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Euclid

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What is Euclid?

Euclid is a WHO. Euclid of Alexandria was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I - approximately 330 B.C. to 260 B.C.

Euclid

Many scholars worked and taught in Alexandria, and that's where Euclid wrote The Elements. The Elements is divided into thirteen books which cover plane geometry, arithmetic and number theory, irrational numbers, and solid geometry.

Euclid is a short geometry primer that includes a few of the most basic and well known principles of plane geometry with emphasis on practical use, enjoyment, and beauty.

The topics are presented in simple, clear, visual, informal terms; no attempt is made to be "mathematically rigorous." Each topic includes a self quiz and a final exam is offered at the end of "the course".

Euclid is an apps project of Ms Math Master that includes elementary lesson drills and simple puzzles.

Copyright © iXB 2015 iXoraBrown Web Design Lakewood Ranch, Florida. All rights reserved. Euclid v1.0.0

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Contents

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Lines and Angles

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Lines and Parts

Points and Lines

Point and line are undefined ideas of Euclidean Geometry. Points have position; lines are infinite in both directions; segments and rays are parts of lines, the first being definite and the second extending infinitely from a single point. An angle is formed by two rays.

Sun's Rays

The Sun Safety Alliance [www.sunsafetyalliance.org] is dedicated to reducing the incidence of skin cancer from the sun's rays and creating awareness of this important health issue.

Angle Classification

Types of Angles

An angle is measured in degrees which represents the amount of rotation between the rays. A full rotation is 360∘. It takes the earth about 360 days to make one trip around the sun.

Vertical Angles

Vertical Angles

When two lines intersect, four angles are formed. The opposite angles are called "vertical angles" and VERTICAL ANGLES ARE EQUAL. If two angles "sit" on a straight line they are called a "linear pair" and their sum is 180∘. The sum of all the angles around a point is 360∘.

Perpendicular Lines

Perpendicular Lines

When two lines intersect in right angles, they are called "perpendicular lines." Notice that the vertical angles are 90∘ and still equal and all the pairs sitting on a line add to 180∘.

Self Quiz

  1. Classify the angle.
    Right Angle
  2. If ∠a is 60∘, what is the measure of ∠b?
    Supplementary Angles
  3. What is the sum of ∠a+∠b+∠c+∠d+∠e?
    Five Angles Sitting on a Line

Answers

  1. right angle
  2. 120∘
  3. 360∘
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Parallel Lines

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Parallel Illusion

Parallels and Transversals

Parallel and Transversal

Parallel lines are lines in a plane that do not intersect even when extended. A third line that crosses parallels is called a transversal. This divides the plane into three areas - above and below the parallels [exterior] and between them [interior.]

Parallel and Transversal

Transversals and Angles

Transversal and Angles

A transversal crossing parallel lines creates 8 angles which have "pair" names: alternate interior angles - ∠4, ∠6 and ∠3, ∠5; alternate exterior angles - ∠1, ∠7 and ∠2, ∠8; corresponding angles - ∠1, ∠5 and ∠2, ∠6 and ∠3, ∠7 and ∠4, ∠8.

ALTERNATE INTERIOR ANGLES ARE EQUAL; ALTERNATE EXTERIOR ANGLES ARE EQUAL; CORRESPONDING ANGLES ARE EQUAL

Transversal and Equal Angle Pairs

Other Parallelisms

Step Ladder

LINES PARALLEL TO A THIRD LINE ARE PARALLEL; LINES PERPENDICULAR TO A THIRD LINE ARE PARALLEL.
In a step ladder, each step is parallel to the step immediately below it and the bottom step is parallel to the floor. Yardlines on a football field are parallel and perpendicular to the sidelines.

Football Field

The Parallel Postulate

Parallel Postulate

Euclid correctly postulated [meaning it cannot be proved] that only one line can pass through a point and be parallel to a given line. Contrary assumptions lead to alternate valid and consistent geometries.

