Geometry is a branch of mathematics concerned with shapes, sizes, positions, and properties of space. It arose independently in early cultures and emerged in ancient Greece where Euclid formalized it in his influential Elements text around 300 BC. Elements defined geometry axiomatically and influenced mathematics for centuries. It included proofs of theorems like two triangles being congruent if they share two equal angles and one equal side. Euclid's work defined much of the rules and language of geometry still used today.

2.  Geometry Is a branch of mathematicsconcerned with
questions of shape ,size, relative position of figures,and the
propertiesofspace.Geometryaroseindependentlyinanumberof
earlyculturesasabodyofpracticalknowledgeconcerninglengths,
areas,andvolumes,withelementsofformalmathematicalscience
emergingintheWestasearlyasThales.Bythe3rdcenturyBC,
geometrywasputintoanaxiomaticformbyEuclid,whose
treatment—Euclideangeometry—setastandardformany
centuriestofollow.

3.  Geometry is everywhere. Angles, shapes, lines, line
segments, curves, and other aspects of geometry are every
single place you look, even on this page. Letters
themselves are constructed of lines, line segments, and
curves! Take a minute and look around the room you are
in, take note of the curves, angles, lines and other aspects
which create your environment. Notice that some are two-
dimensional while others are three-dimensional. These
man-made geometrical aspects please us in an aesthetic
way

5. He was born around 624 BC and died around 547 BC. Yes that was a
long time ago, but he made some very major contributions to the field of
geometry. In fact, some consider him the first mathematician. On a visit to
Egypt, he was able to calculate the height of a pyramid. He is credited for
making five notable contributions to the field of geometry, one of which is
named after him.
The first is that the diameter of a circle bisects, or cuts, the circle in half. The
second is that the base angles of an isosceles triangle are equal to each other.
The third is when you have two straight lines intersecting each other, the
opposite or vertical angles are equal to each other. The fourth notable
contribution states that when two triangles have two equal angles and one
equal side, then they are congruent, or equal, to each other. The fifth is
called Thales' Theorem. It states that an angle that is inscribed or drawn
inside a half-circle or semicircle will be a right angle. These five
contributions are credited to Thales because he provided the first written
proof of these theorems.

6.  Pythagoras was born in approximately 569 B.C. His father was Mnesarchus and his
mother was Pythais. Pythagoras spent his early years in Samos. There is little known
about his child hood and all physical descriptions of Pythagoras are said to be
fictitious except for the vivid birthmark on his thigh. It is believed that he had two
brothers and some believe there were three. Pythagoras was extremely well
educated. There were three philosophers that influenced him while he was young.
One of the most important of these man was a man named Pherekydes. The
philosopher that introduced Pythagoras to mathematical ideas was Thales , who
lived in Miletus. It was because of Thales that Pythagoras became interested in math,
astronomy and cosmology. Pythagoras was interested in all principles of
mathematics. He was intrigued by the concept of numbers and basically numbers
themselves. Pythagoras had a theory that all relations were able to be reduced in to
some form of number. Pythagoras also derived a theory on ratios and scales being
produced with the sound of vibrating strings. He made large contributions to music
theory.He studied many different types of numbers ,for example triangles, odd
numbers and perfect squares. He believed that each number had its own personality
traits and were all different and unique. For example ten is the best number because
it contains four consecutive integers (1+2+3+4=10)
 Pythagoras’ greatest contribution to the mathematical society of today is Pythagoras
theorem. It is believed that the theory of a2+b2=c2 was known to the Babylonians
1000 years before Pythagoras but it was he who was able to prove it.

7. The following are a list of theorems
contributed by Pythagoras :-
1.The sum of the angle is a triangles equal to two right
angles.
2.The Pythagorean theorem.
3.Construction figures of a given area and geometrical
algebra.
4.The discovery of irrationals
5.The five regular solids.
6. Pythagoras taught that the earth was a sphere in the
center of the universe.
Pythagoras life came to an end in approximately 475
B.C. Many of his contributions are still used in everyday
math of today's generation.

8.  Euclid was an ancient Greek mathematician who lived in the Greek
city of Alexandria in Egypt during the 3rd century BCE. Euclid is
often referred to as the 'father of geometry' and his book Elements
was used well into the 20th century as the standard textbook for
teaching geometry.
 The most famous work by Euclid is the 13-volume set called
Elements. This collection is a combination of Euclid's own work
and the first compilation of important mathematical formulas by
other mathematicians into a single, organized format. Thus, it
made mathematical learning much more accessible. Elements also
contains a series of mathematical proofs, or explanations of
equations that will always be true, which became the foundation
for Western math.
 Euclid's Elements contains several axioms, or foundational
premises so evident they must be true, about geometry. These
include such basic principles as when two non-parallel lines will
meet, that opposite angles of an isosceles triangle are equal, and
how to find the area of a right triangle. Elements also contains
geometric interpretations of algebra, such as ideas like
a(b+c)=ab+ac. Most important among these is Euclid's algorithm, a
formula for devising the greatest common factor of two integers.

