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When two lines are intersected by a transversal, they formtwo pairs of interior angles.In fig, the pairs of alternate angles are: i ∠EBC, ∠BCH ii ∠FBC, ∠GCB When two parallel lines are in

CONTENTS GEOMETRY AND MENSURATION

GEOMETRY: LINES, ANGLES & TRIANGLES

Properties of two Similar Triangles

Some Important Theorems

Basic Pythagorean Triplets

GEOMETRY: POLYGONS & QUADRILATERALS

SOLVING LINEAR INEQUALITIES IN ONE UNKNOWN

SOLVING QUADRATIC INEQUALITIES

SYSTEM OF INEQUALITIES IN ONE UNKNOWN

Conditions for parallelism and perpendicularity of lines

in terms of their slopes:

Angle between two lines:

Colinearity of three points:

DIFFERENT FORMS OF THE EQUATION OF A LINE:

Horizontal and vertical lines:

LINEAR EQUATION WITH ONE VARIABLE SIMULTANEOUS EQUATIONS

Applications of Linear Equations

GRE Math for Winners

Algebra, Sets, Geometry

& Coordinate Geometry

GEOMETRY AND MENSURATION

The chapters of geometry and mensuration have had theirfair share of questions in GRE For doing well in questionsbased on this chapter, you should familiarize yourself withthe basic formulae and visualizations of the various shapes

of solids and two dimensional figures based on this chapter

GEOMETRY: LINES, ANGLES & TRIANGLES

Background

Geometry and Mensuration are important areas in the GREexamination Over the past few years the trend has beenthat there are around 4 -6 questions based on thesechapters

The points which lie on the same line are called collinear

points The points which do not lie on the same line are

called non-collinear points.

denoted by the symbol ⊥

If two lines are perpendicular to the same line, they areparallel to each other

Some Important Points:

i) A line which is perpendicular to a line segment i.e.,

intersects it at the midpoint of the segment is calledperpendicular bisector of the segment

ii) Every point on the perpendicular bisector of a segment is

equidistant from the two endpoints of the segment.Conversely, if any point is equidistant from the twoendpoints of the segment, then it must lie on theperpendicular bisector of the segment

If PO is the perpendicular bisector of segment AB, then, AP

= PB Also, if AP = PB, then P lies on the perpendicularbisector of segment AB

iii) The ratio of intercepts made by three parallel lines on a

transversal is equal to the ratio of the correspondingintercepts made on any other transversal by the sameparallel lines

If line a | | line b | | line c and line l and line m are twotransversals, then,

PR/RT = QS/SU

ANGLES

An angle is the union of two non-collinear rays with commonorigin The common origin is called the vertex and the tworays are sides of the angle

The angle is generally denoted by the symbol '∠'.

Thus, in the figure, angle ABC is denoted as ∠ABC

Angle is measured in degrees

The shaded region is known as the interior of the angle

A 90° angle is called a right angle

∠ABC in the adjoining figure is a right angle

Adjacent angles

Adjacent angles are the angles with the common vertex, acommon side and disjoint interiors (i.e., their interiorsshould not be common.)

Thus, ∠DBC and ∠ABD are adjacent angles

Vertically opposite angles:

When two lines intersect each other at a common point,they form four angles at the point of intersection Theangles opposite to each other are called vertically oppositeangles The sum of all the four angles is 360o.It must benoted that the two vertically opposite angles are equal

Thus, ∠A = ∠D and ∠B = ∠C

Also, ∠A + ∠B + ∠C + ∠D =360°

∠M + ∠N = 180°.

Therefore, ∠M and ∠N are also supplementary angles

If the two supplementary angles are equal then they areright angles

Thus, ∠C = ∠D = 90°

Linear Pair:

Two angles are said to form a linear pair, if they have acommon side and their other two sides are opposite rays.The sum of the measures of the angles which form a linearpair is 180o The angles that form a linear pair are alwaysadjacent

∠AOC and ∠COB form a linear pair

Bisector of an angle:

A ray is said to be the bisector of an angle if it divides theinterior of the angle into two angles of equal measures.

