Selina Textbook Solutions: Properties of Angles and Lines | Mathematics Class 6 ICSE PDF Download

 Page 1


IMPORTANT POINTS 
1. Property : When two straight lines intersect: 
(i) sum of each pair of adjacent angles is always 180°. 
(ii) vertically opposite angles are always equal. . 
2. Property : If the sum of two adjacent angles is 180°, their exterior arms are always in 
the same straight line. 
Conversely, if the exterior arms of two adjacent angles are in the same straight line ; the 
sum of angles is always 180° 
3. Parallel Lines : Two straight lines are said to be parallel, if they do not meet 
anywhere, no matter how much they are produced in either direction. 
 
4. Concepts of Transversal Lines : When a line cuts two or more lines (parallel or 
non-parallel); it is called a transversal line or simply, a transversal. In each of the 
following figures : PQ is a transversal line. 
 
5. Angles formed by two lines and their transversal line : When a transversal cuts 
two parallel or nonparallel lines; eight (8) angles are formed which are marked 1 to 8 in 
the adjoining diagram. 
These angles can further he distinguished, as given below: 
Page 2


IMPORTANT POINTS 
1. Property : When two straight lines intersect: 
(i) sum of each pair of adjacent angles is always 180°. 
(ii) vertically opposite angles are always equal. . 
2. Property : If the sum of two adjacent angles is 180°, their exterior arms are always in 
the same straight line. 
Conversely, if the exterior arms of two adjacent angles are in the same straight line ; the 
sum of angles is always 180° 
3. Parallel Lines : Two straight lines are said to be parallel, if they do not meet 
anywhere, no matter how much they are produced in either direction. 
 
4. Concepts of Transversal Lines : When a line cuts two or more lines (parallel or 
non-parallel); it is called a transversal line or simply, a transversal. In each of the 
following figures : PQ is a transversal line. 
 
5. Angles formed by two lines and their transversal line : When a transversal cuts 
two parallel or nonparallel lines; eight (8) angles are formed which are marked 1 to 8 in 
the adjoining diagram. 
These angles can further he distinguished, as given below: 
 
(i) Exterior Angles : Angles marked 1, 2, 7 and 8 are exterior angles. 
(ii) Interior Angles : Angles marked 3, 4, 5 and 6 are interior angles. 
(iii) Exterior Alternates Angles : Two pairs of exterior alternate angles are marked as : 
2 and 8 ; and, 1 and 7. 
(iv) Interior Alternate Angles : Two pairs of interior alternate are marked as : 3 and 5 ; 
and 4 and 6. In general, interior alternate angles are simply called as alternate angles 
only. 
(v) Corresponding Angles : Four pairs of corresponding angles are marked as : 1 and 
5 ; 2 and 6 ; 3 and 7 ; and 4 and 8. 
(vi) Co-interior or Conjoined or Allied Angles : Two pairs of co-interior or allied 
angles are marked as : 3 and 6 ; and 4 and 5. 
(vii) Exterior Allied Angles : Two pairs of exterior allied angles are marked as : 2 and 
7 ; and 1 and 8. 
EXERCISE 25 (A) 
Question 1. 
Two straight lines AB and CD intersect each other at a point O and angle AOC = 
50° ; find : 
(i) angle BOD 
(ii) ?AOD 
(iii) ?BOC 
 
Solution: 
(i) ?BOD = ?AOC 
(Vertically opposite angles are equal) 
Page 3


IMPORTANT POINTS 
1. Property : When two straight lines intersect: 
(i) sum of each pair of adjacent angles is always 180°. 
(ii) vertically opposite angles are always equal. . 
2. Property : If the sum of two adjacent angles is 180°, their exterior arms are always in 
the same straight line. 
Conversely, if the exterior arms of two adjacent angles are in the same straight line ; the 
sum of angles is always 180° 
3. Parallel Lines : Two straight lines are said to be parallel, if they do not meet 
anywhere, no matter how much they are produced in either direction. 
 
