Question

Explaination needed

Class 9

Q.1: What are the five postulates of Euclid’s Geometry?

Q.2: If a point C lies between two points A and B such that AC = BC, then prove that AC =1/2 AB. Explain by drawing the figure.

Q.3: If in Q.2, point C is called a mid-point of line segment AB. Prove that every line segment has one and only one mid-point.

Q.4: In the given figure, if AC = BD, then prove that AB = CD.

Q.5: Does Euclid’s fifth postulate imply the existence of parallel lines? Explain.

Q.6: It is known that x + y = 10 and that x = z. Show that z + y = 10.

Note : Explaination needed

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Answer 1

${ \large{ \sf{ \underbrace{\underline{\bigstar \: Answer\:1}}}}}$

$\sf\color{red}{1}$To draw a straight line from any point to any point.

$\sf\color{red}{2}$ To produce a finite straight line continuously in a straight line.

$\sf\color{red}{3}$To describe a circle with any center and distance.

$\sf\color{red}{4}$That all right angles are equal to one another.

$\sf\color{red}{5}$ That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles,

the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

${ \large{ \sf{ \underbrace{\underline{\bigstar \: Answer\:2}}}}}$

${ \small{ \boxed{ \red{ \underline{ \bf \:Given\: that}}}}}$

$\sf\color{red}{C}$ is the lies between both points $\sf\color{red}{A\:and\:B}$

${ \small{ \boxed{ \red{ \underline{ \bf \:And}}}}}$

$\sf{AC= BC}$

Add AC both side we get

$\sf{AC + AC= BC + AC}$

According to definition of Euclid , if equals are added to equals, whole will equal

Here, $\sf\color{red}{(BC + AC)}$ coincides with $\sf\color{red}{AB}$

It is known that things which coincide with one another are equal to one another.

$\bold\pink{\fbox{\sf{2AC\: + \:BC}}}$

$\tt\color{orchid}{Divide\: by\: 2\: we\: get}$

$\bold\blue{\fbox{\sf{AC\:=\: AB/2}}}$

$\therefore\textbf\color{gold}{Hence proved}$

${ \large{ \sf{ \underbrace{\underline{\bigstar \: Answer\:3}}}}}$

Let $\tt\color{orchid}{C\: and\: D}$ are two midpoints of the line segments $\tt\color{orchid}{AB}$

$\small\textbf{According to Euclid’s axioms 4}$

$\sf{AC = BC...(1)}$

$\small\textbf\color{red}{D is also a mid point so that}$

$\sf{AD= DB … (2)}$

$\small\textbf\color{red}{We have AB = AB … (3)}$

$\large\textbf{And we know }$

AB = AC + CB

AB = AD + DB

From equation (iii)

AC + CB = AD + DB

From equation

(1) and (2)plug the value of BC and DB we get

AC +AC= AD +AD

2AC= 2AD

Divide by 2 we get

AC = AD

Both points are on same line so both points will superimpose and D and C are exactly at the same place

Hence midpoint of the lines segment is always unique

${ \large{ \sf{ \underbrace{\underline{\bigstar \: Answer\:4}}}}}$

From the above figure we get that

AC = AB + BC

BD = BC + CD

It is given that AC = BD

Plug the value of AC we get

AB + BC= BC + CD … (1)

According to Euclid’s axiom,when equals are subtracted from equals, the remainders

are also equal.

Subtracting BC from both side in equation (1),

We get

AB + BC − BC= BC + CD − BC

AB= CD

Hence proved

${ \large{ \sf{ \underbrace{\underline{\bigstar \: Answer\:5}}}}}$

$\huge\bold{Yes.}$

Euclid’s fifth postulate is imply for parallelism of lines because if a straight line l falls on two straight lines m and n such that sum of the interior angles on one side of l is two right angles, then by Euclid’s fifth postulate the line will not meet on this side of l.

${ \large{ \sf{ \underbrace{\underline{\bigstar \: Answer\:6}}}}}$

$\bold{We\: have}$

$\bold\blue{\fbox{\sf{x+y=10\: (i)}}}$

$\bold{And,}$

$\bold\blue{\fbox{\sf{x=z\: (ii)}}}$

$\bold{Applying \:Euclid’s\: axiom,}$

$\small\color{purple}{“if\: equals\: are \:added\: to\: equals,\:the\: wholes\: are \:equal”}$

${ \small{ \boxed{ \red{ \underline{ \bf \:We\: get,}}}}}$

$\bold{From\: Eq.\: (i) \:and\: (ii)}$

$\sf{x+y=z+y .\:(iii)}$

$\sf\color{violet}{From \:Eqs. \:(i) \:and\: (iii)}$

$\bold\pink{\fbox{\sf{z+y=10}}}$

Thankyou :)

Refer the attachments for better understanding :)

Answer 2

Step-by-step explanation:

Q-1:-

The five postulates of Euclid’s Geometry:-

1.To draw a straight line from any point to any point.

2.A terminated line can be produced indefinitely.

3.To describe a circle with any radius

4.That all right angles equal to one another.

5. If a straight line falling on two straight lines makes the interior angles in the same side of it together is less than two right angles then the two straight lines if produced indefinitely meet on that side on which the sum of the angles is less than two right angles.

Q-2:-

A____________C___________B

Let AC be a given line

Then we have AB = AC + BC

Given that AC = BC

now AB = AC+AC

=>AB = 2AC

=> AC = AB/2

Q:3-

If point C is called a mid-point of line segment

then AC= BC

and AC = BC = AB/2

=>2AC = 2BC = AB

It has only one mid point in a line segment.

Q-5:-

It satisfies the Parallel lines also

If two Parallel lines falls on a line then the interior angles on the same side to the third line(transversal) are supplementary that is 180°

or two right angles.

Q-6:-

Given that.

x + y = 10 -------(1)

and x = z.

Put x = z in the above equation then

=> z+y = 10

Hence, Proved