Parallel Lines And Transversals
Class 8th Mathematics (new) MHB Solution
Practice Set 2.1- In the adjoining figure, each angle is shown by a letter. Fill in the boxes with the…
- Observe the angles shown in the figure and write the following pair of angles.(1)…
Practice Set 2.2- In the adjoining figure, if line m || line n and line p is a transversal then find x.…
- In the adjoining figure, if line a || line b and line l is a transversal then find x.…
- In the adjoining figure line p || line q. Line t and line s are transversals. Find the…
- In the adjoining figure. line p || line q. line l || line m. Find measures of ∠a, ∠b,…
- In the adjoining figure, line a || line b. line l is a transversal. Find the measures…
- In the adjoining figure, line p || line l || line q. Find ∠x with the help of the…
Practice Set 2.3
- In the adjoining figure, each angle is shown by a letter. Fill in the boxes with the…
- Observe the angles shown in the figure and write the following pair of angles.(1)…
- In the adjoining figure, if line m || line n and line p is a transversal then find x.…
- In the adjoining figure, if line a || line b and line l is a transversal then find x.…
- In the adjoining figure line p || line q. Line t and line s are transversals. Find the…
- In the adjoining figure. line p || line q. line l || line m. Find measures of ∠a, ∠b,…
- In the adjoining figure, line a || line b. line l is a transversal. Find the measures…
- In the adjoining figure, line p || line l || line q. Find ∠x with the help of the…
Practice Set 2.1
Question 1.In the adjoining figure, each angle is shown by a letter. Fill in the boxes with the help of the figure.
Corresponding angles.
(1) ∠p and [ ] (2) ∠q and [ ]
(3) ∠r and [ ] (4) ∠ s and [ ]
Interior alternate angles.
(5) ∠s and [ ] (6) ∠w and [ ]
Answer:• Given: Line q is transversal is to line m and line l.
• To find corresponding angles of
1) ∠ p
2) ∠ q
3) ∠ r
4) ∠ s
• Explanation:
If we go by the definition, the definition of corresponding angels tells us, if the arms on the transversal of a pair of angles are in the same direction and the other arms are on the same side of the transversal, then it is called a pair of corresponding angles.
So, now in the above given figure we have say, line q making transversal to line m and line l.
1) For ∠p, ∠w is the angle which is in the same side and same direction of transversal so ∠w is the corresponding angle to ∠p.
2) For ∠q, ∠x is the angle which is in the same side and same direction of transversal so ∠x is the corresponding angle to ∠q.
3) For ∠r, ∠y is the angle which is in the same side and same direction of transversal so ∠r is the corresponding angle to ∠y.
4) For ∠s, ∠z is the angle which is in the same side and same direction of transversal so ∠s is the corresponding angle to ∠z.
Now for Interior Alternate angles
Pairs of angles which are on the opposite sides of transversal and their arms on the transversal show opposite directions is called a pair of alternate angles.
When these angels are in the inner side they are called Interior alternate angels.
5) For ∠s the angel which is in the inner side as well as on the opposite side of transversal and it’s arm show opposite direction is ∠x. So ∠s and ∠x form pair of Interior Alternate angel.
6) For ∠w the angel which is in the inner side as well as on the opposite side of transversal and it’s arm show opposite direction is ∠r. So ∠w and ∠r form pair of Interior Alternate angel.
Question 2.Observe the angles shown in the figure and write the following pair of angles.
(1) Interior alternate angles
(2) Corresponding angles
(3) Interior angles
Answer:• Given: Line q is transversal is to line m and line l.
• To find: (1) Interior alternate angles
(2) Corresponding angles
(3) Interior angles
(1) Now for Interior Alternate angles
Pairs of angles which are on the opposite sides of transversal and their arms on the transversal show opposite directions is called a pair of alternate angles.
When these angels are in the inner side they are called Interior alternate angels.
1) For ∠b the angle which is in the inner side as well as on the opposite side of transversal and it’s arm show opposite direction is ∠h. So ∠b and ∠h form pair of Interior Alternate angel.
2) For ∠c the angel which is in the inner side as well as on the opposite side of transversal and it’s arm show opposite direction is ∠e. So ∠c and ∠e form pair of Interior Alternate angel.
