Vikasana - CET 2012

BRIDGE COURSE

VIKASANA

GEOMETRY-BASICS

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Vikasana - CET 2012

An exact position or location

You are here.

In fact, you cannot see a

true point.

Point P

P

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A series of points that go

endlessly in both directions.

Because a line has no width or height,

you cannot see a true line.

A

B

Line AB

Ray is the part of a line that has one

endpoint and goes on endlessly

in the other direction.

C

DRay CD

A part of a line that has

two endpoints.

E

F

Line Segment EF

INTERSECTING LINES: Lines that cross

PARALLEL LINES: Lines that never cross and are

always the same distance apart

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Perpendicular Lines

Two lines that intersect to form four right angles

Ray

Intersecting Lines

Parallel Lines

Line Segment

X

Y

Z

X

Z

Y

∠XYZ

90 °°°°

c

R

S TU

V

∠∠∠∠RST=90

�

∠∠∠∠USR =

90�

∠∠∠∠USV =

90�∠∠∠∠TSV = 90�

V

R

TU

c

S

UT ⊥ RV

135°°°°

obtuse

45°°°° acute

180 °°°° is a straight angle

Parallel lines have the same slope

or steepness.

These 2 lines are not parallel, but

they are not intersecting either.

These lines are called skew lines.

S

K

E

W

E

R

intersecting

perpendicular

90º

parallelskew

The figure formed when two rays share the same endpoint

Right Angle:An angle that forms a square corner

Acute Angle:An angle less

than a right angle

Obtuse Angle:An angle greater than a right angle

If we look around us, we will see angles

everywhere.

Angles In Daily Life

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Common

endpoint

B C

B

A

Ray BC

Ray BA

Ray BA and BC are two non-collinear rays

When two non-collinear rays join with a

common endpoint (origin) an angle is formed.

Common endpoint is called the vertex of the

angle. B is the vertex of ∠∠∠∠ABC.

Ray BA and ray BC are called the arms of ∠∠∠∠ABC.

Z

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To name an angle, we name any point on one ray,

then the vertex, and then any point on the other

ray.

For example: ∠∠∠∠ABC or ∠∠∠∠CBA

We may also name this angle only by the single

letter of the vertex, for example ∠∠∠∠B.

A

BC

Naming An Angle

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An angle divides the points on the plane into three

regions:

A

BC

F

R

P

T

X

Interior And Exterior Of An Angle

• Points lying on the

angle (An angle)• Points within the

angle (Its interior

portion. )• Points outside the

angle (Its exterior

portion. )

Angles are accurately measured in degrees.

Protractor is used to measure and draw angles.

Measurement Of An Angle

The unit used to

measure angles.

70

There are four main types of angles.

Straight angle

Right angle Acute angle Obtuse angle A

BC

A

BC

A

BC

BA C

Types Of Angles

Acute angle: An angle whose measure is less than 90

degree.

Acute AngleObtuse AngleStraight Angle Right Angle

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Examples Of

Acute Angle

Right angle: An angle whose measure is 90 degree.

Right Angle Acute AngleStraight Angle Obtuse Angle

Examples Of

Right Angle

Obtuse angle: An angle whose measure is greater than 90

degree and less than 180 degree.

Obtuse AngleAcute AngleStraight Angle Right Angle

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Examples Of

Obtuse Angle

Straight angle: An angle whose measure is 180

degree.

Straight Angle Acute AngleRight Angle Obtuse Angle

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Examples Of Straight Angle

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A

BC

D

E F

P

Q R

Which of the angles below is a right angle,

less than a right angle

and greater than a right angle?

Right angle

Greater than a right angle

Less than a right angle

1.

2.

3.

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Pairs Of Angles : Types

• Adjacent angles

• Vertically opposite angles• Complimentary angles• Supplementary angles•Linear pairs of angles

•Congruent angles

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Two angles that have the same measure are called

congruent angles.

Congruent angles have the same size and shape.

A

BC

300

D

EF

300

D

EF

300

Congruent Angles

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Adjacent Angles-

Adjacent angles are angles

which have a common

side and a common

vertex but no

interior points in common.

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Adjacent AnglesTwo angles that

have a common

vertex and a

common ray are

called adjacent

angles. C

D

B

A

Common ray

Common vertex

Adjacent Angles∠∠∠∠ABD and ∠∠∠∠DBC

Adjacent angles do not overlap each other.

D

EF

A

B

C

∠∠∠∠ABC and ∠∠∠∠DEF are not adjacent angles

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� Vertically Opposite Angles- Vertically Opposite Angles are the angles opposite to each other when two lines cross.

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Vertically Opposite Angles

Vertically opposite

angles are pairs of

angles formed by two

lines intersecting at

a point.

∠∠∠∠APC = ∠∠∠∠BPD

∠∠∠∠APB = ∠∠∠∠CPD

A

DB

C

P

Four angles are formed at the point of intersection.

Point of intersection ‘P’ is the common vertex of the four

angles.

Vertically opposite angles are congruent.

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If the sum of two angles is 900, then they are called

complimentary angles.

600

A

BC

300

D

EF

∠∠∠∠ABC and∠∠∠∠DEF are complimentary because

600 + 300 = 900∠∠∠∠ABC + ∠∠∠∠DEF

Complimentary Angles

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700

D

EF

300

p

QR

If the sum of two angles

is more than 900 or less than 900,

then they are not

complimentary angles.

∠DEF and ∠PQR are not complimentary because

700 + 300 = 1000

∠∠∠∠DEF + ∠∠∠∠PQR

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If the sum of two angles

is 1800 then they are

Called supplementary

angles.

