Geometry

Some Definitions, Axioms And Postulates:
A solid has three dimensions, a surface has two, a line has one and a point has none. Euclid summarized these statements as definitions. Some of them are:
(i) A point is that which has no parts
(ii) A line is a breadth less length
(iii) The end of a line are points
(iv) A straight line is a line which lies evenly with the points on itself.
(v) A surface is that which has length and breadth only.
(vi) A plane surface is a surface which lies evenly with straight lines on itself.

Basic Geometrical Concepts:
Axioms: The basic facts which are taken for granted, without proof, are called axioms.

(i) Things which are equal to the same thing are equal to one another.
If a = b, b = c => a = c

(ii) The equals are added to equals, the whole are equal
a + c = b + c

(iii) If equals are subtracted from equals, the remainders are equal
If a = b => a – c = b – c

(iv) Things which are double of the same things are equal to one another.
If a = b => 2a = 2b

(v) Things which are halves of the same things are equal to one another.
If a = b => a/2 = b/2

(vi) Things which are greater than the same thing are greater than one another.
If a > b, b > c => a > c

(vii) Things which coincide with one another are equal to one another.

(viii) The whole is greater than the part.

Postulates :

Postulate – 1 : A straight line may be drawn from any one point to any another point.


Postulate – 2 : A terminated line can be produced indefinitely


Postulate – 3 : A circle can be drawn with any centre and any radius.


where O is the centre of the circle and OA = (r) radius of the circle.

Postulate – 4 : All right angles are equal to one another


the measurement of Because each and every right angle is always 90o.

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



e.g. Line PQ falls on lines AB and CD such that the sum of interior angles ?1 + ?2 < 180o is on the left side of PQ. So, the lines AB and CD will eventually intersect on the left side of PQ.

Theorem: Two distinct lines cannot have more than one point in common given.


Two distinct intersecting lines cannot be parallel to the same line.


Euclid’s fifth postulate is very significant in the history of Mathematics. By implication, we can see that no intersection of lines will take place when the sum of the measures of the interior angles on the same side of the falling line is exactly 180o.

2. Lines And Angles

Basic Terms And Definitions:
I A line: When two or more than two points are joined end point, it is called aline. It is denoted by


II A line Segment: A part (or portion) of a line with two end points is called a line segment e.g. AB is a line segment and denoted by AB.



III A ray: A part of line with one end point is called a ray; e.g. PQ is a ray and denoted by


IV Collinear points : If three or more points lie on the same line, they are called collinear points. i.e. A, B, C, D, E and F are collinear points.


V Non-collinear points : If three or more points do not lie on the same line, they are called non-collinear points. i.e. P, Q, R, S, T, U and V are non-collinear points.


VI An Angle: When two rays originate from the same end point, an angle is formed ; e.g. ?AOB is an angle and OA and OB are called the arms of an angle ?AOB. The measurement of an angle is degree.


VII Vertex: The end point of the arms of an angle is called the vertex of an angle; e.g. O is the vertex of an angle ?AOB.


Kinds Of Angles:

(i) An acute angle: The angles between 0o and 90o are called acute angles. i.e. 0o < acute angle < 90o; e.g. ?AOB is an acute angle.


(ii) A right angle: A right angle is exactly equal to 90°, i.e., right angle = 90° e.g. ?POQ is 90° (a right angle)


(iii) An obtuse angle: An angle greater than 90o but less than 180o is called an obtuse angle, i.e. 90o < obtuse angle < 180o e.g. ?MOP is an obtuse angle.




(iv) Straight angle: A straight angle is equal to 180°, i.e. a straight angle is 180° or is 2 x 90° = 2 right angles; e.g. ?COD is a straight angle.


(v) Reflex angle: An angle which is greater than 180° but less than 360° is called a reflex angle, i.e., 180° < reflex angle < 360°; e.g. ?EOF is a reflex angle.