(2) Corresponding angles
If we go by the definition, the definition of corresponding angels tells us, if the arms on the transversal of a pair of angles are in the same direction and the other arms are on the same side of the transversal, then it is called a pair of corresponding angles.
So, now in the above given figure we have say, line q making transversal to line m and line l.
1)For ∠a, ∠e is the angle which is in the same side and same direction of transversal so ∠a is the corresponding angle to ∠e.
2)For ∠b, ∠f is the angle which is in the same side and same direction of transversal so ∠b is the corresponding angle to ∠f.
3)For ∠d, ∠h is the angle which is in the same side and same direction of transversal so ∠d is the corresponding angle to ∠h.
4)For ∠c, ∠g is the angle which is in the same side and same direction of transversal so ∠c is the corresponding angle to ∠g.
(3) Interior angles
A pair of angles which are on the same side of the transversal and inside the given lines is called a pair of interior angles.
So, we get only two such pairs of angels.
1) ∠b has ∠e on the same side of transversal and inside the given line. So ∠b and ∠e form pair of interior angels.
2) ∠c has ∠h on the same side of transversal and inside the given line. So ∠c and ∠h form pair of interior angels.
In the adjoining figure, each angle is shown by a letter. Fill in the boxes with the help of the figure.
Corresponding angles.
(1) ∠p and [ ] (2) ∠q and [ ]
(3) ∠r and [ ] (4) ∠ s and [ ]
Interior alternate angles.
(5) ∠s and [ ] (6) ∠w and [ ]
Answer:
• Given: Line q is transversal is to line m and line l.
• To find corresponding angles of
1) ∠ p
2) ∠ q
3) ∠ r
4) ∠ s
• Explanation:
If we go by the definition, the definition of corresponding angels tells us, if the arms on the transversal of a pair of angles are in the same direction and the other arms are on the same side of the transversal, then it is called a pair of corresponding angles.
So, now in the above given figure we have say, line q making transversal to line m and line l.
1) For ∠p, ∠w is the angle which is in the same side and same direction of transversal so ∠w is the corresponding angle to ∠p.
2) For ∠q, ∠x is the angle which is in the same side and same direction of transversal so ∠x is the corresponding angle to ∠q.
3) For ∠r, ∠y is the angle which is in the same side and same direction of transversal so ∠r is the corresponding angle to ∠y.
4) For ∠s, ∠z is the angle which is in the same side and same direction of transversal so ∠s is the corresponding angle to ∠z.
Now for Interior Alternate angles
Pairs of angles which are on the opposite sides of transversal and their arms on the transversal show opposite directions is called a pair of alternate angles.
When these angels are in the inner side they are called Interior alternate angels.
5) For ∠s the angel which is in the inner side as well as on the opposite side of transversal and it’s arm show opposite direction is ∠x. So ∠s and ∠x form pair of Interior Alternate angel.
6) For ∠w the angel which is in the inner side as well as on the opposite side of transversal and it’s arm show opposite direction is ∠r. So ∠w and ∠r form pair of Interior Alternate angel.
Question 2.
Observe the angles shown in the figure and write the following pair of angles.
(1) Interior alternate angles
(2) Corresponding angles
(3) Interior angles
Answer:
• Given: Line q is transversal is to line m and line l.
• To find: (1) Interior alternate angles
(2) Corresponding angles
(3) Interior angles
(1) Now for Interior Alternate angles
Pairs of angles which are on the opposite sides of transversal and their arms on the transversal show opposite directions is called a pair of alternate angles.
When these angels are in the inner side they are called Interior alternate angels.
1) For ∠b the angle which is in the inner side as well as on the opposite side of transversal and it’s arm show opposite direction is ∠h. So ∠b and ∠h form pair of Interior Alternate angel.
2) For ∠c the angel which is in the inner side as well as on the opposite side of transversal and it’s arm show opposite direction is ∠e. So ∠c and ∠e form pair of Interior Alternate angel.
(2) Corresponding angles
If we go by the definition, the definition of corresponding angels tells us, if the arms on the transversal of a pair of angles are in the same direction and the other arms are on the same side of the transversal, then it is called a pair of corresponding angles.
So, now in the above given figure we have say, line q making transversal to line m and line l.
1)For ∠a, ∠e is the angle which is in the same side and same direction of transversal so ∠a is the corresponding angle to ∠e.
