Class 7 Mathematics · Part 1 · Chapter 5

5. Parallel and Intersecting Lines

Answers to all Figure it Out and in-text questions on linear pairs, vertically opposite, corresponding, alternate and interior angles. CBSE / NCERT Class 7 Ganita Prakash.

5.1 Across the Line

?Can two straight lines intersect at more than one point?

No. Two different straight lines can meet at only one point.

?What patterns do you observe among the four angles formed by two intersecting lines?

The opposite angles are equal: ∠a = ∠c and ∠b = ∠d.

Two angles next to each other add up to 180°: ∠a + ∠b = 180°, ∠b + ∠c = 180°, and so on.

?If ∠a is 120°, find ∠b, ∠c and ∠d. Is ∠a = ∠c and ∠b = ∠d always true?

∠b = 180° − 120° = 60°, ∠c = 180° − 60° = 120°, ∠d = 180° − 120° = 60°.

Yes, it is always true. ∠a + ∠b = 180° and ∠c + ∠b = 180°, so ∠a = ∠c. In the same way, ∠b = ∠d.

FIOFigure it Out: List all the linear pairs and vertically opposite angles in Fig. 5.3.

Linear pairs: ∠a and ∠b, ∠b and ∠c, ∠c and ∠d, ∠d and ∠a.

Vertically opposite angles: ∠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? What is each angle?

Yes. The four angles add up to 360°, so each angle is 360° ÷ 4 = 90°. Such lines are perpendicular.

5.3 Between Lines

?Describe how the line segments in Fig. 5.5 meet or cross each other.

AB and CD intersect at the point X, which is near the middle of both. The angle between them is about 60°.

IJ and LM intersect at the point Y, near the middle of both. The angle between them is about 80°.

FG and FH meet at the endpoint F at an angle of 115.3°.

ST and UV do not meet. OP and QR do not meet.

Measure the angles in your book with a protractor. Small differences are fine.

?Are ST and UV likely to meet if extended? Are OP and QR likely to meet if extended?

ST and UV come slightly closer to each other towards the right. So, they are likely to meet if extended.

OP and QR stay the same distance apart. So, they are not likely to meet. They look parallel.

?Name some parallel lines you can spot in your classroom.

Opposite edges of the blackboard, opposite edges of a door, lines on a notebook page, and the edges of a bench.

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

a, h and i; c and g; d and f; b and e.

5.4 Parallel and Perpendicular Lines in Paper Folding

?Activity 2: folding a square sheet

The opposite edges are parallel to each other. The adjacent edges are perpendicular to each other.

After one horizontal fold, we see 3 parallel lines (2 edges and 1 fold). The new line is perpendicular to the vertical sides.

After the second fold: 5 parallel lines. After the third fold: 9 parallel lines.

Pattern: 3, 5, 9, 17, … Each fold doubles the number of strips, and the lines are always one more than the strips.

A vertical fold is perpendicular to the horizontal lines.

Yes. Fold one of the other two corners onto the centre of the sheet. This crease is parallel to the diagonal.

?Are a, b and c parallel to p, q and r respectively? Why or why not?

Yes. a and p are made by folding the edges to the centre line, so they are parallel to the centre line and to each other.

b and q are both edges of the sheet that are perpendicular to these lines, so they are parallel.

c and r are made by folding opposite corners in the same way, so they make the same angle with the edges. So, they are parallel.

FIOFigure it Out

1. Draw some lines perpendicular to the lines on the dot paper.

Draw these in your book. For a line going 1 dot right and 2 dots up, a perpendicular line goes 2 dots right and 1 dot down.

2. Mark parallel and perpendicular lines in Fig. 5.11.

(a) Perpendicular lines meet at a right angle, like the corner of a set square.

(b) Parallel lines stay the same distance apart and do not meet.

3. Draw different sets of parallel lines on the dot paper.

Draw these in your book. For example, all segments that go 2 dots right and 1 dot up are parallel.

4. Draw lines parallel to the given segments.

(a) Yes, some were challenging.

