CBSE Class 7 Mathematics Parallel and Intersecting Lines Assignment Set 01

Short Answer Questions

Question 1. Two straight lines cross each other at a point on a sheet of paper.
(i) What name is given to such a pair of lines, and what is the point where they meet called?
(ii) How many angles are formed at that point?
(iii) Name the two special pairs of angles that are formed there.
Answer:
(i) They are called intersecting lines, and the meeting point is the point of intersection.
(ii) Four angles are formed at that point.
(iii) Linear pairs (two adjacent angles making a straight angle) and vertically opposite angles (two angles facing each other).
In simple words: Crossing lines meet at one point and make four angles, forming linear pairs and vertically opposite pairs.

Teacher's Note:
1. Give a short meaning with each name.
2. There are four linear pairs and two vertically opposite pairs.

Question 2. Two straight lines intersect at a point and one of the four angles formed measures 125°. Find the measures of the remaining three angles, writing the reason for each one.
Answer:
Let ∠a = 125°, and let ∠b, ∠c, ∠d be the other angles taken in order.
∠b = 180° − 125° = 55°, as ∠a and ∠b form a linear pair.
∠c = 125°, as ∠c is vertically opposite ∠a.
∠d = 55°, as ∠d is vertically opposite ∠b.
Check: 125° + 55° + 125° + 55° = 360°, the complete angle.
In simple words: The opposite angle is also 125°, and the two neighbouring angles are 55° each.

Teacher's Note:
1. A reason is needed for every angle.
2. The four angles around a point always total 360°.

Question 3. (i) When is a pair of intersecting lines said to be perpendicular to each other?
(ii) Two lines intersect in such a way that all four angles formed are equal. Reason out why each angle must measure 90°.
Answer:
(i) Two intersecting lines are perpendicular when they meet at a right angle (90°).
(ii) The four angles together make a full turn of 360°. If all four are equal, each one is 360° ÷ 4 = 90°, so the lines are perpendicular.
In simple words: Four equal angles share 360°, so each is 90°.

Teacher's Note:
1. Use 360° for angles around a point.
2. Equal linear pair angles also give 90° each.

Question 4. (i) Write two differences between a pair of parallel lines and a pair of intersecting lines.
(ii) Two line segments drawn on a page do not meet within the page. Describe how you would use a transversal to decide whether they are parallel.
Answer:
(i) Intersecting lines meet at a point, while parallel lines never meet however far they are extended.
Intersecting lines form four angles at the meeting point, while parallel lines form no angle with each other and stay the same distance apart everywhere.
(ii) Draw one straight line across both segments as a transversal and measure a pair of corresponding angles. If they are equal, the segments are parallel; if not, they are not parallel.
In simple words: Parallel lines never meet; to check, compare corresponding angles made by a crossing line.

Teacher's Note:
1. Not meeting on the page does not prove the lines are parallel.
2. Corresponding angles give a reliable test.

Question 5. A square sheet of newspaper is folded horizontally into half, and the fold is then opened out.
(i) What is the new line formed called, and how is it related to the two horizontal edges of the sheet?
(ii) How is this new line related to the two vertical edges of the sheet?
(iii) How many parallel lines can now be seen on the sheet?
Answer:
(i) The new line is a crease. It is parallel to the two horizontal edges, because the fold puts the top edge exactly on the bottom edge, so the crease stays at the same distance from both.
(ii) It is perpendicular to the two vertical edges, meeting each at 90°.
(iii) 3 parallel lines: the two horizontal edges and the crease.
In simple words: The crease is parallel to the top and bottom edges and perpendicular to the side edges.

Teacher's Note:
1. Count the edges along with the crease.
2. Folding is an easy way to make parallel lines.

Question 6. (i) What is a transversal?
(ii) How many angles are formed when a transversal crosses a pair of lines?
(iii) At the most, how many different measures can these angles have? Give the reason.
Answer:
(i) A transversal is a line that cuts a pair of lines at two different points.
(ii) Eight angles, four at each point of intersection.
(iii) At most four. The eight angles form four pairs of vertically opposite angles, and the two angles in each pair are equal.
In simple words: A transversal makes 8 angles, but they can have only up to 4 different sizes.

Teacher's Note:
1. Vertically opposite angles are the key to part (iii).
2. With parallel lines only two measures appear.

Question 7. In the figure below, the transversal t cuts the lines l and m, and the eight angles formed are numbered.
(i) Write all four pairs of corresponding angles.
(ii) Write all four pairs of vertically opposite angles.
Answer:
(i) Corresponding angles: ∠1 and ∠5; ∠2 and ∠6; ∠3 and ∠7; ∠4 and ∠8.
(ii) Vertically opposite angles: ∠1 and ∠3; ∠2 and ∠4; ∠5 and ∠7; ∠6 and ∠8.
In simple words: Corresponding angles sit in the same corner at each crossing; vertically opposite angles face each other at one crossing.

