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The Elements of Geometry

    Introduction

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    Euclid’S Elements of Geometry

    Exercise 3.1

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The Elements of Geometry > Exercise 3.1

Exercise 3.1

1. Answer the following:

(1)

1) how many dimensions does a solid have ?

A) A solid shape have dimensions namely , and .

(2)

2)how many books are there in euclid's elements?

A) books

(3)

3)write the number of faces of a cube and a cuboid?

A)There are faces in cube and cuboid

(4)

4) what is the sum of the interior angles of a triangle ?

A)Sum of interior angle of triangle is degree

(5)

5) write three undefined terms of geometry.

A)These words are , and , and are referred to as the "three undefined terms of geometry".

2) State whether the following statements are true or false? Also give reasons for your answers.

a) Only one line can pass through a given point.

because, we can draw infinite number of lines through a given point as shown below.

b) All right angles are equal.

A right angle is always 90 degrees, regardless of where it is drawn. Since all right angles measure 90 degrees, they are equal.

c) Circles with same radii are equal.

Two circles are considered equal if they have the same radius because their size and shape are identical.

d) A line segment can be extended on its both sides endlessly to get a straight line.

A line segment has two endpoints, but when extended infinitely on both sides, it becomes a straight line, which has no endpoints.

e) From the figure, AB > AC

3) In the figure given below, show that length AH > AB + BC + CD

AH = + + + + + +

= AB + BC + CD is a of AH

According to Euclid's axiom the whole thing is greater then a part

Therefore AH AB + BC + CD

4) If a point Q lies between two points P and R such that PQ = QR, prove that PQ = 12 PR Q lies between P and R

Thus PR = PQ +

Now, we are given PQ =

Put QR =

PR = PQ + PQ

PR = PQ

1/2 = PQ

PQ = 12 PR

5) Draw an equilateral triangle whose sides are 5.2 cm. each

6. What is a conjecture? Give an example of it.

Solution

A conjecture is a statement that is believed to be true based on observations. A conjecture might not have any proof.

For example If I write a set of even numbers from 2 to 10 and ask someone to tell me the next number, they will likely say .

2, 4, 6, 8, ,

This is a conjecture based on observation, which is believed to be , but it does not have any proof.

7. Mark two points P and Q. Draw a line through P and Q. Now how many lines which are parallel to PQ, can you draw?

8. In the adjacent figure, a line n falls on lines l and m such that the sum of the interior angles 1 and 2 is less than 180°, then what can you say about lines l and m.

In the given figure, l, m and are three lines each other

Given ∠1 + ∠2 < ° or Using Euclids Postulate 5

We can conclude that line l and m are not to each other. Since ∠1 + ∠2 °.From Euclid Postulate 5. We can say that Line l, m will eventually on the right of the line

9. In the adjacent figure, if ∠1 = ∠3,∠2 = ∠4 and ∠3 = ∠4, write the relation between ∠1 and ∠2 using an Euclid’s postulate.

Given ∠1 = ∠,∠2 = ∠ and ∠3 = ∠

Therefore Euclids 1st postulates states that things that are equal to same things are also to one another

Therefore ∠1 = ∠

In the adjacent figure, we have BX = 1/2 AB, BY = 1/2 BC and AB = BC. Show that BX = BY

Given BX = 12 , BY = 12 , AB =

Euclids postulate states that, things which are halves of the same thing are to one another

Here AB and has same length

and BX and BY are of AB and BC respectively

Hence BX =