(b) The slanting segments like e, f, g and h were harder.

5. Which line is parallel to line a — b or c?

Line c. It stays the same distance from a. Line b comes closer to a, so it will meet a if extended.

5.5 Transversals

?Is it possible for all the eight angles to have different measurements? What about the five angles ∠6, ∠5, ∠4, ∠3 and ∠2?

No. Vertically opposite angles are equal: ∠1 = ∠3, ∠2 = ∠4, ∠5 = ∠7, ∠6 = ∠8. So, there are at most 4 different measures.

No. ∠2 and ∠4 are vertically opposite, so they are equal.

?Activity 3: How many distinct angles are formed when t meets l at 60°? What do you observe about l and m?

Two distinct angles: 60° and 120°.

Lines l and m are parallel, because the corresponding angles are equal (60°).

?What notation is used in Fig. 5.19 to show that l and m are parallel? Are all corresponding angles equal?

An arrow mark (>) on each line. Yes, all the corresponding angles are equal.

5.7 Drawing Parallel Lines

FIOFigure it Out: Draw a line parallel to l through point A using your geometry box.

1. Place one edge of a set square along the line l.

2. Place a ruler along the other edge of the set square and hold it firmly.

3. Slide the set square along the ruler until its first edge passes through A.

4. Draw a line along this edge. This line through A is parallel to l, because both make the same angle with the ruler (equal corresponding angles).

?Why are lines l and m (made by paper folding) parallel to each other?

t is perpendicular to l, and m is perpendicular to t.

So, l and m both make 90° with t. These are equal corresponding angles. Hence, l and m are parallel.

5.8 Alternate Angles

?Is there a relation between the interior angles ∠3 and ∠6 (Example 3)?

Yes. ∠3 + ∠6 = 50° + 130° = 180°.

Reason: ∠2 = ∠6 (corresponding angles) and ∠2 + ∠3 = 180° (linear pair). So, ∠3 + ∠6 = 180° always.

FIOFigure it Out

1. Find the angles marked in Fig. 5.30.

a = 48° (alternate angles)

b = 52° (alternate angles)

c = 81° (alternate angle of 81°; also 180° − 99°)

d = 99° (interior angles: 180° − 81°)

e = 69° (alternate angles)

f = 48° (interior angles: 180° − 132°)

g = 122° (corresponding angles)

h = 75° (alternate angles)

i = 54° (alternate angles)

j = 97° (corresponding and vertically opposite angles)

2. Find the angle a in Fig. 5.31.

(i) The transversal makes 42° with the upper line. a and 42° are interior angles on the same side, so a = 180° − 42° = 138°.

(ii) a = 180° − 62° = 118°.

(iii) The slanting line makes 180° − 110° = 70° with the middle line. So, a = 70° + 35° = 105° (corresponding angles).

(iv) The vertical line makes 90° with the base and the slanting line makes 67°. So, a = 90° − 67° = 23° (alternate angles).

3. Find x and y in Fig. 5.32.

(i) The slanting line makes 90° − 65° = 25° with the lower line. x = 25° (vertically opposite). y = 180° − 25° = 155°.

(ii) x = 78° − 53° = 25°.

4. In Fig. 5.33, find ∠GEH, ∠HEF and ∠FED.

∠GEH = ∠ABC = 45° (corresponding and vertically opposite angles).

∠FED = ∠IKJ = 78° (corresponding and vertically opposite angles).

∠HEF = 180° − 45° − 78° = 57° (angles on a straight line).

5. In Fig. 5.34, find x and y.

The slanting line makes 55° with the vertical lines.

y = 180° − 55° = 125° and x = 125° (corresponding angles).

6. Find ∠NOP in Fig. 5.35.

Draw lines through N and O parallel to LM and PQ.

At N: 96° = 40° + 56°, so NO makes 56° with the parallel line.

At O: ∠NOP = 56° + 52° = 108°.

5.9 Parallel Illusions

?What causes these illusions?

The many short slanting lines around the long lines trick our eyes. The long lines look tilted, but they are parallel. A ruler can check this.

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