Line and angle
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Transcript of Line and angle
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
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