\(\angle \)2 = \(\angle \)4          ,,,,,,,(1) | Given
\(\angle \)1= \(\angle \)4          ........(2) | Given
\(\angle \)3 =\(\angle \)2         .......(3) | Given
From (1), \(\angle \)4 = \(\angle \)2           ......(4)
From (3) and (4),
\(\angle \)3 = \(\angle \)4               .....(5)
I Things which are equal to the same thing are equal to one another
From (2) and (4),
\(\angle \)1 = \(\angle \)3
I Things which are equal to the same thing are equal to one another

Yes, If a straight line 'I' falls on two lines 'm' and 'n' such that sum of the interior angles on one side of I is two right angles, then by Euclid's fifth postulate, the lines will not meet on this side of I. Also we know that the sum of the interior angles on the other side of the line I will be two right angles too. Thus, they will not meet on the other side also.

∴ The lines 'rn' and 'n' never meet, i.e., they are parallel.

AB = CD(Given)
\(\Rightarrow\)AB + BC = BC + CD
\(\Rightarrow\)AC = BD
Euclid's axiom used: If equals are added to equals, the wholes are equal.

Things which are equal to the same thing are equal to one another.

(i) False
(ii) False
(iii) True
(iv) False

Euclid used the term postulate for the assumptions that were specific to geometry and otherwise called axioms.

Given, ㄥ2 = ㄥ1 and ㄥ3 = ㄥ4    ......(i)
Now,  ㄥ2 = ㄥ1
ㄥ2 + ㄥ3 = ㄥ1 + ㄥ3    [if equals are added to equals, then the wholes are equal ]
⇒ ㄥ2 + ㄥ3 = ㄥ2 + ㄥ3      [from Eq. (i)]
⇒ ㄥ2 + ㄥ3 + 30° = ㄥ2 + ㄥ3 + 30° [if equals are added to equals, then the wholes are equal]
⇒ ㄥ2 + ㄥ3 + 30° =ㄥ2 +ㄥ3 + θ ⇒ 30° = θ
[since, if equals are subtracted from equals, then the remainders are equal]
ஃ  θ = 30°

Given, QX = \(\frac{1}{2}\) XY       ...(i)
PX=\(\frac{1}{2}\)XZ        ....(ii)
and    QX= PX    ......(iii)
From Eqs. (i), (ii) and (iii), we get
\(\frac{1}{2}\)XY =\(\frac{1}{2}\)XZ
ஃ  XY = XZ     [by axiom 6]

Given,   AB = BC
and   BX = BY
On subtracting Eq. (ii) from Eq. (i), we get
AB - BX = BC - BY
⇒ AX = CY

(i) \(\overleftrightarrow { EF } ,\overleftrightarrow { GH } ,R\)
(ii) \(\overleftrightarrow { AB } ,\overleftrightarrow { EF } ,\overleftrightarrow { GH } ,R\)
(iii)  \(\overrightarrow { RB } ,\overrightarrow { RH } ,\overrightarrow { RG } \)
(iv)  \(\overline { RQ } ,\overline { RP } \)

In the given figure, AC coincides with AB + BC.
Also, Euclid's axiom 4 says that things which coincide with one another, are equal to one another. So, it can be deduced that
AB + BC = AC

We know that, two angles are said to be complement of each other, if their sum is 90°. Let the angle be x, then its complement will be (90 ° - x).
According to statement, we get x = 25° + (90°-x)
⇒  x = 115 ° - x
⇒  2x = 115 ° ⇒ x = \(\frac{115^\circ}{2}=57\frac{1^{\circ}}{2}\)

Euclid's axiom 2

In the given figure,
PR = PQ   [radii ofthe same circle]    ... (i)
and QP = QR    [radii of the same circle] ... (ii) 
From Eqs. (i) and (ii), PR = QR   [by axiom 1] ... (iii) 
From Eqs. (i), (ii) and (iii), PQ = QR = PR   [by axiom 1]
Hence proved.

