NCERT Solutions for Class 7 Maths Ganita Prakash Chapter 5 Parallel and Intersecting Lines

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Section 5.1 Across the Line

How many angles do they form?
See SolutionFour angles are formed.

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Can two straight lines intersect at more than one point?
See SolutionNo, two distinct straight lines can intersect at most one point.

What patterns do you observe among these angles?
See SolutionThe patterns we observed among these angles are:
Vertically opposite angles are equal:
For example, in the figure, ∠a = ∠c and ∠b = ∠d.
Linear pairs add up to 180°:
Adjacent angles like ∠a + ∠b or ∠b + ∠c form a straight angle, which means their sum is always 180°.
These patterns are true for any pair of intersecting lines and form the basis of angle relationships in geometry.

In Fig. 5.2, if ∠a is 120°, can you figure out the measurements of ∠b, ∠c and ∠d, without drawing and measuring them?

See SolutionYes.
∠a + ∠b = 180 (linear pair)
=> ∠b = 180 – 120 = 60
∠c = ∠a = 120 (vertically opposite).
∠d = ∠b = 60 (vertically opposite).
So: ∠b = 60, ∠c = 120, ∠d = 60.

Is this always true for any pair of intersecting lines?
See SolutionYes, vertically opposite angles formed by any pair of intersecting lines are always equal.

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Figure it Out

List all the linear pairs and vertically opposite angles you observe in Fig. 5.3:

See Solution Linear Pairs (adjacent angles on a straight line, sum = 180 degrees):
∠a and ∠b
∠b and ∠c
∠c and ∠d
∠d and ∠a
Pairs of Vertically Opposite Angles (angles opposite at the intersection, are equal):
∠a and ∠c
∠b and ∠d

 5.2 Perpendicular Lines

Can you draw a pair of intersecting lines such that all four angles are equal? Can you figure out what will be the measure of each angle?

See Solution Yes, this is possible if all four angles are equal, each angle must be a right angle (90 degrees). We say such lines as perpendicular lines.

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Which pairs of lines appear to be parallel in Fig. 5.6 below?

See Solution(Based on visual inspection of Figure 5.6 on the page)
Line segment ‘a’ appears parallel to line segment ‘i’ and ‘h’.
Line segment ‘c’ appears parallel to line segment ‘g’.
Line segment ‘b’ appears parallel to line segment ‘e’.
Line segment ‘d’ appears parallel to line segment ‘f’.

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 5.4 Parallel and Perpendicular Lines in Paper Folding

Activity 2
Take a plan square of paper (use a newspaper for this activity).
How would you describe the opposite edges of the sheet? They are ______ to each other.
See AnswerParallel

How would you describe the adjacent edges of the sheet? The adjacent edges are ______ to each other. They meet at a point. They form right angles.
See AnswerPerpendicular

Fold the sheet horizontally in half. A new line is formed (see Fig. 5.7).
See AnswerThe two original top/bottom edges and the new horizontal fold.

How many parallel lines do you see now? How does the new line segment relate to the vertical sides?

See Solution It is perpendicular to the vertical sides.

Make on one more horizontal fold in the folded sheet. How many parallel lines do you see now?
See Solution 5 parallel lines (the two original edges plus the three horizontal folds).

What will happen if you do it once more? How many parallel lines will you get? Is there a pattern? Check if the pattern extends further, if you make another horizontal fold.
See Solution Doing it once more (a third horizontal fold) would result in 9 parallel lines (2 original edges + 7 folds).
The pattern relates to powers of 2; after n folds, we have 2ⁿ⁻¹ fold lines, plus the 2 original edges for 2ⁿ⁺¹ total parallel lines (starting from n=1 fold).
We can check if the pattern continues.

Make a vertical fold in the square sheet. This new vertical line is ______ to the previous horizontal lines.
See SolutionPerpendicular

Fold the sheet along a diagonal. Can you find a fold that creates a line parallel to the diagonal line?
See Solution This requires performing the activity. It is possible to create such a fold (e.g., by folding an edge onto the diagonal or folding the paper in half parallel to the first diagonal fold).

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Here is another activity for you to try.
•Take a square sheet of paper, fold it in the middle and unfold it.
•Fold the edges towards the centre line and unfold them.
•Fold the top right and bottom left corners onto the creased line to create triangles. Refer to Fig. 5.8.
•The triangles should not cross the crease lines.
•Are a, b and c parallel to p, q and r respectively? Why or why not?
See SolutionFrom Figure 5.8, we can see the final result shows:
Line segments a, b and c (the fold lines/creases)
Line segments p, q and r (the edges of the triangular folds)
No, they are not parallel.
Reason:
Lines a, b and c are vertical crease lines running from top to bottom of the paper
Lines p, q and r are the diagonal edges of the triangular folds created when we fold the corners
Since the crease lines run vertically and the triangle edges run diagonally, they intersect each other rather than being parallel. Parallel lines never intersect and always maintain the same distance apart, but here we can see that the vertical crease lines and diagonal triangle edges cross each other at various points.
The triangular folds create diagonal lines that cut across the vertical crease lines at angles, which is why the instruction specifies that “the triangles should not cross the crease lines” – to maintain the proper geometric relationship for this folding exercise.

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Figure it Out

1. Draw some lines perpendicular to the lines given on the dot paper in Fig. 5.10.

Answer:

2. In Fig. 5.11, mark the parallel lines using the notation above (single arrow, double arrow etc.). Mark the angle between perpendicular lines with a square symbol.

(a) How did you spot the perpendicular lines?
(b) How did you spot the parallel lines?
Answer:
Lines are shown in the following figures.

See Explanation(a) Perpendicular lines are typically spotted by identifying lines that meet at a 90-degree angle (a right angle). On dot or grid paper, this looks like the corner of a square or rectangle aligned with the grid lines or lines whose slopes are negative reciprocals of each other.
(b) Parallel lines are spotted by identifying lines that have the same direction or slope and always remain the same distance apart. On dot paper, they form opposite sides of shapes like parallelograms, rectangles or trapezoids or they connect dots with the same horizontal and vertical displacement.