(vi) Complementary angles: Two angles whose sum is 90o are called complementary angles i.e., ?x + ?y = 90o
e.g. ?ABD + ?DBC = 90°


[Complementary angles]
40° + 50° = 90°
[Complementary angles]
60° + 30° = 90°
[Complementary angles]
70° + 20° =90°
[Complementary angles]
80° + 10° = 90°
[Complementary angles]
45° + 45° = 90°
[Complementary angles]

(vii) Supplementary angles: Two angles whose sum is 180o are called supplementary angles i.e., ?x + ?y = 180o
[Two right angles = 2× 90° = 180°]
e.g. ?ABD + ?DBC = 180°
[supplementary angles]
90° + 90° = 180°
[supplementary angles]
100° + 80° = 180°
[supplementary angles]
110° + 70° = 180°
[supplementary angles]
120° + 60° = 180° [supplementary angles]
130° + 50° = 180° [supplementary angles]
90° + 90° = 180° [supplementary angles]

(viii) Adjacent angles: If two angles have a common vertex and a common arm, they are called adjacent angles, i.e. ?ABD and ?DBC have common arm BD and also common vertex B, so, they are djacent angles. e.g. ?x and ?y are adjacent angles.


(ix) Linear pair of angles: If the non common arms QP and QR in the given figure, from a line, then the angles
?PQS and ?SQR are called linear pair of angles.


(x) Vertically opposite angles: When two lines intersect each other at a point, they make two pairs of vertically opposite angles such type of angles are also equals.
e.g. ?AOC = ?BOD [Vertically opposite angles]


?COB = ?AOD [Vertically opposite angles]

Intersecting Lines And Non-intersecting Lines:
(i) Intersecting Lines: If two lines intersect each other at any point, they are called intersecting lines.
e.g. AB and CD are intersecting lines because they intersect each other at a point O.


(ii) Non-intersecting (parallel) lines: If two lines never intersect each other and the distance between them is always equal (same), they are called non-intersecting (parallel) lines, i.e., parallel lines do not intersect even at infinity e.g. CD || MN


Theorem: If two lines intersect each other, then vertically opposite angles are equal.




Parallel Lines and a TRANSVERSAL:
If l || m and t is a transversal.


Then, (i) Corresponding angles:
?1 = ?5
?2 = ?6
?4 = ?8
?3 = ?7
(ii) Alternate interior angles:
?4 = ?6
?3 = ?5
(iii) Alternate exterior angles:
?1 = ?7
?2 = ?8
(iv) Interior angles on the same side of the transversal:
6X
?4 + ?5 = 180o and ?3 + ?6 = 180o





Corresponding Angles Axiom:
Axiom 1: If a transversal intersects two parallel lines, then each pair of corresponding angles is equal.
If l || m and t is a transversal, then corresponding angles:
?1 = ?5
?2 = ?6
?4 = ?8
?3 = ?7

Axiom 2: If a transversal intersects two lines such that a pair of corresponding angles is equals, then the two lines are parallel to each other.
If transversal PS intersects two lines AB and CD such that

?AQP = ?CRQ [Pair or corresponding angles]
or ?BQP = ?DRQ [Pair of corresponding angles]
then, AB || CD
Theorem: If a transversal intersects two parallel lines, then each pair of alternate interior angles is equal. If transversal PS intersects two parallel lines AB and CD respectively,
Then, ?AQR = ?QRD [Pair of alternate interior angles]
And ?BQR = ?CRQ [Pair of alternate interior angles]
Theorem: If a transversal intersect two lines such that a pair of alternate interior angles is equal, then the two lines are parallel






If PS transversal intersect two lines AB and CD such that

?BQR = ?CRQ [A pair of alternate interior angles]
Then, B || CD

Theorem: If a transversal intersects two parallel lines, then each pair of interior angles on the same side of the transversal is supplementary.
If a transversal t intersects two parallel lines AB and CD at P and Q points respectively, Then, ?APQ + ?CQP = 180o [Pair of interior angles]
And ?BPQ + ?DQP = 180o [Pair of interior angles]

