Lines&Angl

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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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An exact position or location

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RAY: A part of a line, with one endpoint, that continues without end in one direction

LINE: A straight path extending in both directions with no endpoints

LINE SEGMENT: A part of a line that includes two points, called endpoints, and all the points between them

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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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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.

What Is An Angle ?

Common endpoint is called the vertex of the angle. B is the vertex of ÐABC.

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

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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: ÐABC or ÐCBA

We may also name this angle only by the single letter of the vertex, for example ÐB.

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. )

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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

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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 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 AnglesVertically opposite angles are pairs of angles formed by two lines intersecting at a point.

ÐAPC = ÐBPD

ÐAPB = ÐCPD

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

ÐABC and ÐDEF are complimentary because

600 + 300 = 900

ÐABC + ÐDEF

Complimentary Angles

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

ÐPQR and ÐABC are supplementary, because

1000 + 800 = 1800

RQ

PA

BC

1000 800

ÐPQR + ÐABC

Supplementary Angles

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Two adjacent supplementary angles are called linear pair of angles.

A

600 1200

PC D

600 + 1200 = 1800

ÐAPC + ÐAPD

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 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 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.

ÐGPB = ÐPQE

ÐGPA = ÐPQD

ÐBPQ = ÐEQF

ÐAPQ = ÐDQF

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

ÐBPQ = ÐDQP

ÐAPQ = ÐEQP

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

L

P

Q

G

F

Line L

6001200

1200600

ÐBPQ + ÐEQP = 1800

ÐAPQ + ÐDQP = 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 the adjacent angles so formed is

180°.

PX Y

QGiven:  The ray PQ stands on the line XY.

To Prove:  ∠QPX  + ∠YPQ = 1800 

Construction:  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 in measure

To Prove : ÐA = Ð C and ÐB = ÐD

Ð A + Ð B = 1800 ………………. Straight line

‘l’

Ð B + Ð C = 1800 ……………… Straight line

‘m’

Þ Ð A + Ð B = Ð B + Ð C

Þ ÐA = ÐC

Similarly ÐB = ÐD

A

BC

D

lm

Proof:

Given: ‘l’ and ‘m’ be two lines intersecting at O as. They lead to two pairs of vertically opposite 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 are parallel 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 = ∠ QRCSimilarly, ∠ AQR = ∠ QRD. Hence the theorem is proved

BA

C D

p

S

Q

R

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If a transversal intersects two lines such that a pair of alternate interior angles is equal, then the two lines are parallel.Given: The transversal PS intersects lines AB and CD at points Q and R respectively where ∠ BQR = ∠ QRC.To Prove: AB || CD

Proof: ∠ BQR = ∠ PQA --------- (1) Vertically opposite anglesBut, ∠ BQR = ∠ QRC ---------------- (2) Given from (1) and (2),

∠ PQA = ∠ QRCBut they are corresponding angles.So, AB || CD

BA

C D

P

S

Q

R

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Lines which are parallel to the same line are parallel 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 are equal. Therefore, We can say that

Line m || Line n NM Spirit

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