GEOMETRY- LINES AND ANGLES

GEOMETRY- LINES AND ANGLES
Lines and Angles
Angle:     An angle is the union of two different rays having the same initial point (or)  Amount of rotation. The common point is called the vertex. The size of an angle is typically measured in degrees
Representing an angle: 
        Generally we use three capital letters to represent an angle. The above angle is represented by \(\angle AOB\) (or) \(\angle BOA\) and is read as ‘angle AOB’ or ‘angle BOA’. Note that the middle letter denotes the vertex of the angle. The symbol \(\angle\) stands for angle. It can also be represented by \(\angle O\). Some times we use numbers or lower case letter to denote angles.
Consider the following angles:

Interior and exterior of an angle: 
An angle divides the plane into three regions.
i) Point belongs to the angle: 
    If any point (like P) lying on one of its arms, then that ‘P’ belongs to the angle

ii)  Interior of an angle:  In the following figure, ‘S’ does not belong to the angle. Note that the points ‘S’ and ‘P’ are on the same side of the \(\mathop {OB}\limits^{\xrightarrow{{}}} \) and the points ‘S’ and ‘P’ are also on the same side of the \(\mathop {OA}\limits^{\xrightarrow{{}}} \). Such a point is said to be in the interior of the angle.

iii) Exterior of an angle: In the following figure, ‘G’ does not belong to the angle. It is not in the interior of the angle. We say that it is in the exterior of the angle.  

Magnitude of an angle: The magnitude of an angle is the amount of rotation through which one of the arms must be rotated about the vertex to bring it to the position of the other.
Measure of an angle: To find out the magnitude to a given angle, we need a standard unit angle. Then we can compare the given angle with the unit angle and say its measure. 
i) A quarter turn of a ray \(\mathop {OA}\limits^{\xrightarrow{{}}} \) about O describes an angle which is called a right angle.

ii) Units for measuring angle: 
    A right angle is divided into 90 equal parts and each part is called a degree. Degree is the unit for measuring an angle.
    One degree is written as 1°. One degree is divided into 60 equal parts and each part is called a minute. One minute is divided into 60 equal parts and each part is called a second. 1 minute is denoted by 1¹ . 1 second is denoted by 1¹¹.
        1⁰ = 60¹    (read as 60 minutes)
        1¹ = 60¹¹    (read as 60 seconds).

GEOMETRY- LINES AND ANGLES

GEOMETRY- LINES AND ANGLES
Types of angles:
Acute angles:An angle whose measure is less than 90° and greater than O° is called an acute angle i.e., If \(\theta \) is an acute angle, then \([{0^o} < \theta < {90^o}\)
Right angle: An angle whose measure is 90° is called a right angle.
Obtuse angle: An angle whose measure is greater than 90° and less than 180° is called obtuse angle.
Straight angle: An angle whose measure is 180° is called a straight angle.
    Note: A straight angle = Two right angles.
Reflex angle: An angle whose measure is greater than 180° and less than 360° is called a reflex angle.
Complete angle:An angle whose measure is 360° is called a complete angle
    
Here \(\angle AOB\) is 360°.
Note: A complete angle = Four right angles.
Zero angle:    If the measure of the angle is zero, it is called a zero angle.
Operations with angles:

Consider \(\angle AOB\)  and \(\angle BOC\) in the above figure. Both \(\angle AOB\)  and \(\angle BOC\) have a common arm \(\mathop {OB}\limits^{\xrightarrow{{}}} \) and a common vertex ‘O’.     \(\therefore \,\,\,\angle AOC = \angle AOB + \angle BOC\). 
Bisector of the angle:
    i)  
         
   \(\angle PQS\,\,\,and\,\,\,\angle SQR\)have the same measure and congruent. So \(\overrightarrow {QS} \) is called the bisector of \(\angle PQR\). A ray which divides an angle into two congruent angles is called the bisector of the angle.
    ii)
                                                                
 In the above figure, \(\angle AOE\) is divided into four congruent angles.
Pairs of angles:
i) Adjacent angles:    
    Two angles in a plane are called adjacent angles, if they have a common vertex, a common side and their interiors do not have a common point.
ii) Linear pair

    In the above figure  are adjacent angles,  are opposite  rays. Such a pair of adjacent angles is called a ‘linear pair’ (or) the pair of adjacent angles whose non common arms are opposite rays is called a ‘linear pair’. Linear pair forms 180°.
iii) Vertically opposite angles: Consider two lines  intersecting at ‘O’.

Consider the  . They have a common vertex “O’ but do not have common arm. Such angles are called vertically opposite angles.  are also vertical opposite angles, which are equal.
iv) Supplementary angles: 
    Two angles are said to be supplementary if the sum of their measure is 180°.
    Ex: The measure of the supplementary angle of 20⁰ is 180⁰ – 20⁰= 160⁰
Complementary angles: 
    If the sum of the measures of two angles is equal to 90°, then they are called Complementary angles.
    Ex: The measure of the complementary angle of 600 is 90⁰ – 60⁰ = 30⁰