2)For ∠b, ∠f is the angle which is in the same side and same direction of transversal so ∠b is the corresponding angle to ∠f.
3)For ∠d, ∠h is the angle which is in the same side and same direction of transversal so ∠d is the corresponding angle to ∠h.
4)For ∠c, ∠g is the angle which is in the same side and same direction of transversal so ∠c is the corresponding angle to ∠g.
(3) Interior angles
A pair of angles which are on the same side of the transversal and inside the given lines is called a pair of interior angles.
So, we get only two such pairs of angels.
1) ∠b has ∠e on the same side of transversal and inside the given line. So ∠b and ∠e form pair of interior angels.
2) ∠c has ∠h on the same side of transversal and inside the given line. So ∠c and ∠h form pair of interior angels.
Practice Set 2.2
Question 1.Choose the correct alternative.
In the adjoining figure, if line m || line n and line p is a transversal then find x.
A. 135°
B. 90°
C. 45°
D. 40°
Answer:• Given: Line m || line n and line p is a transversal
• To find: The value of x.
Now in the given figure we have 3x and x. 3x and x form a pair of interior angle.
Now by the property of interior angels we know that, each pair of interior angles formed by two parallel lines and their transversal is of supplementary angles i.e. 180°.
∴ x+3x=180(∵ Property of interior angles.)
⇒ 4x=180
⇒ 
∴ The value of x is 45°.
Question 2.Choose the correct alternative.
In the adjoining figure, if line a || line b and line l is a transversal then find x.
A. 90°
B. 60°
C. 45°
D. 30°
Answer:• Given: Line a || line b and line l is a transversal
• To find: Value of x.
Now from the figure we can see
∠GHB=∠CHF (∵ opposite angles are same)
⇒ ∠GHB=2x
∠AGB+∠BHG=180° (∵ (∵ Property of interior angles.)
∴ 4x+2x=180°
⇒ 6x=180°
⇒ 
∴ The value of x is 30°
Option(D)
Question 3.In the adjoining figure line p || line q. Line t and line s are transversals. Find the measure of ∠x and ∠y using the measures of angles given in the figure.
Answer:• Given: Line p || line q, line t and line s are transversals.
• To find: The measure of ∠x and ∠y.
Here we can see
∠KLD=∠HLD (∵ Opposite angles are equal)
⇒ ∠ KLD=70°
∠KLI+∠JIL=180° (∵ (∵ Property of interior angles.)
∴ 70+Y=180°
⇒ Y=110°
∴ The value of y is 110°
Also,
∠ BKL+∠ JKL =180° (Linear pair)
⇒ x+∠JKL = 180°
∴ ∠ JKL= 180-x…(1)
∠ KJI+∠ AJI =180° (Linear pair)
⇒ 40+∠KJI = 180°
∴ ∠ KJI= 140°…(2)
Now,
∠KJI+∠JKL=180° (∵ Property of interior angles.)
140+180-x=180 (From 1 and 2)
⇒-x=180-180-140
⇒ -x=-140°
∴ x=140°
∴ The value of x is 140°.
Question 4.In the adjoining figure. line p || line q. line l || line m. Find measures of ∠a, ∠b, and ∠c, using the measures of given angles. Justify your answers.
Answer:• Given: Line p || line q, line l || line m.
• To find: The measure of ∠a, ∠b and ∠c.
Now in this figure
∠CIJ+∠AJI=180° (∵ Exterior angles are supplementary)
⇒ 80° +a=180°
⇒ a = 100°
Also,
∠AJI+∠IJL=180° (Linear pair)
⇒100° +∠IJL=180°
∴ ∠IJL=80°
∠BLK=∠LJI=b(corresponding angles are equal)
∴ b=80°
∠EIK=∠CIJ=80° (Opposite angles are equal)
∠GKD=∠EIK=c(corresponding angles are equal)
∴ c=80°
∴ Values of a,b and c are 100° , 80° ,80° respectively.
Question 5.In the adjoining figure, line a || line b. line l is a transversal. Find the measures of ∠x, ∠y, ∠z using the given information.
Answer:• Given: Line a || line b, line l is transversal.
• To find: The measure of ∠x, ∠y and ∠z.