Teacher's Note:
1. Corresponding pairs are at different crossings.
2. Vertically opposite pairs are at the same crossing.

Question 8. (i) Which two instruments from your geometry box are used together to draw a pair of parallel lines?
(ii) Which property of angles makes us sure that the two lines drawn in this way are really parallel?
Answer:
(i) A ruler (scale) and a set square. The set square slides along the fixed edge of the ruler.
(ii) Sliding does not change the slant of the set square, so both lines make equal corresponding angles with the same line. Equal corresponding angles mean the lines are parallel.
In simple words: Slide a set square along a ruler; equal corresponding angles make the lines parallel.

Teacher's Note:
1. The ruler acts as the transversal.
2. Hold the ruler firmly so it does not move.

Question 9. A transversal cuts a pair of parallel lines l and m. One of the interior angles formed at l measures 72°. Write, with a reason for each, the measure of
(i) its corresponding angle at m,
(ii) its alternate interior angle at m,
(iii) the interior angle on the same side of the transversal at m.
Answer:
(i) 72°, because corresponding angles on parallel lines are equal.
(ii) 72°, because alternate angles on parallel lines are equal.
(iii) 180° − 72° = 108°, because same-side interior angles add up to 180°.
In simple words: Corresponding and alternate angles stay 72°; the same-side interior angle is 108°.

Teacher's Note:
1. Two rules give 'equal', one rule gives 'add to 180°'.
2. Always write the reason with the answer.

Question 10. A transversal cuts two parallel lines. A pair of alternate interior angles so formed measure (3x − 5)° and (2x + 15)°. Find the value of x and the measure of each of these two angles.
Answer:
Alternate interior angles on parallel lines are equal, so 3x − 5 = 2x + 15.
3x − 2x = 15 + 5, so x = 20.
3 × 20 − 5 = 55 and 2 × 20 + 15 = 55. Each angle is 55°.
In simple words: Make the two alternate angles equal: x = 20, and each angle is 55°.

Teacher's Note:
1. Getting the same value from both expressions is a built-in check.
2. Move x terms to one side, numbers to the other.

Question 11. A transversal t cuts two lines p and q. The angle it makes with p at one position is 118°, and the angle it makes with q in the corresponding position is 115°. Are p and q parallel to each other? Justify your answer.
Answer:
No, p and q are not parallel.
If a transversal cuts parallel lines, the corresponding angles must be exactly equal.
Here the corresponding angles are 118° and 115°, which are not equal.
So the lines are not parallel; extended far enough, they will meet on one side.
In simple words: 118° and 115° are not equal, so the lines are not parallel.

Teacher's Note:
1. A 3° difference is enough to rule out parallel lines.
2. Lines that are not parallel meet somewhere.

Question 12. A square sheet is folded horizontally into half again and again. The creases together with the two original horizontal edges are counted as parallel lines. One fold gives 3 parallel lines and two folds give 5 parallel lines.
(i) How many parallel lines will three folds and four folds give?
(ii) Write the rule that gives the number of parallel lines after n folds.
Answer:
(i) Each new fold halves every strip. 1 fold gives 2 strips (3 lines), 2 folds give 4 strips (5 lines), 3 folds give 8 strips, so 9 lines, and 4 folds give 16 strips, so 17 lines.
(ii) After n folds there are 2ⁿ equal strips, so the number of parallel lines is 2ⁿ + 1. Check: n = 2 gives 4 + 1 = 5.
In simple words: The strips double with every fold, and there is always one more line than strips.

Teacher's Note:
1. Lines = strips + 1.
2. Test the rule on the given cases first.

Question 13. In the figure below, l ∥ m and t is a transversal. Find the values of x and y, giving the reason for each.

Answer:
x = 62: the 118° angle and x° are interior angles on the same side of the transversal, so 118 + x = 180.
y = 118: the 118° angle and y° are alternate interior angles on the parallel lines l and m, so they are equal.
Check: x° and y° form a linear pair, and 62° + 118° = 180°.
In simple words: x is the supplement of 118°, and y equals 118° as an alternate angle.

Teacher's Note:
1. Both x and y are inside the parallel lines.
2. The linear-pair check confirms both answers.

Question 14. A student measures the perpendicular distance between two straight lines at three different places, taking the readings in order from the left end to the right end.
(i) For the first pair the readings are 4.0 cm, 4.0 cm and 4.0 cm. What can be said about this pair of lines?
(ii) For the second pair the readings are 4.0 cm, 3.5 cm and 3.0 cm. What can be said about this pair, and on which side will the two lines meet if they are extended?
Answer:
(i) The distance is the same everywhere, so the lines are parallel; they will never meet.
(ii) The distance keeps decreasing from left to right, so the lines are not parallel. They come closer towards the right, so they will meet on the right side if extended.
In simple words: Same gap means parallel; a shrinking gap means the lines meet on the side where it shrinks.

