Deepesh kumar

9th – f 13

Maths

G.L.T SARASWATI BAL MANDIR

UNIT OF ANGLE MEASUREMENT

वि�क्ला�नां�� कला�षष्टचा�: तत� षष्टया� भा�ग उच्यात� |तत्त्रि��शत�� भा��द्रा�शिश: भागणो� दृा�दश�� त� | |

सू!या"सिसूध्द�न्त: १२८60 VIKATA = 1 KALA 60KALA = 1 BHAGA30BHAGA = 1 RASI

याह सू�स्क+ त में- सू!या"सिसूध्द�न्त: में- दिदया� जा� चा0क� ह� |

In the modern notation vikala is a

second ,Kala is a minute , and

bhaga is a degree .It is Indeed

remarkable that ancient system

of measurement of angles is identical to the present system .

SANSKRIT MATHEMATICS Bakhshali Manuscript , Satapatha Brahmana ,

Baudhayana. INDIAN MATHEMATICS

Aryabhata, Brahmagupta, Mahavira  Bhaskara II, Madhava of

Sangamagrama and Nilakantha Somayaji .

INTERNATIONAL MATHEMATICSC. F. Gauss , N. Lobachevsky ,

J. Bolyai , B. Riemann

Lines

&

Angles

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.

GEOMETRICAL TERMS

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

INTERSECTING LINES: Lines that cross

PARALLEL LINES: Lines that never cross and are always the same distance apart

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.

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.

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. )

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

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

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

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.

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

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

Two adjacent supplementary angles are called linear pair of angles.

A

600 1200

PC D

600 + 1200 = 1800

ÐAPC + ÐAPD

LINEAR PAIR OF ANGLES

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.

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

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.

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

ÐBPQ + ÐEQP = 1800

ÐAPQ + ÐDQP = 1800