much is the circumference of a circle with diameter d = 4m

6.28 m

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What is the area of a 2-dimensional figure?

The surface inside of its sides

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Which are the units used for the area?

Area units (the standard is square meters)

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If we have a triangle with a base b=4m and height h=2m, how much is its area?

4 m2

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If we have a circle with radius r=3m, how much is its area (considering π=3.14)?

28,26 m2

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Which name has a 2-dimensional figure with five sides?

Pentagon

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Does an oval have all of its points equally distances to the center of the figure?

Yes

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What are perpendicular lines?

Perpendicular lines are lines that intersect at right angles.

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What angle do perpendicular lines intersect at?

Right angles (90°)

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How can you determine if two lines are perpendicular using a protractor?

By measuring the angle between the two lines. If the angle is 90° then the lines are perpendicular.

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