10.  His third axiom would then be “if x=y, and if a=b, then x – a = y – b.”
 Euclid’s Fourth Axiom: Coincidental Equality
 The fourth axiom seems to be the most obvious reference to geometry. If
two shapes “coincide,” then one fills out the exact shape and volume of
the second.
 Simple cases include angles that are equal, straight line segments of the
same length, and triangles of the same size and shape.
 Consider drawing a triangle, and then constructing a second triangle in a
way that copies the angles and lengths from the first triangle. Then, cut
out the second triangle and lay it over the first. If these triangles precisely
overlap, then they “coincide,” and are equal to one another.
 Euclid’s Fifth Axiom: Part of the Whole
 Euclid’s fifth axiom states that “x + a > x.” To a modern mathematician,
this would not be true if ‘a’ had the value “zero,” or if ‘a’ were a negative
number. For example, if ‘a’ were a geometric shape with no area, such as a
line that has no thickness, then adding a line segment “beside” the edge of
a square, ‘x’, would not increase the area of the square.
 A more complete formula to cover our modern sensibilities would be “if a
> zero, then x+a > x.”

11.  Euclid’s First Postulate: a Line Segment
between Points
Euclid’s first postulate states that any two points can
be joined by a straight line segment. It does not say that
there is only one such line; it merely says that a straight
line can be drawn between any two points.
 Euclid’s Second Postulate: Extend a Straight Line
Euclid’s second postulate allows that line segment to be
extended farther in that same direction, so that it can
reach any required distance. This could result in an
infinitely long line.

12.  Euclid’s Third Postulate is
Central to Circles
The third postulate starts with an arbitrary line
segment, and an arbitrary point, which is not
necessarily on the line segment. First, use the
compass to note the end points of the line segment,
then, put the sharp spike of the compass on the
arbitrary point, and finally, draw the circle with the
same radius as the line segment.

13.  Euclid’s Fourth Postulate: All
Right Angles are Equal to one
another
 Euclid was probably thinking of right angles as made by constructing
one line perpendicular to another. Any two such right angles are
“equal” to one another.
 Euclid’s fourth postulate states that, “if x and y are both right
angles, then x=y.”
 This may be more profound if the angles are oriented differently:
opening to the left or right, up or down, or towards some other
direction.
 Euclid did not measure angles in degrees or radians, and he did not
use a protractor. Instead, he usually discusses “how many angles in a
diagram add up to some number of right angles.” For example, in
Book I, Proposition 13 basically states that when a straight line
“stands on” another straight base line, the sum of the two angles on

14.  Euclid’s Fifth Postulate: the
Parallel Postulate
 A parallelogram demonstrates parallel lines
 Euclid’s fifth postulate is the longest, and is now
called the “parallel postulate.”
 The fifth postulate has been the subject of much
debate and labour over the centuries -can it be
proven from the other postulates and axioms?
Eventually mathematicians realized that the fifth
postulate defines plane geometry, the geometry for
a flat surface, and it cannot be derived from the
other Euclidean axioms.
 This postulate’s explanation needs diagrams.
Consider this paralellogram, with the interior angles on
the base marked in green. The sum of those two interior
angles is 180 degrees. The two blue slanted lines are
parallel; they will never meet even if extended infinitely far.

16. Answer:AC = CB (Given)
Also AC + AC = BC + AC. (Equals are added to equals)
∵ things which coincide with one another are equal to one
another.
∴ AC + BC coincides with AB
⇒ 2AC = AB
⇒ AC = ½AB.

17. Answer: Euclid's postulate 5 states, "The whole is greater than
the part." It is considered 'universal truth', because it holds true in
every field.
Consider the following cases:
Case I: Consider a group of numbers 15, 8, 4, 2, 1 such that 15 = 8 + 4
+ 2 + 1 and 15 is greater than any of its part (8, 4, 2, 1)
Case II: Consider a circle, consisting of six sectors (a, b, c, d, e and f).
The area of a circle as a whole is greater than that of any sector (its
part).

18. Answer:
(i) For every line L and for every point P not lying on L, there
exists a unique line M passing through P and parallel to L.
If we draw perpendicular both from L and M i.e. AB and XY.
The perpendicular distances are equal i.e. AB = XY.
(ii) Two distinct intersecting lines cannot be parallel to the
same line.
Q & A from Exam. papers and other books

19. Answer: Let there be two such mid points C and D. Then
using above said theorem (see answer 4), we can prove
AC = ½AB ... (I)
and AD = ½AB ... (II)
From I and II, we have
∴ AC = AD = ½AB
AC and AD can be equal only if D coincides with C. Therefore,
C is the unique
mid-point.

20. YASH – 43
YASH LAKRA – 45
ADITAY HOODA – 03
TUSHAR CHAUHAN – 40
AJAY CHAUHAN – 04