Chapter 2 - Parallel lines and transversals
Practice set 2.1
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 □
Ans: ∠w
(2) ∠q and □
Ans: ∠x
(3) ∠r and □
Ans: ∠y
(4) ∠s and □
Ans: ∠z
Interior alternate angles.
(5) ∠s and □
Ans: ∠x
(6) ∠w and □
Ans: ∠r
2. Observe the angles shown in the figure and write the following pair of angles.
(1) Interior alternate angles
Ans:
∠c and ∠e
∠b and ∠h
(2) Corresponding angles
Ans:
∠a and ∠e
∠b and ∠f
∠c and ∠g
∠d and ∠h
(3) Interior angles
Ans:
∠c and ∠h
∠b and ∠e
Practice set 2.2
1. Choose the correct alternative.
(1) 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°
Ans: Option (C) : 45°
Solution:
line m || line n and line p is a transversal
m∠BFG + m∠FGD = 180° …[Interior angles]
∴ 3x + x = 180°
∴ 4x = 180°
∴ x = \(\large \frac {180}{4}\)
∴ x = 45°
(2) 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°
Ans: Option (D) : 30°
Solution:
line a || line b and line l is a transversal.
m∠UVS ≅ m∠PUV …[Alternate angles]
∵ m∠PUV = 4x
∵ m∠UVS = 4x
m∠UVS + m∠WVS = 180° …[Angles in a linear pair]
∴ 4x + 2x = 180°
∴ 6x = 180°
∴ x = \(\large \frac {180}{6}\)
∴ x = 30°
2. In the adjoining figure line p || line q. Line t and line s are transversals. Find measure of ∠x and ∠y using the measures of angles given in the figure.
Solution:
Consider ∠z as shown in figure.
line p || line q and line t is a transversal.
∴ m∠z = 40° …(i) [Corresponding angles]
m∠x + m∠z = 180° …[Angles in a linear pair]
∴ m∠x + 40⁰ = 180° …[From(i)]
∴ m∠x = 180° – 40°
∴ m∠x = 140°
Consider ∠w as shown in the figure.
m∠w + 70° = 180° …[Angles in a linear pair]
∴ m∠w = 180° – 70°
∴ m∠w = 110° …(ii)
line p || line q and line s is a transversal.
m∠y ≅ m∠w …[Alternate angles]
∵ m∠w = 110°
∴ m∠y =110° …[From (ii)]
Ans: The value of m∠x is 140° and m∠y is 110°.
3. 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.
Solution:
line p || line q and line l is a transversal
m∠a + 80° = 180° …[Interior angles]
∴ m∠a= 180° – 80°
∴ m∠a= 100°
line l || line m and line p is a transversal
∴ m∠c = 80° …(i) [Exterior alternate angles]
line p || line q and line m is a transversal.
m∠b ≅ m∠c …[Corresponding angles]
∵ m∠c = 80°
∴ m∠b = 80° …[From (i)]
Ans: The values are;
m∠a = 100°
m∠b = 80°
m∠c = 80°
4. In the adjoining figure, line a || line b. line l is a transversal. Find the measures of ∠ x, ∠ y, ∠ z using the given information.
Solution:
line a || line b and line l is a transversal.
∴ m∠x = 105° …(i) [Corresponding angles]
m∠y = m∠x …[Vertically opposite angles]
∴ m∠y = 105° …[From (i)]
m∠z + 105° = 180° …[Angles in a linear pair]
∴ m∠z = 180° – 105°
∴ m∠z = 75°
Ans: The values are;
m∠x = 105°
m∠y = 105°
m∠z = 75°
5. In the adjoining figure, line p || line l || line q. Find ∠ x with the help of the measures given in the figure.
Solution:
line p || line l and line IJ is a transversal.
m∠IJN = m∠JIH …[Alternate angles]
∴ m∠IJN = 40° …(i)
line l || line q and line MJ is a transversal.
m∠MJN = m∠JMK …[Alternate angles]
∴ m∠MJN = 30° …(ii)
Now,
m∠x = m∠IJN + m∠MJN …[Angle addition property]
∴ m∠x = 40° + 30° …[From (i) and (ii)]
∴ m∠x = 70°
Ans: The measure of m∠x = 70°