Ray OX is the bisector of ∠AOB

∴ ∠AOX = ∠XOB = ½∠AOB

Every point on the angle bisector is equidistant from thesides of the angle Conversely, if any point in the plane ofthe angle is equidistant from the sides of the angle, then itlies on the angle bisector of the angle If OX is the anglebisector of ∠AOB, then AX = XB Also, if AX = XB, then Xlies on the angle bisector of ∠AOB

Corresponding Angles:

When two lines are intersected by a transversal, they formfour pairs of corresponding angles EF and GH areintersected at points B and C by the transversal AD asshown in fig The four pairs of corresponding angles are

i) ∠ABE, ∠BCG

ii) ∠EBC, ∠GCD

iii) ∠ABF, ∠BCH

iv) ∠FBC, ∠HCD

When the two parallel lines are intersected by a transversal,the pairs of corresponding angles so formed are congruent.Congruence is denoted by the symbol

If EF is parallel to GH and AD is the transversal, as shown infigure then:

Alternate Angles:

When two lines are intersected by a transversal, they formtwo pairs of interior angles.

In fig, the pairs of alternate angles are:

i) ∠EBC, ∠BCH

ii) ∠FBC, ∠GCB

When two parallel lines are intersected by a transversal, thepairs of alternate angles so formed are congruent In fig.,i) ∠EBC, ∠BCH

ii) ∠FBC, ∠GCB

Conversely, if the transversal intersects two lines and if onepair of alternate angles is congruent, then the two lines areparallel Hence, when one pair of alternate angles iscongruent then the other pair of alternate and all pairs ofcorresponding angles are congruent

When two parallel lines are intersected by a transversal, thepairs of interior angles so formed are supplementary.

TRIANGLES

The plane figure bounded by the union of three A lines,which join three non-collinear points, is called a triangle

A triangle is denoted by the symbol Δ

The three non-collinear points are called the vertices of the

triangle A, B and C are the vertices of the B C triangle AB,

BC and AC are the sides of the triangle ∠ABC, ∠ACB and

∠BAC are the interior angles of the Δ ABC

∠ABC + ∠ACB + ∠BAC = 180°

Equilateral Triangle:

If all the three sides of a triangle are equal, it is anequilateral triangle In an equilateral triangle, all the anglesare congruent and equal to 60o

Right Triangle:

If one of the angles of a triangle is 90°, it is called a righttriangle The other two angles of the right triangle will beacute and complementary The side opposite to the rightangle is called the hypotenuse

In Δ ABC, AC is the hypotenuse ∠ABC = 90°

Interior Angles of a Triangle:

∠ABC, ∠BAC and ∠ACB are the three interior angles of thetriangle ABC

Exterior angles of a Triangle:

The angle formed by extending one side of a triangle withanother side is called an exterior angle of the triangle Atriangle has six exterior angles

The exterior angles of Δ ABC are ∠FAC, ∠ACD, ∠ECB, ∠CBI,

∠HBA and ∠BAG

Properties of a triangle:

i) The sum of all the three angles of a triangle is 180o

ii) If two angles of a triangle are equal, then lengths of the

sides opposite to the equal angles are equal Conversely, ifthe two sides of a triangle are equal, the angles opposite tothe two sides are equal

iii) The sum of an interior angle and the adjacent exterior

angle is 180°

In the fig., ∠ABC + ∠ABH = 180°

∠ABC + ∠CBI = 180°

iv) The measure of an exterior angle is equal to the sum of

the measures of the opposite interior angles (also calledremote interior angles) of the triangle, not adjacent to it

In the fig., ∠ABH = ∠BAC + ∠BCA

Hence, an exterior angle of a triangle is greater than each ofthe interior angles not adjacent to it

