the e tutor - lines and angles

8/2/2019 The E Tutor - Lines and Angles

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LINES AND ANGLES


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Point

An exact location on a plane is

called a point.

Line

Linesegment

Ray

A straight path on a plane,

extending in both directionswith no endpoints, is called a

line.

A part of a line that has twoendpoints and thus has adefinite length is called a line

segment.

A line segment extendedindefinitely in one directionis called a ray.

Recap Geometrical Terms


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RAY: A part of a line, with one endpoint, that continueswithout end in one direction

LINE: A straight path extending in both directions with noendpoints

LINE SEGMENT: A part of a line that includes twopoints, called endpoints, andall the points betweenthem


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INTERSECTING LINES: Lines that cross

PARALLEL LINES: Lines that never cross and are always

the same distance apart


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Commonendpoint

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.

What Is An Angle ?

Common endpoint is called the vertex of the angle.Bis the

vertexof ABC.

Ray BAand rayBCare called thearms ofABC.


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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

B

C

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.)


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Right Angle:An angle that forms asquare corner

Acute Angle:An angle lessthan a right angle

Obtuse Angle:An angle greaterthan a right angle


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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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Pairs Of Angles : Types

Adjacent angles

Vertically opposite angles

Complimentary angles

Supplementary angles

Linear pairs of angles


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Adjacent AnglesTwo angles that have acommon vertexand acommon rayare called adjacent angles.

C

D

B

A

Common ray

Common vertex

Adjacent AnglesABDand DBC

Adjacent angles do not overlap each other.

D

E F

A

B

C

ABCand DEF are not adjacent angles


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Vertically Opposite AnglesVertically opposite angles are pairs of angles formed bytwolines intersecting at a point.

APC = BPDAPB = CPD

A

DB

C

P

Four anglesare formed at the point of intersection.

Point of intersection P is thecommon vertex of the fourangles.

Vertically opposite angles arecongruent.


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

600

A

BC

300

D

EF

ABCandDEFare complimentary because

600 + 300 = 900

ABC + DEF

Complimentary Angles


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If the sum of two angles is 1800 then they are calledsupplementary angles.

PQRandABCare supplementary, because

1000 + 800 = 1800

RQ

PA

BC

1000 800

PQR + ABC

Supplementary Angles


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Two adjacent supplementary angles are called linear pair

of angles.

A

6001200

PC D

600 + 1200 = 1800

APC + APD

Linear Pair Of Angles


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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 lineM 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 areformed 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 parallel lines are cut by a transversal, pairs ofcorresponding angles are formed.

Four pairs of corresponding angles are formed.

Corresponding pairs of angles are congruent.

GPB = PQEGPA = PQDBPQ = EQFAPQ = DQF

Line MBA

Line ND E

L

P

Q

G

F

Line L


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Alternate Angles

Alternate angles are formed on opposite sides of thetransversal and at different intersecting points.

Line MBA

Line ND E

L

P

Q

G

F

Line L

BPQ = DQPAPQ = 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 linesthat are cut by a transversal, are calledinterior angles.

A pair of interior angles lie on thesame sideof thetransversal.

The measures of interior angles in each pairadd up to 1800.

Interior Angles

Line M

BA

Line ND E

L

P

Q

G

F

Line L

600

1200

1200600

BPQ + EQP = 1800

APQ + DQP = 1800


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Theorems of

Lines and angles


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If a ray stands on a line, then the sum of theadjacent angles so formed is 180.

PX Y

QGiven: The rayPQstands on the line XY.To Prove: QPX + YPQ = 1800Construction: Draw PE perpendicular to XY.Proof: QPX= QPE + EPX

=QPE + 90 .. (i)

YPQ = YPE QPE

=90 QPE . (ii)

(i) + (ii)

QPX +YPQ = (QPE+ 90) + (90 QPE)

QPX + YPQ = 1800

Thus the theorem is proved

E


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Vertically opposite angles are equal inmeasure

To Prove :

=

C and

B =

D

A + B = 1800 . Straight line l B + C = 1800 Straight line m

+ = B + CA = C

Similarly B =D

A

B

C

D

lm

Proof:

Given: l and m be two lines intersecting at O as.They lead to two pairs of verticallyopposite angles, namely,

(i) A and C (ii) B and D.


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If a transversal intersects two lines such that a pair of

corresponding angles is equal, then the two lines areparallel to each other.

Given: Transversal PS intersects parallel lines

AB and CD at points Q and R respectively.

To Prove:

BQR = QRC and AQR = QRD

Proof :

PQA = QRC ------------- (1) Corresponding angles

PQA = BQR -------------- (2) Vertically opposite angles

from (1) and (2)

BQR = QRC

Similarly, AQR = QRD. Hence the theorem is proved

BA

C D

p

S

Q

R

If l i li h h i f


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If a transversal intersects two lines such that a pair ofalternate interior angles is equal, then the two lines areparallel.

Given:The transversal PS intersects lines AB and CDat points Q and R respectively

where BQR = QRC.

To Prove: AB || CD

Proof:

BQR = PQA --------- (1)

Vertically opposite angles

But, BQR = QRC ---------------- (2) Given

from (1) and (2),

PQA = QRC

But they are corresponding angles.

So, AB || CD

BA

C D

P

S

Q

R

Li hi h ll l t th li


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Lines which are parallel to the same line areparallel to each other.

Given: line m || line l and line n || line l.To Prove: l || n || m

Construction:

BA

C D

P

S

R

l

n

m

E F

Let us draw a line t transversal

for the lines, l, m and n

t

Proof:It is given that line m || line l and

line n || line l.

PQA = CRQ -------- (1)

PQA = ESR ------- (2)

(Corresponding angles theorem) CRQ = ESR

Q

But, CRQ and ESR are corresponding angles and they areequal. Therefore, We can say that

Line m || Line n

the e tutor - lines and angles

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