Teacher's Note:
1. Parallel lines keep a constant distance.
2. Follow the direction in which the gap gets smaller.

Question 15. A line l and a transversal t meet at the point X, and the angle between them is 60°. A point Y is then marked on t, and through Y a line m is drawn making an angle of 60° with t in the corresponding position.
(i) What is the relation between l and m? Give the reason.
(ii) Write the measures of all the angles formed at X and at Y.
Answer:
(i) l ∥ m. The 60° angles at X and Y are in the same position with respect to t, so they are corresponding angles. They are equal, so the lines are parallel.
(ii) At X: 60°, 120°, 60°, 120° in order, since each 120° angle is the linear pair of a 60° angle. At Y: also 60°, 120°, 60°, 120°, in the same positions.
In simple words: Equal corresponding angles make l and m parallel, and each crossing has two 60° and two 120° angles.

Teacher's Note:
1. This is exactly how a set square draws a parallel line.
2. 180° − 60° = 120° gives the other angles.

Question 16. Using a protractor, Ravi measures the two angles of a linear pair and gets 89° and 93°.
(i) Give two reasons why his two measures do not add up to exactly 180°.
(ii) Why do we still accept that a linear pair always adds up to 180°?
Answer:
(i) The protractor may not have been placed exactly on the vertex and along the arm, and the drawn pencil lines have thickness, while an ideal geometric line has none.
(ii) The rule comes from reasoning, not measuring: the two angles of a linear pair together make a straight angle, which is 180°, so their sum must be 180°.
In simple words: Measurements have small errors, but reasoning shows the sum is always 180°.

Teacher's Note:
1. 89 + 93 = 182, which is 2° too much.
2. Geometry trusts reasoning over measurement.

Question 17. Two straight lines intersect at a point and form the angles ∠a, ∠b, ∠c and ∠d, taken in order around that point. Without measuring any of them, explain why ∠a must be equal to ∠c.
Answer:
∠a and ∠b lie along one straight line, so they form a linear pair: ∠a + ∠b = 180°.
∠b and ∠c lie along the other straight line, so ∠b + ∠c = 180°.
Both sums are 180°, so ∠a + ∠b = ∠b + ∠c, which gives ∠a = ∠c.
No measuring is used, so this holds for every pair of intersecting lines.
In simple words: ∠a and ∠c each make 180° with the same ∠b, so they must be equal.

Teacher's Note:
1. This is a proof: it works for every figure.
2. Take away the common ∠b from both sides.

Question 18. A transversal cuts the lines l and m. At l the four angles formed measure 65°, 115°, 65° and 115°, and at m the four angles formed also measure 65°, 115°, 65° and 115°.
(i) Does this by itself prove that l ∥ m? Explain your answer.
(ii) What exactly must be checked before saying that l and m are parallel?
Answer:
(i) No. The same two measures appear at both lines, but the lines may slant in opposite directions. Then an angle in a particular position at l could be 65° while the angle in the same position at m is 115°, and the lines would not be parallel.
(ii) We must check that a pair of corresponding angles (angles in the same position at the two crossings) is equal. Checking a pair of alternate angles for equality also works.
In simple words: The same set of angles is not enough; the angles in the same position must be equal.

Teacher's Note:
1. Position matters, not just the list of measures.
2. Draw two lines leaning opposite ways to see this.

Question 19. The lines l and m are both perpendicular to the same line t.
(i) Using corresponding angles, explain why l and m must be parallel to each other.
(ii) If instead it is given that l ∥ m and t ⟂ l, what can you say about t and m?
Answer:
(i) Since l ⟂ t and m ⟂ t, both make 90° with t. Taking t as the transversal, the corresponding angles are both 90°, hence equal, so l ∥ m.
(ii) t makes 90° with l, so the corresponding angle at m is also 90°. Hence t ⟂ m as well.
In simple words: Two lines at right angles to the same line are parallel, and a line perpendicular to one of two parallel lines is perpendicular to the other too.

Teacher's Note:
1. Part (ii) is the reverse of part (i).
2. The symbol ⟂ means 'is perpendicular to'.

Question 20. A carpenter lays a straight rod across the two long edges of a window frame to test them. The rod makes 89° with the first edge and 91° with the second edge, both angles being measured in the same position on the same side.
(i) Are the two long edges parallel? Give the reason.
(ii) What should the second angle have been?
(iii) Suggest one more way in which the carpenter can test the two edges.
Answer:
(i) No. The rod is a transversal and the two angles are corresponding angles. For parallel edges they must be equal, but 89° ≠ 91°.
(ii) It should also have been 89°.
(iii) He can measure the perpendicular distance between the two edges at two or three places; the edges are parallel only if all the readings are the same.
In simple words: Unequal corresponding angles show the edges are not parallel.

Teacher's Note:
1. Same position means corresponding angles.
2. Equal gaps are another test for parallel lines.