Example: ∠ABH > ∠BAC

And, ∠ABH > ∠BCA

v) If two sides of a triangle are not congruent, then the

angle opposite to the greater side is greater

In ΔABC, If AB > AC, then ∠ACB > ∠ABC

Conversely, if two angles of a triangle are not congruent

then the side opposite to the greater angle is greater

In ∠ABC, if ∠ABC > ∠ACB then AC > AB

vi) The sum of any two sides of a triangle is always greater

than the third side

In ΔABC, AB + BC > AC, also AB + AC > BC and AC + BC >AB

vii) A triangle will have at least two acute angles.

Perimeter of a triangle:

The perimeter is the sum of all sides of a triangle If a, b and

c are the length of the sides of ΔABC Perimeter = a + b + c

triangle has three altitudes The point of intersection of the

three altitudes is called the orthocenter.

In ΔABC, the altitudes are AF, CE and BD G is the

centroid of the triangle The centroid divides each median

Area of triangle = 1/2 × base × height

Area of an equilateral triangle = √3/4 × (side)2

Properties of a Triangle:

i) For given perimeter, an equilateral triangle has maximum

area

ii) Area of a triangle = √s(s-a)(s-b)(s-c) where a, b and c arethe sides of the triangle and s is the semi perimeter,

iii) The ratio of the areas of two triangles is equal to the

ratio of the products of their bases and correspondingheights

A(∆ABC)/A(∆PQR) = AD×BC/PS×QR

iv) Triangles of equal heights have areas proportional to

their corresponding bases

vi) Areas of two triangles will be equal if they lie between

the same parallel lines and have same base

i) Side-Angle-Side (SAS):

Two triangles are said to be congruent if two correspondingsides and the angle included between them are equal This

is called SAS rule

In ΔABC and ΔDEF

AB = DE, AC = DF and ∠BAC = ∠EDF

Therefore, ΔABC ΔDEF

ii) Angle-Side-Angle (ASA):

Two triangles are said to be congruent if two pairs ofcorresponding angles and the corresponding included sideare equal It is called ASA rule

Here, BC = EF, ∠B = ∠E, ∠C = ∠F

Therefore, ΔABC = ΔDEF

iii) Side-Side-Side (SSS):

If all the three pairs of corresponding sides of the trianglesare equal, then the two triangles are congruent This iscalled SSS rule.

Here, AB = DE, BC = EF, and AC = DF

Therefore, ΔABC ΔDEF

iv) Hypotenuse Side Test:

Two right triangles are congruent if a corresponding side andthe hypotenuse of the triangles are congruent Here, AC =

Tests for similarity:

It is not necessary to list all the conditions for similarity i.e.,proportionality of sides and congruence of angles, to provethat two triangles are similar If certain select conditions aresatisfied, then the others will necessarily follow Theseselect conditions are called tests for similarity

i) AA Test: For a given correspondence between two

triangles, if the two angles of one triangle are congruent to

the corresponding two angles of the other triangle, then thetwo triangles are similar.

∠ABC ∠PQR

∠ACB ∠PRQ

ΔABC ~ ΔPQR by AA test for similarity

ii) SSS Test: For a given correspondence between two

triangles, if the three sides of one triangle are proportional

to the corresponding three sides of the other triangle, thenthe two triangles are similar

PQ/AB = QR/BC = PR/AC

ΔPQR ~ ΔABC by SSS test for similarity

iii) SAS Test : For a given correspondence between two

triangles, if the two sides of one triangle are proportional tothe corresponding two sides of the other triangle and theangles included between them are congruent, then the twotriangles are similar

AB/PQ = BC/QR and ∠ABC ∠PQR

ΔABC ~ ΔPQR by SAS test for similarity.