∠∠∠∠PQR and ∠∠∠∠ABC are supplementary, because

1000 + 800 = 1800

RQ

PA

BC

1000 800

∠∠∠∠PQR + ∠∠∠∠ABC

Supplementary Angles

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If the sum of two angles

is more than 1800 or less than

1800, then they are not

supplementary angles.

∠∠∠∠DEF and∠∠∠∠PQR are not supplementary because

∠∠∠∠ABC + ∠∠∠∠DEF

1100 + 800 = 1900

D

EF

800

CB

A

1100

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Two adjacent supplementary angles are called

linear pair of angles.A

6001200

P

C D

600 + 1200 = 1800

∠∠∠∠APC + ∠∠∠∠APD

Linear Pair Of Angles

Name the adjacent angles and linear pair of angles

in the given figure:

Adjacent angles:

∠∠∠∠ABD and ∠∠∠∠DBC

∠∠∠∠ABE and ∠∠∠∠DBA

Linear pair of angles: ∠∠∠∠EBA, ∠∠∠∠ABC

C

D

B

A

E

600

300

900

∠∠∠∠EBD, ∠∠∠∠DBC

C

D

B

A

E

600

300

900

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Name the

vertically opposite

angles and adjacent

angles in the given

figure:

A

DB

C

P

Vertically opposite angles: ∠∠∠∠APC and ∠∠∠∠BPD

∠∠∠∠APB and ∠∠∠∠CPD

Adjacent angles: ∠∠∠∠APC and ∠∠∠∠CPD

∠∠∠∠APB and ∠∠∠∠BPD

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A line that

intersects two or

more lines at

different points

is called a transversal.

Line L (transversal)

BALine M

Line NDC

P

Q

G

F

Pairs Of Angles Formed by a Transversal

Line M and line N are parallel lines.Line L intersects line M and line N at point P and Q.Four angles are formed at point P and another four at point Q by the

transversal L.Eight angles are formed in all by the transversal L.

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Pairs Of Angles Formed by a

Transversal•Corresponding angles

•Alternate angles

• Interior angles

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� Corresponding Angles- When two lines are crossed by another line the angles in matching corners are called corresponding angles.

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Corresponding AnglesWhen two parallel lines are cut by a transversal,

pairs of corresponding angles are formed.

Four pairs of corresponding angles are formed.

Corresponding pairs of angles are congruent.

∠∠∠∠GPB = ∠∠∠∠PQE

∠∠∠∠GPA = ∠∠∠∠PQD

∠∠∠∠BPQ = ∠∠∠∠EQF

∠∠∠∠APQ = ∠∠∠∠DQF

Line MBA

Line ND E

L

P

Q

G

F

Line L

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Alternate AnglesAlternate angles are formed on opposite sides of the

transversal and at different intersecting points.

Line MBA

Line ND E

L

P

Q

G

F

Line L

∠∠∠∠BPQ = ∠∠∠∠DQP

∠∠∠∠APQ = ∠∠∠∠EQP

Pairs of alternate angles are congruent.

Two pairs of alternate angles are formed.

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The angles that lie in the area between the two

parallel lines that are cut by a transversal

are called interior angles.

A pair of interior angles lie on the same side of the transversal.The measures of interior angles in each pair add up to 1800.

Interior Angles

Line MBA

Line ND E

P

Q

G

F

Line L

6001200

1200600

∠∠∠∠BPQ + ∠∠∠∠EQP = 1800

∠∠∠∠APQ + ∠∠∠∠DQP = 1800Line N

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Name the pairs of the following angles

formed by a transversal.

Line MBA

Line ND E

P

Q

G

F

Line L

Line MBA

Line ND E

P

Q

G

F

Line L

Line MBA

Line ND E

P

Q

G

F

Line L

500

1300

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Reflex Angle- An angle that is greater then

180° and less than 360° is known as reflex angle.

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More AnglesMore Angles

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� Alternate Interior Angles- When two lines are crossed by another line the pairs of angles on opposite sides of the transversal but inside the

� two lines are called

� Alternate Interior

� Angles.

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� Alternate Exterior Angles- When two lines are crossed by another line the pairs of angles on opposite sides of the transversal but outside

� the two lines are

� called Alternate

� Exterior Angles.

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A plane

is a flat

surface that has

length & width

but no height.

A true plane goes on forever in all

directions.

A true plane goes on forever in all

directions.

A true plane goes on forever in all

directions.

A true plane goes on forever in all

directions.

Planes can intersect.

Planes can be perpendicular.

Planes can be parallel.

intersecting

perpendicular

parallel

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Similar Figures

�Plane figures that have the

same shape are called similar

figures.

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Congruent Figures

�Plane figures that have both the

same size and shape are called

congruent figures.

PointAn exact location on a plane is

called a point.

Line

Line

segment

Ray

A straight path on a plane,

extending in both directions with

no endpoints, is called a line.

A part of a line that has two

endpoints and thus has a definite

length is called a line segment.

A line segment extended

indefinitely in one direction is

called a ray.

Recap Geometrical Terms

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AngleAn angle is formed when two rays share the common point.

Acute

Angle

An angle whose

measure is less than 90 degree.

Right

Angle

An angle whose

measure is 90 degree.

Obtuse

Angle

An angle whose measure is greater than

90 and less than 180 degree.

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Straight Angle

An angle whose measure is 180 degrees.

Reflex Angle

An angle whose measure is greater than 180 and

less than 360 degree.

PlaneA plane is a flat surface that has length and width

but no height.