Theorem: If a transversal intersect two lines such that a pair of interior angles on the same side of the transversal is supplementary, then the two lines are parallel.
If a transversal t intersects two lines AB and CD such that a pair of interior angles on the same side of the transversal is supplementary, i.e.,
?APQ + ?BPQ = 180o [Supplementary]
And ?CQP + ?DQP = 180o [Supplementary]
Then, AB || CD

Lines Parallel To The Same Line
Theorem: Lines which are parallel to the same line are parallel to each other.










Theorem: The sum of the angles of a triangle is 180o.
Theorem: If a side of a triangle is produced, then the exterior angle so formed is equal to the sum of the two interior opposite angles.
?1 + ?2 = ?4
?BAC + ?ABC = ?ACD


3. Triangles
Congruence Of Triangles
Congruent means equal in all the respect or geometrical figures whose shapes and sizes are same Let ABC and DEF be two triangles in which AB = DE, BC = EF, AC = DF and ?A = ?D, ?B = ?E, ?C = ?F
respectively. Then, ?ABC ?DEF


“CPCT” means corresponding parts of congruent triangles.

CRITERIA FOR CONGRUENCE OF TRIANGLES:

Side-angle-Side:
I SAS) Congruence rule: Two triangles are congruent if two sides and the included angle of one triangle are 9
equal to the corresponding sides and the included angle of the other triangle.



If in?sABC and DEF, AB = DE, AC = DF and ?BAC =?EDF
Then, ?ABC ?DEF
It is called SAS congruence rule i.e. side-angle-side]
II Angle-Side-Angle (ASA) Congruence rule: Two triangles are congruent if two angles and the included side of one triangle are equal to two corresponding angles and the included side of other triangle.


Congruence Rule:
i.e. Angle-Side-Angle(ASA) congruence rule may be called Angle-Angle-Side (AAS) congruence rule.

III Side-Side-Side(SSS) congruence rule:
If three sides of one triangle are equal to the three sides of another triangle, then the two triangles are congruent.



If in ?sABC and DEF, AB = DE, BC = EF and AC = DF
Then, ?ABC ?DEF
[It is called SSS congruence rule i.e. side-side-side]



IV Right angle-Hypotenuse-Side (RHS) congruence rule:
If in two right triangles the hypotenuse and one side of one triangle are equal to the hypotenuse and one side of the other triangle, then the two triangles are congruent.


If ABC and DEF are two right triangles in which
?B = ?E = 90o, AC = DF and AB = DE
Then, ?ABC ?DEF
Theorem: Angles opposite to equal sides of an isosceles triangle are equal

Converse of Theorem: The sides opposite to equal angles of a triangle are equal: In ?ABC if ?B = ?C

Then, AB = AC

Inequalities In A Triangle:

Theorem: If two sides of a triangle are unequal, the angle opposite to the longer side is greater (or greater)


Let ABC be a triangle in which AC > AB and AC > BC.
Then, ?B > ?A and ?B > ?C
The side opposite to the largest angle is the longest.

Theorem: The sum of any two sides of a triangle is greater than the third side.

Let ABC be a triangle and AB, BC and AC are its corresponding sides.
Then, AB + BC > AC A. + AC > BC
and AC + BC > AB


4. Quadrilaterials
Quadrilateral is a closed figure with four sides:


Angles’ Sum property of a Quadrilateral:
Theorem: The sum of the angles of a quadrilateral is 360o

Types of Quadrilaterals:

I A Trapezium: In a quadrilateral if one pair of opposite sides is parallel, then it is called a trapezium (Fig.3) i.e. If AB || CD then quadrilateral ABCD is a trapezium.