PROPERTIES OF TWO SIMILAR TRIANGLES

i) If two triangles are similar, then

Ratio of sides = Ratio of heights = Ratio of Medians = Ratio

of angle bisectors

If ΔABC ~ ΔPQR, AB/PQ = AD/PS = BE/QT

ii) The ratio of the areas of two similar triangles is equal to

the ratio of the squares of the corresponding sides

If ΔABC ~ ΔPQR, then A(∆ABC)/A(∆PQR) = (AB)2/(PQ)2 =(BC)2/(QR)2 =(AC)2/(PR)2

SOME IMPORTANT THEOREMS

i) Theorem of Pythagoras: In a right-angled A triangle,

the square of the hypotenuse is equal to the sum of thesquares of the other two sides

In Δ ABC, if ∠ABC = 90o

Then AC2 = AB2 + BC2

Conversely, if the square of one side of a triangle is equal tothe sum of the squares of its remaining two sides, then theangle opposite to the first side is a right angle

ii) Theorem of 45 o - 45 o - 90 o Triangle: If the angles of a

triangle are 45o, 45o and 90o, then the perpendicular sidesare 1/√2 times the hypotenuse

In ∆ABC, AB = BC = 1/√2 AC

iii) Theorem of 30 o - 60 o - 90 o Triangle: If the angles of a

triangle are 30o, 60o and 90o, then the sides opposite to 30o

is half the hypotenuse and the side opposite to 60o is √3/2times the hypotenuse.

In Δ ABC, AB =1/2 AC and BC = √3/2 AC

Ratio of sides: 1: √3: 2

iv) Midpoint Theorem: The segment joining the midpoints

of any two sides of a triangle is parallel to the third side and

is half the length of the third side

If AD = DB and AE = EC, then DE is parallel to BC and DE =1/2BC

v) Basic Proportionality Theorem: If a line is drawn

parallel to one side of a triangle and intersecting the othersides in two distinct points then the other sides are divided

in the same ratio by it

if DE is parallel to BC, then AD/DB= AE/EC

vi) Interior Angle Bisector Theorem: The angle bisector

of any angle of a triangle divides the side opposite to theangle in the ratio of the remaining two sides

In Δ ABC, if BE is the angle bisector of ∠ABC,

Then BA/BC = AE/CE

vii) Exterior Angle Bisector Theorem: The angle bisector

of any exterior angle of triangle divides the side opposite tothe angle (externally) in the ratio of the remaining twosides In ΔABC, ∠DBC is an external angle and BE is theexterior angle bisector

Here, BA/BC= AE/CE

BASIC PYTHAGOREAN TRIPLETS

→ 3, 4, 5 → 5, 12, 13 → 7, 24, 25 → 8, 15, 17 → 9, 40, 41 →

11, 60, 61 → 12, 35, 37 → 16, 63, 65 → 20, 21, 29 → 28, 45,53

These triplets are very important since a lot of questions arebased on them (You should try to commit to memory at

least the first two triplets)

Any triplet formed by either multiplying or dividing one ofthe basic triplets by any positive real number will be anotherPythagorean triplet

Thus, since 3, 4, 5 form a triplet so also will 6, 8 and 10 asalso 9, 16 and 25

Similarity of Right Triangles (RHS)

Two right triangles are similar if the hypotenuse and side ofone is proportional to hypotenuse and side of another (RHS– Right angle hypotenuse side)

Example: Δ ABC is right-angled at A and AD is the altitude

to BC If AB = 8 and AC = 15, find the length of BC andaltitude AD If M is the midpoint of BC, find AM

Example: The semi perimeter of a triangle is 15 The three

sides of the triangle are such that they representconsecutive even numbers Find the area of the triangle

Solution:

Semi perimeter, S = 15

∴ Perimeter = 15 × 2 = 30

Let the shortest side be x

∴ The other sides will be x + 2, x+ 4

∴x + x + 2 + x + 4 = 30 ⇒3x = 24 x = 8

∴ 3 sides are 8, 10, and 12

∴ Area

= √s(s-a)(s-b)(s-c) = √15(7)(5)(3) = √5×3×5×3×7 = 15√7sq.units

Example: In the figure AD which is the bisector of ∠EAC,

intersects BC produced in D If B = 8, AC = 6, BC = 3, findCD

Example: Line is parallel to and p is perpendicular to

both of them If x°/y° = 2/3 find x

Solution:

Third angle of right triangle = y

∴ x + y = 90 ∴x + 3x/2 = 90 ∴ 5x/2 = 90

∴ x = 80/5 = 36°

Example: The vertex angle of an isosceles triangle has a

measure of r° How many degrees are there in the angleformed by the bisectors of the base angles of the triangle?i) 90 + r/2

Sum of all angles of a triangle=180°

Vertex angle of an isosceles triangle is r°

base angle1 + base angle2 + r degrees = 180 degrees

Let the base angle be x (In an isosceles triangle, the baseangles are equal)

GEOMETRY: POLYGONS & QUADRILATERALS

Polygons are plane figures formed by a closed series ofrectilinear (straight) segments

Example: Triangle, Rectangle etc.

Polygons can broadly be divided into two types:

i) Regular polygons: Polygons with all the sides and angles

equal

ii) Irregular polygons: Polygons in which all the sides or

angles are not of the same measure

Polygon can also be divided as concave or convex polygons.Convex polygons are the polygons in which all the diagonalslie inside the figure otherwise it's a concave polygonPolygons can also be divided on the basis of the number ofsides they have

Polygons can also be divided on the basis of the number ofsides they have

PROPERTIES OF POLYGONS:

i) Sum of all the angles of a polygon with 'n' sides = (2n –4)π /2

ii) Sum of all exterior angles = 360°

iii) No of sides = 360°/exterior angle

iv) Perimeter = n × s

SOME IMPORTANT POLYGONS

REGULAR HEXAGON

i) Area = [(3√3)/2] (side)2

ii) A regular hexagon is actually a combination of 6

equilateral triangles all of side 'a'

Hence, the area is also given by: 6 × area of equilateraltriangle of side a

iii) If you look at the figure closely it will not be difficult to

realize that circumradius (R) = a

If the diagonals are d1 d2

Then, the area of the quadrilateral = ½ d1 d2 sin Ɵ1 = ½ d1

d2 sin Ɵ2

ii) Area = 1/2 × diagonal × sum of the perpendiculars to it

from opposite vertices

i) In a convex quadrilateral inscribed in a circle, the product

of the diagonals is equal to the sum of the products of the

opposite sides For example, in the figure:

i) Area = Base (b) x Height (h) = bh

ii) Area = product of any two adjacent sides × sine of the

included angle = ab sin Q

iii) Perimeter = 2 (a + b) where a and b are any two

adjacent sides

Properties :

i) Diagonals of a parallelogram bisect each other.

ii) Bisectors of the angles of a parallelogram form a

rectangle

iii) A parallelogram inscribed in a circle is a rectangle.

iv) A parallelogram circumscribed about a circle is a

rhombus

v) The opposite angles in a parallelogram are equal.

vi) The sum of the squares of the diagonals is equal to the

sum of the squares of the four sides in the figure:

AC2 + BD2 = AB2 + BC2 + CD2 + AD2 = 2(AB2 + DC2)

RECTANGLES:

A rectangle is a parallelogram with all angles 90°

i) Area = Base × Height = b × h

Note: Base and height are also referred to as the length andthe breadth in a rectangle

ii) Diagonal (d) = √b2 + h2 → by Pythagoras theorem

Properties of a Rectangle:

i) Diagonals are equal and bisect each other.

ii) Bisectors of the angles of a rectangle (a parallelogram)

form another rectangle (??? Could not understand)

iii) All rectangles are parallelograms but the reverse is not

i) Diagonals bisect each other at right angles.

ii) All rhombuses are parallelograms but the reverse is not