4. Concepts of Transversal Lines : When a line cuts two or more lines (parallel or 
non-parallel); it is called a transversal line or simply, a transversal. In each of the 
following figures : PQ is a transversal line. 
 
5. Angles formed by two lines and their transversal line : When a transversal cuts 
two parallel or nonparallel lines; eight (8) angles are formed which are marked 1 to 8 in 
the adjoining diagram. 
These angles can further he distinguished, as given below: 
 
(i) Exterior Angles : Angles marked 1, 2, 7 and 8 are exterior angles. 
(ii) Interior Angles : Angles marked 3, 4, 5 and 6 are interior angles. 
(iii) Exterior Alternates Angles : Two pairs of exterior alternate angles are marked as : 
2 and 8 ; and, 1 and 7. 
(iv) Interior Alternate Angles : Two pairs of interior alternate are marked as : 3 and 5 ; 
and 4 and 6. In general, interior alternate angles are simply called as alternate angles 
only. 
(v) Corresponding Angles : Four pairs of corresponding angles are marked as : 1 and 
5 ; 2 and 6 ; 3 and 7 ; and 4 and 8. 
(vi) Co-interior or Conjoined or Allied Angles : Two pairs of co-interior or allied 
angles are marked as : 3 and 6 ; and 4 and 5. 
(vii) Exterior Allied Angles : Two pairs of exterior allied angles are marked as : 2 and 
7 ; and 1 and 8. 
EXERCISE 25 (A) 
Question 1. 
Two straight lines AB and CD intersect each other at a point O and angle AOC = 
50° ; find : 
(i) angle BOD 
(ii) ?AOD 
(iii) ?BOC 
 
Solution: 
(i) ?BOD = ?AOC 
(Vertically opposite angles are equal) 
? ?BOD =50° 
(ii) ?AOD 
?AOD + ?BOD = 180° 
?AOD + 50° = 180° [From (i)] 
?AOD = 180°-50° 
?AOD = 130° 
(iii) ?BOC = ?AOD 
(Vertically opposite angles are equal) 
? ?BOC =130° 
Question 2. 
The adjoining figure, shows two straight lines AB and CD intersecting at point P. 
If ?BPC = 4x – 5° and ?APD = 3x + 15°; find : 
 
(i) the value of x. 
(ii) ?APD 
(iii) ?BPD 
(iv) ?BPC 
Solution: 
Question 3. 
The given diagram, shows two adjacent angles AOB and AOC, whose exterior 
sides are along the same straight line. Find the value of x. 
 
Solution: 
Since, the exterior arms of the adjacent angles are in a straight line ; the adjacent 
angles are supplementary 
? ?AOB + ?AOC = 180° 
? 68° + 3x – 20° = 180° 
? 3x = 180° + 20° – 68° 
Page 4


IMPORTANT POINTS 
1. Property : When two straight lines intersect: 
(i) sum of each pair of adjacent angles is always 180°. 
(ii) vertically opposite angles are always equal. . 
2. Property : If the sum of two adjacent angles is 180°, their exterior arms are always in 
the same straight line. 
Conversely, if the exterior arms of two adjacent angles are in the same straight line ; the 
sum of angles is always 180° 
3. Parallel Lines : Two straight lines are said to be parallel, if they do not meet 
anywhere, no matter how much they are produced in either direction. 
 
4. Concepts of Transversal Lines : When a line cuts two or more lines (parallel or 
non-parallel); it is called a transversal line or simply, a transversal. In each of the 
following figures : PQ is a transversal line. 
 