II A parallelogram: In a quadrilateral if both pairs of opposite sides are parallel and equal, then it is called a parallelogram (Fig.4) i.e., AB || CD and AB = CD; AD || BC and AD = BC, then ABCD is a parallelogram.


III A Rectangle: In a quadrilaterals (parallelogram) if all angles are right angles, then it is called a rectangle (Fig.5) i.e. AB || CD, AB = CD, AD || BC; AD = BC and ÐA = ÐB = ÐC = ÐD = 90o, then ABCD is a rectangle.


IV A Rhombus: In a quadrilaterals (parallelogram) if all sides are equal, then it is called a rhombus (Fig.6), i.e., AB || CD, AD || BC and AB = BC = CD = DA, then ABCD is a rhombus.

V A Square: In a quadrilateral (parallelogram) if all sides are equal and all angles are 90o, then it is called a square (Fig.7) i.e. AB || CD, AD || BC, AB = BC = CD = DA and ?A = ?B = ?C = ?D = 90o VI

A Kite: In a quadrilateral ABCD (Fig.8), if AD = CD and AB = CB, then it is called a kite; i.e., two pairs of adjacent sides are equal but it is not a parallelogram.


Properties Of A Parallelogram:

Theorem: A diagonal of a parallelogram divides it into two congruent triangles.

Theorem: If each pair of opposite sides of a quadrilateral is equal, then it is a parallelogram.


Theorem: In a parallelogram, opposite angles are equal.


Here, ?A = ?C and ?B = ?D
Theorem: If in a quadrilateral, each pair of opposite angles is equal, then it is a parallelogram.



Theorem: The diagonals of a parallelogram bisect each other.


Here, OA = OC and OB = OD
Converse of above Theorem: If the diagonals of a quadrilateral bisect each other, then it is a parallelogram.
Theorem: A quadrilateral is a parallelogram if a pair of opposite sides is equal and parallel Theorem: The line segment joining the mid-points of two sides of a triangle is parallel to the third side.
Converse of above Theorem: The line drawn through the mid-point of one side of a triangle, parallel to another side bisects the third side.

5. Circles
The collection of all the points in a plane which are at a fixed distance from a fixed point n the plane, is called circle.
Centre of the Circle: The fixed point is called the centre of the circle O is the centre of the circle in Fig.1.


Radius of the circle: The fixed distance from the centre and circumference of the circle is called the radius of the circle. OA = OC = r is the radius of the circle. We can draw infinite Radius in a circle and all are equal in length.
Chord of the circle: The line segment which joins two points on the circumference of a circle is known as the chord of the circle. The chord of a circle does not pass through the centre of the circle. CD is a chord of the circle in Fig.2.


Diameter of the circle: The chord, which passes through the centre of the circle, is called a diameter of the circle. We can drawn infinite diameters in a circle and all are equal in length. In Fig.2, AOB is a diameter of the Xe-circle. It is denoted by d.

It is said that a diameter is the longest chord of a circle.
A circle divides the plane on which it lies into following three parts in Fig.3
(i) Interior of the circle: The plane which exists inside of a circle or the region inside of a circle is known as the interior of the circle.
(ii) Circle: The geometrical figure which is surrounded by a circular line segment or a circle is a collection of all those points in a plane that are at given constant distance from a given fixed point in the plane.
(iii) Exterior of the circle: The plane which exists outside of a circle or the region out side of a circle is known as the exterior of the circle.

Arc of a circle: A continuous piece of a circle is called an arc of the circle.
Minor arc: The shorter (smaller) arc of a circle is called minor arc. In Fig.4, PQ is the minor arc.

Major arc: The longer arc of a circle is called major arc. In Fig.4; PRQ is the major arc in Fig.4.

Semi circle: If P and Q are ends of a diameter then both arcs are equal and each is called a semi circle, i.e., PXQ and PYQ are equal arcs having a semi-circle in Fig.5. It is also called semicircular region.

