Selina Textbook Solutions: Angles | Mathematics Class 6 ICSE PDF Download

 Page 1


IMPORTANT POINTS 
1. Ray. A ray is a half-line: It has one end point and other end is open. It can not be 
measured like a line. Here, OA is a ray. 
 
From appoint, infinite numbers of rays can be drawn. 
 
2. Angle: When two rays meet at a point, then an angle is formed. Angle is measured in 
degrees with the help of an instrument known as a protractor. 
The point where the two rays meet is called an initial point or vertex of the angle and the 
two rays which form the angle, are called the sides of the angle e.g., two rays OA and 
OB meet at O. 
 
? Angle AOB is formed. Vertex is always kept between the ends points. 
The anlgle can be denotes as ?AOB, or ?BOA, here sign ‘ ?’ denotes the angle. 
The angle is also denoted with letters A, B, C etc. or numbers 1, 2, 3 etc. also. 
3. Parts of an angl: Angle has three parts : Interior, exterior and the angle itself. 
4. Comparison of Angles: Two angles can be compared with respect to their 
magnitude. Any angle of greater measure is greater. 
5. Kinds of Angles :  
(i) Zero angle: When two rotating rays (sides of angles) coincide each other, then ?ero 
angle is formed. 
 
(ii) Right angle: An angle of 90° is called a right angle. 
Page 2


IMPORTANT POINTS 
1. Ray. A ray is a half-line: It has one end point and other end is open. It can not be 
measured like a line. Here, OA is a ray. 
 
From appoint, infinite numbers of rays can be drawn. 
 
2. Angle: When two rays meet at a point, then an angle is formed. Angle is measured in 
degrees with the help of an instrument known as a protractor. 
The point where the two rays meet is called an initial point or vertex of the angle and the 
two rays which form the angle, are called the sides of the angle e.g., two rays OA and 
OB meet at O. 
 
? Angle AOB is formed. Vertex is always kept between the ends points. 
The anlgle can be denotes as ?AOB, or ?BOA, here sign ‘ ?’ denotes the angle. 
The angle is also denoted with letters A, B, C etc. or numbers 1, 2, 3 etc. also. 
3. Parts of an angl: Angle has three parts : Interior, exterior and the angle itself. 
4. Comparison of Angles: Two angles can be compared with respect to their 
magnitude. Any angle of greater measure is greater. 
5. Kinds of Angles :  
(i) Zero angle: When two rotating rays (sides of angles) coincide each other, then ?ero 
angle is formed. 
 
(ii) Right angle: An angle of 90° is called a right angle. 
 
(iii) Straight Angle: An angle of 180° is called a straight angle. 
 
(iv) Complete Angle: When a ray completes a revolution on rotating it, then a complete 
angle is formed. 
 
(v) Acute angle : An angle less than 90° is called an acute angle. 
 
(vi) Obtuse angle : An angle greater than 90° and less than 180° is called an obtuse 
angle. 
 
(vii) Reflex Angle: An angle greater than the 180° and less than 360° 
Page 3


IMPORTANT POINTS 
1. Ray. A ray is a half-line: It has one end point and other end is open. It can not be 
measured like a line. Here, OA is a ray. 
 
From appoint, infinite numbers of rays can be drawn. 
 
2. Angle: When two rays meet at a point, then an angle is formed. Angle is measured in 
degrees with the help of an instrument known as a protractor. 
The point where the two rays meet is called an initial point or vertex of the angle and the 
two rays which form the angle, are called the sides of the angle e.g., two rays OA and 
OB meet at O. 
 
? Angle AOB is formed. Vertex is always kept between the ends points. 
The anlgle can be denotes as ?AOB, or ?BOA, here sign ‘ ?’ denotes the angle. 
The angle is also denoted with letters A, B, C etc. or numbers 1, 2, 3 etc. also. 
3. Parts of an angl: Angle has three parts : Interior, exterior and the angle itself. 
4. Comparison of Angles: Two angles can be compared with respect to their 
magnitude. Any angle of greater measure is greater. 
5. Kinds of Angles :  
(i) Zero angle: When two rotating rays (sides of angles) coincide each other, then ?ero 
angle is formed. 
 