In, the figure above
∠AGE=z
Also,
∠AGE+∠EGB=180° (Linear pair)
z+105=180°
z=75°
∠GHD=x
∠EGB=∠GHD(corresponding angles are equal)
∴ x=105°
∠DHF=y
Also
∠DHF=∠GHD (Opposite angles are equal)
∠DHF=105°
∴ Values of x, y and z are 105° ,105° ,75° respectively.
Question 6.In the adjoining figure, line p || line l || line q. Find ∠x with the help of the measures given in the figure.
Answer:• Given: line p || line l || line q.
• To find: Value x
Now, In the above figure ∠GHD=x
Also,
∠GHD=∠GHD+∠IHD
∠AGH=∠GHD (∵ Alternate angles are equal)…(1)
∠EIH=∠DHI (∵ Alternate angles are equal)…(2)
From (1) and (2) we get,
∠GHD=40°
∠DHI=30°
We know
∠GHD=∠GHD+∠IHD
∠GHD= 40°+30°
∠GHD= 70°
But,
∠ GHD = x
∴ x=70°
∴ The value of x is 70°.
Choose the correct alternative.
In the adjoining figure, if line m || line n and line p is a transversal then find x.
A. 135°
B. 90°
C. 45°
D. 40°
Answer:
• Given: Line m || line n and line p is a transversal
• To find: The value of x.
Now in the given figure we have 3x and x. 3x and x form a pair of interior angle.
Now by the property of interior angels we know that, each pair of interior angles formed by two parallel lines and their transversal is of supplementary angles i.e. 180°.
∴ x+3x=180(∵ Property of interior angles.)
⇒ 4x=180
⇒
∴ The value of x is 45°.
Question 2.
Choose the correct alternative.
In the adjoining figure, if line a || line b and line l is a transversal then find x.
A. 90°
B. 60°
C. 45°
D. 30°
Answer:
• Given: Line a || line b and line l is a transversal
• To find: Value of x.
Now from the figure we can see
∠GHB=∠CHF (∵ opposite angles are same)
⇒ ∠GHB=2x
∠AGB+∠BHG=180° (∵ (∵ Property of interior angles.)
∴ 4x+2x=180°
⇒ 6x=180°
⇒
∴ The value of x is 30°
Option(D)
Question 3.
In the adjoining figure line p || line q. Line t and line s are transversals. Find the measure of ∠x and ∠y using the measures of angles given in the figure.
Answer:
• Given: Line p || line q, line t and line s are transversals.
• To find: The measure of ∠x and ∠y.
Here we can see
∠KLD=∠HLD (∵ Opposite angles are equal)
⇒ ∠ KLD=70°
∠KLI+∠JIL=180° (∵ (∵ Property of interior angles.)
∴ 70+Y=180°
⇒ Y=110°
∴ The value of y is 110°
Also,
∠ BKL+∠ JKL =180° (Linear pair)
⇒ x+∠JKL = 180°
∴ ∠ JKL= 180-x…(1)
∠ KJI+∠ AJI =180° (Linear pair)
⇒ 40+∠KJI = 180°
∴ ∠ KJI= 140°…(2)
Now,
∠KJI+∠JKL=180° (∵ Property of interior angles.)
140+180-x=180 (From 1 and 2)
⇒-x=180-180-140
⇒ -x=-140°
∴ x=140°
∴ The value of x is 140°.
Question 4.
In the adjoining figure. line p || line q. line l || line m. Find measures of ∠a, ∠b, and ∠c, using the measures of given angles. Justify your answers.
Answer:
• Given: Line p || line q, line l || line m.
• To find: The measure of ∠a, ∠b and ∠c.
Now in this figure
∠CIJ+∠AJI=180° (∵ Exterior angles are supplementary)
⇒ 80° +a=180°
⇒ a = 100°
Also,
∠AJI+∠IJL=180° (Linear pair)
⇒100° +∠IJL=180°
∴ ∠IJL=80°
∠BLK=∠LJI=b(corresponding angles are equal)
∴ b=80°
∠EIK=∠CIJ=80° (Opposite angles are equal)
∠GKD=∠EIK=c(corresponding angles are equal)
∴ c=80°
∴ Values of a,b and c are 100° , 80° ,80° respectively.
Question 5.
In the adjoining figure, line a || line b. line l is a transversal. Find the measures of ∠x, ∠y, ∠z using the given information.
