1In the adjoining figure, each angle is shown by a letter. Fill in the boxes with the help of the figure.
The transversal cuts the two lines at two points; four angles are formed at each point.
Corresponding angles:
(1) ∠p and ∠w
(2) ∠q and ∠x
(3) ∠r and ∠y
(4) ∠s and ∠z
Interior alternate angles:
(5) ∠s and ∠x
(6) ∠w and ∠r
How to find them: corresponding angles sit in the same position at the two points (for example, both upper-left). Interior alternate angles lie between the two lines and on opposite sides of the transversal.
2Observe the angles shown in the figure and write the following pair of angles.
Here the horizontal line is the transversal of the two slanting lines.
(1) Interior alternate angles
∠c and ∠e∠b and ∠h
(2) Corresponding angles
∠a and ∠e∠b and ∠f∠c and ∠g∠d and ∠h
(3) Interior angles
∠b and ∠e∠c and ∠h
The interior angles are the four angles between the two slanting lines: ∠b, ∠c, ∠e and ∠h. Pairs on the same side of the transversal are interior angles; pairs on opposite sides are interior alternate angles.
Advertisement
Practice Set 2.2
1Choose 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°
3x and x are interior angles on the same side of line p.
3x and x are interior angles, so they are supplementary.
3x + x = 180° ∴ 4x = 180° ∴ x = 45°
Answer: (C) 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°
The angle 4x at line a has a corresponding angle at line b (shown in green).
The angle above line l and to the right of line b corresponds to 4x, so it also measures 4x. This angle and 2x form a linear pair.
4x + 2x = 180° ∴ 6x = 180° ∴ x = 30°
Answer: (D) 30°
2In the adjoining figure line p ∥ line q. Line t and line s are transversals. Find the measures of ∠x and ∠y using the measures of angles given in the figure.
Given angles in brown; angles found on the way in green.
To find ∠x (transversal t):
The 40° angle at line p and the angle above line q to the right of line t are corresponding angles, so that angle = 40°.
That 40° angle and ∠x form a linear pair.
∴ m∠x = 180° − 40° = 140°
To find ∠y (transversal s):
The 70° angle at line q and the angle below line p to the right of line s are corresponding angles, so that angle = 70°.
That 70° angle and ∠y form a linear pair.
∴ m∠y = 180° − 70° = 110°
Answer: m∠x = 140°, m∠y = 110°
3In the adjoining figure, line p ∥ line q and line l ∥ line m. Find the measures of ∠a, ∠b and ∠c, using the measures of given angles. Justify your answers.
Given angle in brown; the angle found on the way in green.
∠a:
Line p ∥ line q and line l is a transversal. The 80° angle at line p and the angle above line l to the right of line q are corresponding angles, so that angle = 80°.
That angle and ∠a form a linear pair. ∴ m∠a = 180° − 80° = 100°
∠b:
Line l ∥ line m and line q is a transversal. The angle above line l to the right of line q is 80° (found above). The angle above line m to the right of line q corresponds to it, so that angle is also 80°.
∠b is vertically opposite to that angle.
∴ m∠b = 80°
∠c:
Line p ∥ line q and line m is a transversal. ∠c and ∠b are corresponding angles (both below line m, on the left of the transversal).
∴ m∠c = m∠b = 80°
Answer: 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 and ∠z using the given information.
Line a ∥ line b; line l is the transversal.
∠z and the 105° angle form a linear pair. ∴ m∠z = 180° − 105° = 75°
∠x and the 105° angle are corresponding angles (both above line l, on the left of lines a and b). ∴ m∠x = 105°
∠y and ∠x are vertically opposite angles. ∴ m∠y = m∠x = 105°
Answer: 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.
Line l splits ∠x into two parts: 40° and 30°.
Name the vertex of ∠x as O, and the points where its arms meet line p and line q as A and B.
Line p ∥ line l and seg OA is a transversal. The 40° angle at A and the upper part of ∠x are interior alternate angles. ∴ upper part = 40°
Line l ∥ line q and seg OB is a transversal. The 30° angle at B and the lower part of ∠x are interior alternate angles. ∴ lower part = 30°
m∠x = 40° + 30° = 70°
Answer: m∠x = 70°
Practice Set 2.3
1Draw a line l. Take a point A outside the line. Through point A draw a line parallel to line l.
Step 1: Draw line l. Take a point A outside the line.
Step 2: Place one edge of set-square B along line l. Place set-square A so that it touches set-square B along its other edge.
Step 3: Hold set-square A firmly and slide set-square B along its edge until the edge that was on line l passes through point A.
Step 4: Draw a line through A along that edge of set-square B. Name it line m.
Step 5: Line m ∥ line l.
Line m through A is parallel to line l.
2Draw a line l. Take a point T outside the line. Through point T draw a line parallel to line l.
Step 1: Draw line l. Take a point T outside the line.
Step 2: Draw seg TM ⊥ line l, with M on line l.
Step 3: Take another point N on line l.
Step 4: Draw seg NQ ⊥ line l on the same side as T, such that l(NQ) = l(MT).
Step 5: Draw the line through T and Q. Name it line m. Line m ∥ line l.
TM = QN, and both are perpendicular to line l, so line TQ is parallel to line l.
Questions 1 and 2 ask for the same construction. Either method works for both; one method is shown for each so that both are practised.
3Draw a line m. Draw a line n which is parallel to line m at a distance of 4 cm from it.
Step 1: Draw line m.
Step 2: Take two points A and B on line m.
Step 3: Draw perpendiculars to line m at A and at B.
Step 4: On these perpendiculars, on the same side of line m, mark points P and Q such that AP = BQ = 4 cm.
Step 5: Draw line PQ and name it line n. Line n ∥ line m, at a distance of 4 cm from it.
Line n is parallel to line m at a distance of 4 cm.