Euclidean Geometry Elliptic Geometry Hyperbolic Geometry

Self Quiz

  1. Name the angle pair.
    Corresponding Angles
  2. True or False? Equal Alternate Interior Angles
  3. What geometry is suggested by the surface of a chip?

    Potato Chips

Answers

  1. corresponding angles
  2. true
  3. hyperbolic
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Triangles

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Triangles

Triangle Classification

Triangles Types

A triangle [three sides-three angles] has two names; one identifies the nature of its sides and the other describes its angles.

A triangle with two equal sides is isosceles and no equal side is scalene. If all three angles are less than 90∘, it's acute. If one angle is 90∘, it's a right triangle and if one angle is greater than 90∘, it's obtuse. A triangle cannot contain more than one right or one obtuse angle.

If all three sides are equal, it's equilateral. All equilateral triangles are acute and equiangular. These properties produce seven general triangle classifications.

Angle Sum Theorem

Triangle Angles Sum is 180 Degrees

THE SUM OF THE THREE ANGLES OF EVERY TRIANGLE IS 180∘.

Ready for a simple proof? - see following figure.

Only one line passes through A parallel to BC. [Euclid's Parallel Postulate]
Therefore, the yellow angles are equal and the green angles are equal. [Alternate interior angles created by parallels and a transversal]
But [angles on the line], ∠y + ∠z + ∠x = 180∘. [Angles that sit on a line add to 180∘]
Therefore [inside the triangle], ∠y + ∠z + ∠x = 180∘ [The angles of the triangle are the same as those on the line]

Triangle Angles Sum is 180 Degrees

Since the angle sum is 180∘, each angle in the equilateral-equiangular triangle must be 60∘.

Equilateral Triangle

Pons Asinorum

IF TWO SIDES OF A TRIANGLE ARE EQUAL, THE ANGLES OPPOSITE THEM ARE EQUAL or
THE BASE ANGLES OF AN ISOSCELES TRIANGLE ARE EQUAL.
If AB = AC, then ∠B = ∠C.

Pons Asinorum

Pons Asinorum is latin for "bridge of asses". Lengend claims that the proof of this theorem "in the old days" required a diagram that resembled a bridge, and that the proof was so difficult that only the scholars could understand it and "pass over the bridge", leaving the "asses" behind.

Triangles are the strongest shape because of they can bear weight without geometric distortion.


Bridge Gable

Pascal's Triangle

Pascal's Triangle

The numbers in Pascal's triangle provide the coefficients in the binomial expansion.

Binomial Expansion

Exterior Angle Theorem

Exterior Angle

An exterior angle of a triangle equals the sum of the two remote angles. Can you see why?

  1. A + B = 180, why?
  2. [C + D] + B = 180, why?
  3. therefore, A + B = [C + D] + B, why?
  4. therefore, A = C + D, why?

Self Quiz

  1. Provide the answers to "why?" in the previous section.
  2. Classify a triangle with two equal sides.
  3. Question 3

Answers

    1. angles on a line
    2. triangle angle sum
    3. both quantities equal 180
    4. subtract B from both sides
  1. Isosceles
  2. 50∘
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Pythagorean Theorem

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Pythagorean Theorem Diagram

The Pythagorean Theorem

The Pythagorean Theorem is probably the most well known geometric theorem. It says that in a right triangle, the sum of the squares of the sides, called legs, will equal the square of the hypotenuse [the slant side.] A² + B² = C².

Pythagorean Theorem Diagram Example: if the legs of a right triangle are 3 and 4, then 3x3=9, and 4x4=16, and 9+16=25, and so the hypotenuse is 5.

Archeologists use the Pythagorean Theorem in field excavations. When they start a dig, they place a rectangular grid over the site surface. To lay out an accurate grid system, archeologists use the Theorem, X² + Y² = D². After deciding how long the baselines [X- and Y- axis] should be, the proper length of the diagonal is calculated using the Pythagorean theorem to make sure the quadrant is a rectangle. Corner stakes will then be placed to mark the accurate location of the site.