Circumference: The length of the complete circle is called the circumference of the circle. It is denoted by C in Fig.6,

i.e. Circumference of the circle (C) = 2?r ; where ? = 22/7 or 3.14
Semi Circumference: Half length of the complete circle is called the semi-circumference of the circle. Both semi-circumferences of the circle are equal in length in Fig.7,



i.e. Semi circumference = ?.r

Segment of the circle: The region between a chord and either of its arcs is called a segment of the circle.
Minor Segment: The smaller region between a chord and smaller arc is called the minor segment of the circle, i.e. PXQ is the minor segment of the circle in Fig.8


Major segment: The bigger region between a chord and bigger arc is called the major segment of the circle, i.e., PYQ is the major segment of the circle in Fig.8.

Minor sector: When a circle is divided by its two Radius, the smaller region of the circle is called minor sector, e.g., OAXB is the minor sector of the circle in Fig.9.


Major Sector: When a circle is divided by its two Radius, the bigger region of the circle is called major sector, e.g. OAYB is the major sector of the circle Fig.9.
Theorem: Chords of a circle subtend equal angles at the centre.


Converse Theorem: Prove that if the angles subtended by the chords of a circle at the centre are equal, then the chords are equal.



Theorem: Prove that the perpendicular from the centre of a circle to a chord bisects the chord.



Converse Theorem: The line drawn through the centre of a circle to bisect a chord is perpendicular to the chord.



Theorem: There is one and only one circle passing through three non-collinear points.

Theorem: The length of the perpendicular from a point to a line is the distance of the line from the point.



Out of these line segments, the perpendicular from P to AB i.e. PM will be the least. So, this least length PM has to be the distance of AB from P.

Theorem: Equal chords of a circle are equidistant from the centre.



Converse Theorem: Prove that chords equidistance from the centre of a circle are equal in length



Angles Subtended By An Arc Of A Circle:
Theorem: If two chords of a circle are equal, then their corresponding arcs are congruent and conversely, if two arcs are congruent, then their corresponding chords are equal.

Let AB and CD be two chords of a circle with centre O.
Then, AXB = CYD
Converse: If AXB = CYD in a circle with centre O, then chord AB = chord CD
The angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.

Here, ?BOC = 2 x ?BAC

Theorem: Angles in the same segment of a circle are equal
Here, ?BAD = ?BCD
Theorem: Angle in a semicircle is a right angle.

Here, ?BAC = 90o
Theorem: If a line segment joining two points subtends equal angles at two other points lying on the same side of the line containing the line segment; four points lie on a circle (i.e., they arc concyclic). In Fig.21

Here, A, B, C, D are concylic.
Cyclic Quadrilateral: A quadrilateral is called cyclic if all the four vertices of it lie on a circle.


Theorem: The sum of either pair of opposite angles of cyclic quadrilateral is 180o.

Here, ?A + ?C = 180o and ?B + ?D = 180o.
Converse Theorem: If the sum of a pair of opposite angles of a quadrilateral is 180o, the quadrilateral is cyclic.
Secant: A line which intersects a circle in two distinct points is called a secant of the circle, e.g. in figure PQ is the secant of a circle ABCD with centre O in figure.

Tangent: A tangent to a circle is a line that intersects the circle in exactly one point. i.e., PQ is a tangent of a circle ABCD with centre O. And the touching point (point of contact) of the tangent PQ be R in figure. We can also say that there is only one tangent at a point of the circle, i.e., the common point of the tangent and the circle is called the point of contact and the tangent is said to touch the circle at the common point.


Theorem: The tangent at any point of a circle is perpendicular to the radius through the point of contact.


Important:
(i) At any point on a circle there can be one and only one tangent.
(ii) The line containing the radius through the point of contact is also called the ‘ normal’ to the circle at the point.

Theorem: The lengths of tangents drawn from an external point to a circle are equal.
So, if PA and PB are two tangents from a point P to a circle with centre O , then PA = PB