5. Angles formed by two lines and their transversal line : When a transversal cuts 
two parallel or nonparallel lines; eight (8) angles are formed which are marked 1 to 8 in 
the adjoining diagram. 
These angles can further he distinguished, as given below: 
 
(i) Exterior Angles : Angles marked 1, 2, 7 and 8 are exterior angles. 
(ii) Interior Angles : Angles marked 3, 4, 5 and 6 are interior angles. 
(iii) Exterior Alternates Angles : Two pairs of exterior alternate angles are marked as : 
2 and 8 ; and, 1 and 7. 
(iv) Interior Alternate Angles : Two pairs of interior alternate are marked as : 3 and 5 ; 
and 4 and 6. In general, interior alternate angles are simply called as alternate angles 
only. 
(v) Corresponding Angles : Four pairs of corresponding angles are marked as : 1 and 
5 ; 2 and 6 ; 3 and 7 ; and 4 and 8. 
(vi) Co-interior or Conjoined or Allied Angles : Two pairs of co-interior or allied 
angles are marked as : 3 and 6 ; and 4 and 5. 
(vii) Exterior Allied Angles : Two pairs of exterior allied angles are marked as : 2 and 
7 ; and 1 and 8. 
EXERCISE 25 (A) 
Question 1. 
Two straight lines AB and CD intersect each other at a point O and angle AOC = 
50° ; find : 
(i) angle BOD 
(ii) ?AOD 
(iii) ?BOC 
 
Solution: 
(i) ?BOD = ?AOC 
(Vertically opposite angles are equal) 
? ?BOD =50° 
(ii) ?AOD 
?AOD + ?BOD = 180° 
?AOD + 50° = 180° [From (i)] 
?AOD = 180°-50° 
?AOD = 130° 
(iii) ?BOC = ?AOD 
(Vertically opposite angles are equal) 
? ?BOC =130° 
Question 2. 
The adjoining figure, shows two straight lines AB and CD intersecting at point P. 
If ?BPC = 4x – 5° and ?APD = 3x + 15°; find : 
 
(i) the value of x. 
(ii) ?APD 
(iii) ?BPD 
(iv) ?BPC 
Solution: 
Question 3. 
The given diagram, shows two adjacent angles AOB and AOC, whose exterior 
sides are along the same straight line. Find the value of x. 
 
Solution: 
Since, the exterior arms of the adjacent angles are in a straight line ; the adjacent 
angles are supplementary 
? ?AOB + ?AOC = 180° 
? 68° + 3x – 20° = 180° 
? 3x = 180° + 20° – 68° 
? 3x = 200° – 68° ? 3x =132° 
x = ° = 44° 
Question 4. 
Each figure given below shows a pair of adjacent angles AOB and BOC. Find 
whether or not the exterior arms OA and OC are in the same straight line. 
 
 
Solution: 
(i) ?AOB + ?COB = 180° 
Since, the sum of adjacent angles AOB and COB = 180° 
(90° -x) + (90°+ x) = 180° 
? 90°-x + 90° + x = 180° 
? 180° =180° 
The exterior arms. OA and OC are in the same straight line. 
(ii) ?AOB + ?BOC = 97° + 83° = 180° 
? The sum of adjacent angles AOB and BOC is 180°. 
? The exterior arms OA and OC are in the same straight line. 
(iii)?COB + ?AOB = 88° + 112° = 200° ; which is not 180°. 
? The exterior amis OA and OC are not in the same straight line. 
Page 5


IMPORTANT POINTS 
1. Property : When two straight lines intersect: 
(i) sum of each pair of adjacent angles is always 180°. 
(ii) vertically opposite angles are always equal. . 
2. Property : If the sum of two adjacent angles is 180°, their exterior arms are always in 
the same straight line. 
Conversely, if the exterior arms of two adjacent angles are in the same straight line ; the 
sum of angles is always 180° 
3. Parallel Lines : Two straight lines are said to be parallel, if they do not meet 
anywhere, no matter how much they are produced in either direction. 
 
4. Concepts of Transversal Lines : When a line cuts two or more lines (parallel or 
non-parallel); it is called a transversal line or simply, a transversal. In each of the 
following figures : PQ is a transversal line. 
 