Pythagoras Who?

Pythagoras was an influential Greek mathematician and philosopher, best known for the theory to which he gave his name.

Pythagoras

Very little is known about Pythagoras's life. He is thought to have been born on the Greek island of Samos approximately 569 BC.

He travelled widely in his youth, visiting Egypt and Persia. He settled in the city of Crotone in southern Italy. There he began teaching and soon had a clutch of students who lived a structured life of study and exercise, inspired by a philosophy based around mathematics. This circle came to be known as the Pythagoreans.

Pythagoreans believed that everything could be reduced to numbers: the whole universe had been built using mathematics. The truth behind the everyday reality we experience lay in numbers.

Find the Third Side

Right Triangle
Given A=6 and B=4, find C.
A² + B² = C²
6² + 4² = C²
36 + 16 = C²
C² = 52
C = √ 52 = 7.2

Given A=8 and C=12, find B.
A² + B² = C²
8² + B² = 12²
B² = 12² - 8²
B² = 144 - 64
B = √ 80 = 8.9

Visual Proof

Visual Proof
Study the grid. Find the right triangle outlined in red. See the squares built on the sides - yellow, blue, and green. Count the number of triangles in each square - yellow (4), blue (4) and green (8). Therefore, the green square equals the sum of the yellow square + blue square.

Self Quiz

  1. What is the name of a right triangle's slant side?

  2. Is this triangle a right triangle? Why or why not?
    Right Triangle
  3. Find the missing side.
    Right Triangle

Answers

  1. hypotenuse
  2. yes, 25 + 144 = 169
  3. 41
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The Rectangle

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Rectangle

Definition

Rectangle

Geometric figures have several classifications, one is based on the number of sides it has. A figure with four sides [four angles] is called a quadrilateral. The rectangle is a quadrilateral with four right angles and opposite sides equal. The sides are usually referred to as length and width or base and height. The rectangle is probably the most comman and useful quadrilateral.

Perimeter

Rectangle

The distance around any geometric polygon [figure with straight sides] is the sum of its sides.

For a rectangle that would be:
P = L + W + L + W or
P = 2L + 2W.

The perimeter of the rectangle below is:

Rectangle P = 2L + 2W
P = 2(8) + 2(4)
P = 16 + 8
P = 24 cm

We would be using perimeter to frame a painting or buy fencing or calculate how far it is to walk around the gym. Perimeter is 1-dimensional and is measured in linear units such as inches, feet or meters.

Yard with Fence

Area

The area of a rectangle is the number of square units inside the rectangle.

Rectangle

To calculate the number of squares, multiply the the number of columns [Length or Base] by the number of rows [Width or Height.]

A = LW
A = 5 X 3
A = 15 sq

Area is 2-dimensional: it has a length and a width. Area is measured in square units such as square inches, square feet or square meters.

We would be using area to calculate how much carpet to buy for a room or how much wall paper or paint is needed for the walls.

The Golden Ratio

Golden Rectangle

The golden rectangle has the property that when a square is removed a smaller rectangle of the same shape remains. Thus a smaller square can be removed, and so on, with a spiral pattern resulting.

The golden rectangle is a rectangle whose side lengths are in the golden ratio, which is approximately 1.618 and represented by the Greek letter phi. The common 3 X 5 card comes pretty close to this ratio.

Greek letter phi

Some artists and architects believe the Golden Ratio makes the most pleasing and beautiful shape. Many buildings and artworks have the Golden Ratio in them, such as the Parthenon in Greece.


The Parthenon

Self Quiz

  1. A 2-meter wide path is constructed around the outside of a rectangular garden that is 50 m by 100 m.

    Rectangle
    What is the area of the garden?
  2. What is the perimeter of the garden?
  3. What is the area of the pathway?

Answers

  1. 5000 sq m
  2. 300 m
  3. 616 sq m
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Final Exam

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The Final Exam is multiple choice. There are 10 questions with four choices each. Click your choice for each question, then click to get your score. Good Luck.


Final Exam