Given Three lines l, m and n, such that l || n and m II n.
To prove    l II m
Proof If possible, let l is not parallel to m, then I and m intersect in a unique point, say P. Thus, through a point P outside n, there are two lines I and m, both parallel to n. This is contradiction to the parallel line axiom. So, our supposition is wrong.
ஃ  l || m 
Hence proved.

(i) equal to one another
(ii) wholes are equal
(iii) the remainders are equal
(iv) equal to one another
(v) part
(vi) equal to one another
(vii) equal to one another
(viii) products are equal
(ix) quotients are equal
(x) 'Trichotomy law'

Yes, we need to have an idea about the terms, point, line, ray, angle, plane, circle and quadrilateral, etc. before defining the required terms.
Point: A small dot made by a sharp pencil on the surface of a paper gives an idea about a point. It has no dimensions. It has only a position.
Line: A line is an idea that it should be straight and that it should extend indefinitely in both the directions. It has no end points and has no definite length.
Note: In geometry a line means "The line in its totality and not a portion of it. Whereas a physical example of a perfect line is not possible. A line extends indefinitely in both directions, so we cannot draw or show it wholly on a paper. That is why we mark arrow heads on its both ends, indicating that it extends indefinitely in both directions.
Ray: A part of line which has only one end point and extends indefinitely in one direction. A ray has no definite length.
Angle: Two rays having a common end-point form an angle.
Plane: Plane is a surface such that every point of the line joining any two points on it, lies on it.
Circle: A circle is the set of all those points in a given plane which are equidistant from a fixed point is the same plane. The fixed point is called the centre of the circle.
Quadrilateral: A closed figure made of four line segments is called a quadrilateral.
Definitions of the required terms are given below:
(i) Parallel Lines: Two lines 'I' and 'm' in a plane are said to be parallel, if they have no common point and we write them as l ‖ m.

Note: The distance between two parallel lines always remains the same.
(ii) Perpendicular Lines: Two lines 'p ' and 'q' lying in the same plane are said to be perpendicular if they form a right angle and we write them as p 丄 q.

(iii) Line Segment: A line segment is a part of line having a definite length. It has two end-points. In the figure a line segment is shown having end points 'A' and 'B'. It is written \(\overline{\mathrm{AB}}\) as or \(\overline{\mathrm{BA}}\).

(iv) Radius of a circle: The distance horn the centre to a point on the circle is called the radius of the circle. In the figure, P is centre and Q is a point on the circle, then PQ is the radius.

(v) Square: A quadrilateral in which all the four angles are right angles and all the four sides are equal is called a square. In the figure PQRS is a square.

(i) False
[If we mark a point O on the surface of a paper and using pencil and ruler, we can draw indefinite number of straight lines passing through O.]
(ii) False
[∵ In the following figure, there are many straight lines passing through 'P'. There are many lines, passing through 'Q'. But there is one and only one line which is passing through 'P' as well as 'Q'.]

(iii) True
[∵ The postulate 2 says that "A terminated line can be produced indefinitely."]
(iv) True
[∵ Superimposing the region of one circle on the other, we find them coinciding. So, their centres and boundaries coincide. Thus, their radii will coincide.]
(v) True
[∵ According to Euclid's axiom, things which are equal to the same thing are equal to one another.]

Given, AC = BD       ...(i)
From figure, it is clear that
AC = AB + BC and BD = BC + CD.
On putting these values in Eq. (i), we get
AB + BC = BC + CD
On subtracting BC from both sides, we get
AB + BC - BC = BC + CD - BC
⇒ AB = CD    [by axiom 3]
Hence proved

(i) False, because from a single point, infinite number of lines can pass.

(ii) False, because from two distinct points, only one straight line can pass. [by postulate axiom]

(iii) True, it is Euclid's postulate 2.

(iv) True, because radii of congruent (equal) circles are always equal. In other words, if we superimpose the region bounded by one circle on the other circle, then they coincide. Then, their centres and boundaries also a>incide. Therefore, their radii will be same.

(v) True,given that AB = PQ (i) and PQ = XY (ii)
From Eqs. (i) and (ii), AB = XY