5. Angles formed by two lines and their transversal line : When a transversal cuts 
two parallel or nonparallel lines; eight (8) angles are formed which are marked 1 to 8 in 
the adjoining diagram. 
These angles can further he distinguished, as given below: 
 
(i) Exterior Angles : Angles marked 1, 2, 7 and 8 are exterior angles. 
(ii) Interior Angles : Angles marked 3, 4, 5 and 6 are interior angles. 
(iii) Exterior Alternates Angles : Two pairs of exterior alternate angles are marked as : 
2 and 8 ; and, 1 and 7. 
(iv) Interior Alternate Angles : Two pairs of interior alternate are marked as : 3 and 5 ; 
and 4 and 6. In general, interior alternate angles are simply called as alternate angles 
only. 
(v) Corresponding Angles : Four pairs of corresponding angles are marked as : 1 and 
5 ; 2 and 6 ; 3 and 7 ; and 4 and 8. 
(vi) Co-interior or Conjoined or Allied Angles : Two pairs of co-interior or allied 
angles are marked as : 3 and 6 ; and 4 and 5. 
(vii) Exterior Allied Angles : Two pairs of exterior allied angles are marked as : 2 and 
7 ; and 1 and 8. 
EXERCISE 25 (A) 
Question 1. 
Two straight lines AB and CD intersect each other at a point O and angle AOC = 
50° ; find : 
(i) angle BOD 
(ii) ?AOD 
(iii) ?BOC 
 
Solution: 
(i) ?BOD = ?AOC 
(Vertically opposite angles are equal) 
? ?BOD =50° 
(ii) ?AOD 
?AOD + ?BOD = 180° 
?AOD + 50° = 180° [From (i)] 
?AOD = 180°-50° 
?AOD = 130° 
(iii) ?BOC = ?AOD 
(Vertically opposite angles are equal) 
? ?BOC =130° 
Question 2. 
The adjoining figure, shows two straight lines AB and CD intersecting at point P. 
If ?BPC = 4x – 5° and ?APD = 3x + 15°; find : 
 
(i) the value of x. 
(ii) ?APD 
(iii) ?BPD 
(iv) ?BPC 
Solution: 
Question 3. 
The given diagram, shows two adjacent angles AOB and AOC, whose exterior 
sides are along the same straight line. Find the value of x. 
 
Solution: 
Since, the exterior arms of the adjacent angles are in a straight line ; the adjacent 
angles are supplementary 
? ?AOB + ?AOC = 180° 
? 68° + 3x – 20° = 180° 
? 3x = 180° + 20° – 68° 
? 3x = 200° – 68° ? 3x =132° 
x = ° = 44° 
Question 4. 
Each figure given below shows a pair of adjacent angles AOB and BOC. Find 
whether or not the exterior arms OA and OC are in the same straight line. 
 
 
Solution: 
(i) ?AOB + ?COB = 180° 
Since, the sum of adjacent angles AOB and COB = 180° 
(90° -x) + (90°+ x) = 180° 
? 90°-x + 90° + x = 180° 
? 180° =180° 
The exterior arms. OA and OC are in the same straight line. 
(ii) ?AOB + ?BOC = 97° + 83° = 180° 
? The sum of adjacent angles AOB and BOC is 180°. 
? The exterior arms OA and OC are in the same straight line. 
(iii)?COB + ?AOB = 88° + 112° = 200° ; which is not 180°. 
? The exterior amis OA and OC are not in the same straight line. 
Question 5. 
A line segment AP stands at point P of a straight line BC such that ?APB = 5x – 
40° and ?APC = .x+ 10°; find the value of x and angle APB. 
Solution: 
AP stands on BC at P and 
?APB = 5x – 40°, ?APC = x + 10° 
 
(i) ?APE is a straight line 
?APB + ?APC = 180° 
? 5x – 40° + x + 10° = 180° 
? 6x-30°= 180° 
?6x= 180° + 30° = 210° 
x = ° = 35° 
(ii) and ?APB = 5x – 40° = 5 x 35° – 40° 
= 175 ° – 140° = 135° 
EXERCISE 25 (B) 
Question 1. 
Identify the pair of angles in each of the figure given below : 
adjacent angles, vertically opposite angles, interior alternate angles, 
corresponding angles or exterior alternate angles.