ANGLE & IT'S TYPES
ANGLE : An angle is the figure formed by two rays joined together at a common point called vertex .
Two rays acts as sides of angle.
If OA is initial arm &OB is final arm as in fig1.The rotation of initial arm is in anticlockwise direction .
Then the angle formed is known as positive angle .
If OB is initial arm & OA is final arm as in fig2.The rotation of initial arm is in clockwise direction. Then the angle formed is known as Negative angle.
Fig.1 Fig.2
Angle formed by two ray generally lie in the same plane.
Intersection of two planes also formed the angle,which is known as Dihedral angles.
The angles are measured in
1)Degree (Sexagesimal system)
2) Radian (Circular system).
There are different types of angles in geometry.
1)Right angle : An angle who measure is 90 degree is known as right angle.
2)Acute angle : An angle who measure is less than 90 degree is known as acute angle.
3)Obtuse angle : An angle who measure is more than 90 degree is known as obtuse angle.
4)Straight angle :When two rays join together in opposite direction at a common point they form a straight angle.
A straight angle is also called as flat angle .
It's measure is 180 degree .
5)Zero angle : An angle whose measure is zero degree is called a zero angle.
It is formed when both the rays which formed the side of angle lie in the same line or they overlap each other or they coincide with each other .
Ray OA & RayOB Coincide with each other .
6)Reflex angle : A reflex angle is an angle which is more than 180 degree but less than 360 degree.
If one angle is more than 180 degree than other angle formed in a plane can be either acute or obtuse
i.e opposite of reflex angle is either acute or obtuse .
7)Directed angle : The ordered pair of rays (OA,OB) together with the ROTATION OF THE RAY OA to the position of the ray OB is called the directed angle AOB.
If the rotation of the initial ray is anticlockwise then the measure of directed angle is cosidered positive and if the rotation is clockwise then the measure of the angle is considered as negative.
8)One rotation angle :After one complete rotation if the initial ray OA coincide with the final ray i.e terminal ray OB then the angle so formed is known as one rotation angle. For example 360 Degree.
9)Standard angle : If the vertex of the angle is at the origin and the initial ray is along positive direction of x -axis of the coordinate system , such angle is known as standard angle.
Angle POX , AngleXOQ , Angle XOR are standard angles .
10)Quadrantal Angle : A directed angle in standard position whose terminal ray lie along X-axix or Y-axis is called a quadrantal angle.
Angle POQ , Angle POR , Angle POS , Angle POQ (Clockwise) are quadrantal angles.
11)Co- terminal angles : Directed angles of different amount of rotation having the same position of initial and final rays are known as Co-terminal angles.
Angle AOB (30 degree) , Angle AOB (-330 degree) , Angle AOB (390 degree) are Co-terminal angles.
12)Supplementary angle : When sum of measure of two angle is 180 Degree then those angles are known as supplementary angle. Eg. If one angle is 60 Degree then it's supplementary angle is 120 Degree.
13) Complementary Angle : When sum of measure of two angle is 90 Degree then those angles are known as Complementary angles.
14) Adjacent angles : Two angles are said to be adjacent if 1)They have common vertex 2)they have one side common & 3) both the angles are on opposite side of common side.
Angle CBD & angle DBE are adjacent angle .
Angle ABC & Angle CBD are adjacent angle .
But Angle ABC & angle DBE are not adjacent angle .
15)Vertically opposite angles : Two angles are said to be vertically opposite if they are formed when two lines intersect at a common point & they are formed in opposite direction.
Angle AOC & Angle DOB
Angle AOD & Angle COB are vertically opposite angles.
16 ) FOR TWO PARALLEL LINE :
If the two line are parallel (i.e. two line never intersect each other) & a transeverse intersect the two parallel lines at two different points then following angles are formed .
1)Alternate angles
2)Corresponding angles
3)interior angle
4)Exterior angle
5)Vertcally opposite angle
from fig.
1)Alternate angles are of equal measures
& they are :
Angle c & angle e
Angle b & angle h
2)Corrosponding angles are ...
Angle d & angle h
angle c & angle g
angle a & angle e
angle b & angle f
3)Interior angles are :
Angle c , angle h , angle b , angle e
Aso angle c+ angle h =180degree
& angle b+ angle e = 180 degree
4) Exterior angles are :
Angle d & angle g
Angle a & angle f
5) Vertically opposite angles are :
angle d & angle b
angle a & angle c
angle h & angle f
angle e & angle g
17) Plane Angle : An Angle between two Intersecting lines is known as plane angle in the same plane .
It is measured in Radian.
It is defined in two dimension .
In case of circle it is define as angle subtended by arc at the centre of the circle .
18) Solid Angle : It is 3 Dimensional angular volume .
It is measured in steradians. If the surface covers the entire sphere then the solid angle subtended at the center is 4 pi radian . Solid angle concept is generally used in physics.
19) Space Angle :
It is a angle between a line and a Plane in the space .
It is also angle between two plane in the space.
It is also an angle between two line in the space.
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