(ii) Right angle: An angle of 90° is called a right angle. 
 
(iii) Straight Angle: An angle of 180° is called a straight angle. 
 
(iv) Complete Angle: When a ray completes a revolution on rotating it, then a complete 
angle is formed. 
 
(v) Acute angle : An angle less than 90° is called an acute angle. 
 
(vi) Obtuse angle : An angle greater than 90° and less than 180° is called an obtuse 
angle. 
 
(vii) Reflex Angle: An angle greater than the 180° and less than 360° 
 
Note: 
1 ° = 60 minutes (60') 
1' = 60 seconds (60?) 
6. Pairs of angles : 
(i) Adjacent Angles: Two angles with same vertex and one common arm and the other 
arms lying in opposite sides of it are called adjacent angles, ?AOB and ?BOC are 
adjacent angles. 
 
(ii) Linear Pair: A linear pair is a pair of adjacent angles whose sum is equal to 180°. 
?AOB and ?BOC are a linear pair as ?AOB + ?BOC = 180°. 
 
(iii) Complementary Angle: Two angles whose sum is 90° are called complementary 
angles. ?ABC and ?PQR are complementary angles as ?ABC + ?PQR = 30° + 60° = 
90°. 
 
(iv) Supplementary Angles: Two angles whose sum is 180°, are called supplementary 
Page 4


IMPORTANT POINTS 
1. Ray. A ray is a half-line: It has one end point and other end is open. It can not be 
measured like a line. Here, OA is a ray. 
 
From appoint, infinite numbers of rays can be drawn. 
 
2. Angle: When two rays meet at a point, then an angle is formed. Angle is measured in 
degrees with the help of an instrument known as a protractor. 
The point where the two rays meet is called an initial point or vertex of the angle and the 
two rays which form the angle, are called the sides of the angle e.g., two rays OA and 
OB meet at O. 
 
? Angle AOB is formed. Vertex is always kept between the ends points. 
The anlgle can be denotes as ?AOB, or ?BOA, here sign ‘ ?’ denotes the angle. 
The angle is also denoted with letters A, B, C etc. or numbers 1, 2, 3 etc. also. 
3. Parts of an angl: Angle has three parts : Interior, exterior and the angle itself. 
4. Comparison of Angles: Two angles can be compared with respect to their 
magnitude. Any angle of greater measure is greater. 
5. Kinds of Angles :  
(i) Zero angle: When two rotating rays (sides of angles) coincide each other, then ?ero 
angle is formed. 
 
(ii) Right angle: An angle of 90° is called a right angle. 
 
(iii) Straight Angle: An angle of 180° is called a straight angle. 
 
(iv) Complete Angle: When a ray completes a revolution on rotating it, then a complete 
angle is formed. 
 
(v) Acute angle : An angle less than 90° is called an acute angle. 
 
(vi) Obtuse angle : An angle greater than 90° and less than 180° is called an obtuse 
angle. 
 
(vii) Reflex Angle: An angle greater than the 180° and less than 360° 
 
Note: 
1 ° = 60 minutes (60') 
1' = 60 seconds (60?) 
6. Pairs of angles : 
(i) Adjacent Angles: Two angles with same vertex and one common arm and the other 
arms lying in opposite sides of it are called adjacent angles, ?AOB and ?BOC are 
adjacent angles. 
 
(ii) Linear Pair: A linear pair is a pair of adjacent angles whose sum is equal to 180°. 
?AOB and ?BOC are a linear pair as ?AOB + ?BOC = 180°. 
 
(iii) Complementary Angle: Two angles whose sum is 90° are called complementary 
angles. ?ABC and ?PQR are complementary angles as ?ABC + ?PQR = 30° + 60° = 
90°. 
 
(iv) Supplementary Angles: Two angles whose sum is 180°, are called supplementary 
angles. ?ABC and ?PQR are supplementary angles, because 
?ABC + ?PQR = 130° + 50° = 180°. 
 
(v) Vertically Opposite Angles: When two lines intersect each other, then the pairs of 
opposite angles so formed are called vertically opposite angles. 
?1 and ?2 are vertically opposite angles. Similarly, ?3 and ?4 are vertically opposite 
angles. 
 