Answer:
• Given: Line a || line b, line l is transversal.
• To find: The measure of ∠x, ∠y and ∠z.
In, the figure above
∠AGE=z
Also,
∠AGE+∠EGB=180° (Linear pair)
z+105=180°
z=75°
∠GHD=x
∠EGB=∠GHD(corresponding angles are equal)
∴ x=105°
∠DHF=y
Also
∠DHF=∠GHD (Opposite angles are equal)
∠DHF=105°
∴ Values of x, y and z are 105° ,105° ,75° respectively.
Question 6.
In the adjoining figure, line p || line l || line q. Find ∠x with the help of the measures given in the figure.
Answer:
• Given: line p || line l || line q.
• To find: Value x
Now, In the above figure ∠GHD=x
Also,
∠GHD=∠GHD+∠IHD
∠AGH=∠GHD (∵ Alternate angles are equal)…(1)
∠EIH=∠DHI (∵ Alternate angles are equal)…(2)
From (1) and (2) we get,
∠GHD=40°
∠DHI=30°
We know
∠GHD=∠GHD+∠IHD
∠GHD= 40°+30°
∠GHD= 70°
But,
∠ GHD = x
∴ x=70°
∴ The value of x is 70°.
Practice Set 2.3
Question 1.Draw a line l. Take a point A outside the line. Through point A draw a line parallel to line l.
Answer:Here we need draw a line l then take a point A outside the line and then through that point draw another line parallel to l.
Steps of construction:
1) Draw a line segment of any length. Mark it as CD.
2) Now from any point say P on that segment draw perpendicular at any distance above or below and name that point A.
3) Now take another perpendicular of same length as of AP, and in same direction.
4) Draw a line through those points.
5) This line is parallel to given line l.
Question 2.Draw a line l. Take a point T outside the line. Through point T draw a line parallel to line l.
Answer:Here we need draw a line l then take a point T outside the line and then through that point draw another line parallel to l.
Steps of construction:
1) Draw a line segment of any length. Mark it as CD.
2) Now from any point say P on that segment draw perpendicular at any distance above or below and name that point T.
3) Now take another perpendicular of same length as of TP, and in same direction.
4) Draw a line through those points.
5) This line is parallel to given line l.
Question 3.Draw a line m. Draw a line n which is parallel to line m at a distance of 4 cm from it.
Answer:Method : Draw a line parallel to line l at a distance 4 cm.
Steps of construction :
(1) Draw line l.
(2) Take two points A and B on the line l.
(3) Draw perpendiculars to the line l from points A and B.
(4) On the perpendicular lines take points P and Q at a distance of 4cm from
A and B respectively.
(5) Draw line PQ.
(6) Line PQ is a line parallel to the line l at a distance 4cm.
Draw a line l. Take a point A outside the line. Through point A draw a line parallel to line l.
Answer:
Here we need draw a line l then take a point A outside the line and then through that point draw another line parallel to l.
Steps of construction:
1) Draw a line segment of any length. Mark it as CD.
2) Now from any point say P on that segment draw perpendicular at any distance above or below and name that point A.
3) Now take another perpendicular of same length as of AP, and in same direction.
4) Draw a line through those points.
5) This line is parallel to given line l.
Question 2.
Draw a line l. Take a point T outside the line. Through point T draw a line parallel to line l.
Answer:
Here we need draw a line l then take a point T outside the line and then through that point draw another line parallel to l.
Steps of construction:
1) Draw a line segment of any length. Mark it as CD.
2) Now from any point say P on that segment draw perpendicular at any distance above or below and name that point T.
3) Now take another perpendicular of same length as of TP, and in same direction.
4) Draw a line through those points.
5) This line is parallel to given line l.
Question 3.
Draw a line m. Draw a line n which is parallel to line m at a distance of 4 cm from it.
Answer:
Method : Draw a line parallel to line l at a distance 4 cm.
Steps of construction :
(1) Draw line l.
(2) Take two points A and B on the line l.
(3) Draw perpendiculars to the line l from points A and B.
(4) On the perpendicular lines take points P and Q at a distance of 4cm from
A and B respectively.
(5) Draw line PQ.
(6) Line PQ is a line parallel to the line l at a distance 4cm.
