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

Complete NCERT Solutions for Class 7 Maths Chapter 5 “Parallel and Intersecting Lines” from the Ganita Prakash textbook, covering angle relationships, transversals, and construction methods. These Class 7 Mathematics Chapter 5 solutions are also useful as quick revision notes before exams.

5.1 Across the Line

Can two straight lines intersect at more than one point?
Answer: No, two distinct straight lines can intersect at only one point. If they never meet, they are parallel. If they appear to intersect at more than one point, the two “lines” are actually the same line.

Is this always true for any pair of intersecting lines?
Answer: Yes. When two lines intersect, they form four angles at the point of intersection. The angles directly opposite each other (vertically opposite angles) are always equal in measure.

Figure It Out (Page 108): Identifying linear pairs and vertically opposite angles from a diagram — this question depends entirely on a textbook figure and cannot be reproduced without it. In general, a linear pair consists of two adjacent angles on a straight line that add up to 180°, while vertically opposite angles are formed by two intersecting lines and are always equal.

5.3 Between Lines

Are line segments ST and UV likely to meet if extended?
Answer: If two lines are not parallel, they will eventually intersect at some point if extended far enough. So ST and UV are likely to meet when extended, since they are not parallel.

Are line segments OP and QR likely to meet if extended?
Answer: No. Parallel lines never meet no matter how far they are extended, so OP and QR will not meet.

Which pairs of lines are parallel? (Diagram-dependent — the specific line labels a, b, c… depend on the textbook figure.) Answer method: Two lines are parallel when they maintain a constant distance apart and never meet, no matter how far extended.

Figure It Out (Pages 113-114)

Q1-Q4 are hands-on drawing/construction exercises (drawing perpendicular lines, marking parallel lines with arrow notation, drawing parallel lines freehand on dot paper) that must be done on paper/geometry box tools and cannot be reproduced in text. Key method notes:

• To spot perpendicular lines in a figure, check if they intersect at a 90° angle.
• To spot parallel lines in a figure, check that the lines never intersect at any point, however far extended.

Q5. Which line is parallel to line a — line b or line c? Answer: Line a is parallel to line c, because these two lines maintain the same distance apart everywhere and never meet, no matter how far they are extended.

Figure It Out (Page 119) — Drawing a Parallel Line with Geometry Tools

Can you draw a line parallel to l, passing through point A, using tools from your geometry box?
Tools needed: Ruler, set-square (right-angled triangle), pencil.

Method:

1. Place the set square so one side lies along line l.
2. Hold a ruler firmly against the other side of the set square (the ruler stays fixed).
3. Slide the set square along the ruler until one edge reaches point A.
4. Draw a line along that edge of the set square through point A.
5. This new line is parallel to line l and passes through point A.

Making Parallel Lines Through Paper Folding

Method: Starting with line l (a crease) and point A: fold a perpendicular to l through A (call this crease t), then fold a perpendicular to t through A again (call this line m). Lines l and m will be parallel.

Why are lines l and m parallel?
Answer: Line t is perpendicular to line l, and line m is also perpendicular to line t. Since two lines that are both perpendicular to the same line are parallel to each other, l and m must be parallel.

Figure It Out (Pages 123-125): Finding Angles with Parallel Lines and Transversals

These problems use the key angle-relationship rules for a transversal crossing parallel lines:

• Alternate interior angles are equal.
• Corresponding angles are equal.
• Co-interior (same-side interior) angles add up to 180°.
• Vertically opposite angles (formed by two intersecting lines) are equal.
• Linear pair angles (adjacent angles on a straight line) add up to 180°.

Q1. Using these rules on the given figure (diagram-dependent inputs), the marked angles work out to: a = 48° (alternate interior angles), b = 52° (alternate angles), c = 81° (180° − 99°, co-interior angles), d = 99° (180° − 81°, co-interior angles), e = 69° (alternate interior angles), f = 48° (180° − 132°, co-interior angles), g = 122° (corresponding angles), h = 15°, i = 54°, and j = 97° (all via alternate interior angles). Note: the exact given angle values in the original figure are image-based; the method (identify the angle relationship, then apply equality or the 180° sum rule) is what matters for solving similar problems.

Q2. Four sub-figures, each solved by chaining angle rules:

• (i) Given angle 1 = 42°: angle 2 = 180° − 42° = 138° (linear pair). Since a and angle 2 are alternate angles (lines l, m parallel, t transversal), a = 138°.
• (ii) Given angle 1 = 62°: angle 2 = 180° − 62° = 118° (linear pair). Angle 2 and angle 3 are corresponding angles (l parallel to m), so angle 3 = 118°. Angle 3 and a are corresponding angles (s parallel to t), so a = 118°.
• (iii) Given angle 1 = 110° at an intersection of lines s and l: since l parallel to m with s as transversal, angle 2 = 110°. Angle 3 = angle 2 − 35° = 110° − 35° = 75°. Angle 3 = angle 4 = 75° (corresponding angles). So a = 180° − 75° = 105° (linear pair).
• (iv) Using angles on a straight line: angle1 + angle2 + 67° = 180°, with angle1 = 90°, so angle2 = 180° − 67° − 90° = 23°. Since a and angle2 are alternate angles (l parallel to t), a = 23°.

Q3. Two sub-figures:

• (i) Lines l and m are perpendicular, so angle2 = 90°. Using the linear pair angle2 + 65° + x = 180°: x = 180° − 90° − 65° = 25°. Since t and m intersect, x = angle1 = 25° (vertically opposite). Since l is parallel to m with t as transversal, y = angle2 + 65° = 90° + 65° = 155°. So x = 25°, y = 155°.
• (ii) l is parallel to m with s as transversal: angle3 = 78° (alternate angles). l is parallel to m with t as transversal: angle1 = 53° (alternate angles). So angle2 = angle3 − angle1 = 78° − 53° = 25°. Since s and t intersect, x = angle2 = 25° (vertically opposite angles).

Q4. Given angle ABC = 45° and angle IKJ = 78°: since IA and HC intersect at B, angle KBE = angle ABC = 45° (vertically opposite). Since JF and IA intersect at K, angle BKE = angle IKJ = 78° (vertically opposite). By corresponding angles, angle GEH = angle KBE = 45° and angle FED = angle BKE = 78°. Since angle GEH + angle HEF + angle FED = 180° (linear pair): angle HEF = 180° − 45° − 78° = 57°.

Q5. Given AB parallel to CD, CD parallel to EF (so AB parallel to EF), EA perpendicular to AB, and angle BEF = 55°: since EF parallel to CD with DE as transversal, y + 55° = 180° (co-interior angles), so y = 125°. Since AB parallel to CD with BD as transversal, x = y = 125° (corresponding angles).

Q6. This problem (finding angle NOP using auxiliary parallel lines drawn through N and O) depends on specific angle values from the textbook figure that could not be reliably reconstructed from the source text due to internal inconsistencies in the available working. The correct method is given by the hint: draw a line through N parallel to LM, and a line through O parallel to PQ, then use alternate-angle relationships to split angle NOP into two parts and add them together.

Parallel Illusions

This section explores optical illusions involving lines that appear non-parallel but are actually parallel. In each case, other slanted or radiating lines in the image visually distort perception, tricking the brain into perceiving straight, evenly-spaced parallel lines as curved, converging, or irregular. Measuring the lines directly with a ruler (or checking that the perpendicular distance between them stays constant) confirms they are truly parallel — this is why our eyes can be deceived by surrounding visual patterns, a phenomenon called an optical illusion.

Extra Questions: Class 7 Maths Chapter 5
Revision Notes: Class 7 Maths Chapter 5

Written by Satish

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