EXERCISE 24 (A) 
Question 1. 
For each angle given below, write the name of the vertex, the names of the arms 
and the name of the angle. 
 
Solution: 
Page 5


IMPORTANT POINTS 
1. Ray. A ray is a half-line: It has one end point and other end is open. It can not be 
measured like a line. Here, OA is a ray. 
 
From appoint, infinite numbers of rays can be drawn. 
 
2. Angle: When two rays meet at a point, then an angle is formed. Angle is measured in 
degrees with the help of an instrument known as a protractor. 
The point where the two rays meet is called an initial point or vertex of the angle and the 
two rays which form the angle, are called the sides of the angle e.g., two rays OA and 
OB meet at O. 
 
? Angle AOB is formed. Vertex is always kept between the ends points. 
The anlgle can be denotes as ?AOB, or ?BOA, here sign ‘ ?’ denotes the angle. 
The angle is also denoted with letters A, B, C etc. or numbers 1, 2, 3 etc. also. 
3. Parts of an angl: Angle has three parts : Interior, exterior and the angle itself. 
4. Comparison of Angles: Two angles can be compared with respect to their 
magnitude. Any angle of greater measure is greater. 
5. Kinds of Angles :  
(i) Zero angle: When two rotating rays (sides of angles) coincide each other, then ?ero 
angle is formed. 
 
(ii) Right angle: An angle of 90° is called a right angle. 
 
(iii) Straight Angle: An angle of 180° is called a straight angle. 
 
(iv) Complete Angle: When a ray completes a revolution on rotating it, then a complete 
angle is formed. 
 
(v) Acute angle : An angle less than 90° is called an acute angle. 
 
(vi) Obtuse angle : An angle greater than 90° and less than 180° is called an obtuse 
angle. 
 
(vii) Reflex Angle: An angle greater than the 180° and less than 360° 
 
Note: 
1 ° = 60 minutes (60') 
1' = 60 seconds (60?) 
6. Pairs of angles : 
(i) Adjacent Angles: Two angles with same vertex and one common arm and the other 
arms lying in opposite sides of it are called adjacent angles, ?AOB and ?BOC are 
adjacent angles. 
 
(ii) Linear Pair: A linear pair is a pair of adjacent angles whose sum is equal to 180°. 
?AOB and ?BOC are a linear pair as ?AOB + ?BOC = 180°. 
 
(iii) Complementary Angle: Two angles whose sum is 90° are called complementary 
angles. ?ABC and ?PQR are complementary angles as ?ABC + ?PQR = 30° + 60° = 
90°. 
 
(iv) Supplementary Angles: Two angles whose sum is 180°, are called supplementary 
angles. ?ABC and ?PQR are supplementary angles, because 
?ABC + ?PQR = 130° + 50° = 180°. 
 
(v) Vertically Opposite Angles: When two lines intersect each other, then the pairs of 
opposite angles so formed are called vertically opposite angles. 
?1 and ?2 are vertically opposite angles. Similarly, ?3 and ?4 are vertically opposite 
angles. 
 
EXERCISE 24 (A) 
Question 1. 
For each angle given below, write the name of the vertex, the names of the arms 
and the name of the angle. 
 
Solution: 
(i) In figure (i) O is the vertex, OA, OB are its arms and name of the angle is ?AOB 
or ?BOA or simply ?O. 
(ii) In figure (ii) Q is the vertex, QP and QR its arms and the name of the angle is ?PQR 
or ?RQP or simply ?Q. 
(iii) In figure (iii), M is the vertex, MN and ML and its anus, and name of the angle is 
?LMN or ?NML or simply ?M. 
P .Q . Name the angles marked by letters a, b, c, x and y. 
 
Solution: 
a = AOE, b = ?AOB, c = ?BOC d = ?COD e= ?DOE 
Question 2. 
Name the points : 
(i) in the interior of the angle PQR, 
(ii) in the exterior of the angle PQR. 
 
Solution: 
(i) a, b and x 
(ii) d, m, n, s, and t. 
Question 3. 
In the given figure, figure out the number of angles formed within the arms OA 
and OE.