Chapter 5 exercise answers: Parallel and Intersecting Lines

Open NCERT p. 108Matches NCERT’s answer

1. In Fig. 5.3, two straight lines cross at one point, making four angles: ∠a, ∠b, ∠c and ∠d, going around the point.
2. Angles that sit next to each other, sharing an arm, and whose other two arms form a straight line, add up to 180° — these are linear pairs.
3. Going around the crossing, the neighbouring pairs are ∠a & ∠b, ∠b & ∠c, ∠c & ∠d, and ∠d & ∠a.
4. Angles that face each other across the point (not next to each other) are vertically opposite angles, and they are always equal.
5. The two pairs that face each other are ∠a & ∠c, and ∠b & ∠d.

AnswerLinear pairs: ∠a & ∠b, ∠b & ∠c, ∠c & ∠d, ∠d & ∠a. Vertically opposite angles: ∠a & ∠c, ∠b & ∠d.

Watch this explained “Left and right are opposite too”, 6:53 into The four angles at a crossing: vertically opposite and linear pairs · हिंदी में देखें

Open NCERT p. 113One way to think about it

1. Fig. 5.10 is dot paper, so describe each given segment by counting dots from one end to the other: so many across, and so many up or down.
2. The four segments are: a horizontal one (2 across), a slanting one (2 across, 2 up), a vertical one (2 down) and a steep one (2 across, 3 down).
3. A perpendicular direction is found by swapping the two counts and reversing one of the directions (up becomes down, or down becomes up).
4. Horizontal segment: any vertical line is perpendicular to it, for example the line straight down through one of its dots.
5. Vertical segment: any horizontal line is perpendicular to it, for example the line straight across through one of its dots.
6. Slanting segment (1 across, 1 up at each step): draw a line going 1 across and 1 down, dot by dot.
7. Steep segment (2 across, 3 down): draw a line going 3 across and 2 up — start at any dot, count 3 dots to the right and 2 dots up, and join the two dots.
8. Check each corner with the right-angle corner of a set square: it should fit exactly.

In shortMany drawings are correct; for example: vertical lines for the horizontal segment, horizontal lines for the vertical segment, lines going 1 across and 1 down for the segment going 1 across and 1 up, and lines going 3 across and 2 up for the steep segment going 2 across and 3 down.

Watch this explained “Making one without measuring”, 5:48 into Perpendicular lines as the case where all four are equal · हिंदी में देखें

Open NCERT p. 113One way to think about it

(a) How did you spot the perpendicular lines?

1. Fig. 5.11 is drawn on squared paper and every corner sits on a grid point, so each edge can be described by counting squares: so many across and so many up or down.
2. Wherever a horizontal edge meets a vertical edge, the corner is a right angle, so mark it with a small square.
3. Two slanting edges are perpendicular when their counts are swapped and one direction reversed, for example 1 across 1 up and 1 across 1 down.
4. Check any doubtful corner with the right-angle corner of a set square.

In shortBy counting squares: a horizontal edge meeting a vertical edge makes a right angle, and two slanting edges are perpendicular when their across/up counts are swapped with one direction reversed. A set square confirms each one.

(b) How did you spot the parallel lines?

1. Count the step of each edge. All horizontal edges are parallel to one another, and all vertical edges are parallel to one another, whatever their lengths.
2. Slanting edges are parallel when their across and up counts are in the same ratio and go the same way, for example 1 across 1 up, 2 across 2 up and 3 across 3 up.
3. Give each family of parallel edges its own mark: one arrow on every edge of the first family, two arrows on every edge of the second family, and so on.

In shortBy counting squares along each edge: edges whose across/up counts are in the same ratio (all the horizontal ones, all the vertical ones, all the 1-across-1-up ones, and so on) are parallel, and each family gets its own arrow mark.

Watch this explained “Writing the decision down”, 6:49 into What "parallel" actually claims, and why it is hard to check · हिंदी में देखें

Open NCERT p. 113One way to think about it

1. Pick any across/up step between dots, for example 2 across and 1 up.
2. Draw several segments with exactly that same step (or a whole-number multiple of it, such as 4 across and 2 up), starting from different dots and running to different lengths.
3. All of these segments are one "set" of parallel lines, because counting squares shows they keep the same direction.
4. Repeat with a different step (say 1 across and 3 up) to get a second, differently-angled set of parallel lines on the same dot paper.

In shortAny family of segments that all share the same across/up step between dots (lengths and starting dots can differ) is a valid set of parallel lines; drawing a few such families, each with its own step, answers the question.

Watch this explained “Counting, instead of looking”, 5:59 into What "parallel" actually claims, and why it is hard to check · हिंदी में देखें

Open NCERT p. 114One way to think about it

(a) Did you find it challenging to draw some of them?

1. Segments a, b, c and d all slant at 45° (as many dots across as up or down), so a line parallel to them is easy to draw: step one dot across and one dot up or down, again and again.
2. Segments e, f, g and h have uneven steps, and copying their slant by eye usually comes out a little off.

In shortYes, some of them are hard to draw by eye: the 45° segments are easy, the others are not.

(b) Which ones?

1. Count the step of each segment between its end dots.
2. a: 1 across, 1 down; b: 2 across, 2 down; c: 3 across, 3 up; d: 4 across, 4 up — all at 45°, so these are easy.
3. e: 1 across, 3 down and f: 1 across, 6 down are steep; g: 5 across, 2 up and h: 10 across, 3 down are shallow. These are the hard ones.

In shorte, f, g and h — the steep and shallow segments that are not at 45°.

(c) How did you do it?

1. Instead of judging the slant by eye, count the segment's step between its end dots; for example, g goes 5 across and 2 up.
2. From a new starting dot, count exactly the same step (5 across, 2 up) and join the two dots.
3. The new segment repeats the same step, so it points in exactly the same direction and is parallel to the given one.

In shortBy counting the across/up step of the given segment and repeating exactly that step from another dot.

Watch this explained “Counting, instead of looking”, 5:59 into What "parallel" actually claims, and why it is hard to check · हिंदी में देखें

Open NCERT p. 114Matches NCERT’s answer

1. Looking alone cannot settle this: b and c both lean almost the same way as a.
2. Use the line already drawn from the shared end of b and c down to the lower end of a. It cuts line a and line c (and line b), so it is a transversal.
3. At the lower end of a, measure or trace the angle between a and this transversal. At the top, measure the angle between c and the transversal, and the angle between b and the transversal. These are alternate angles.
4. For c the two alternate angles come out equal, as nearly as a protractor can tell (about 30° each); for b the angle at the top is clearly bigger (about 33°).
5. Equal alternate angles mean the lines are parallel, so c is parallel to a, and b is not.

AnswerLine c is parallel to line a. The transversal drawn from the top point to the lower end of a makes equal alternate angles (about 30°) with a and with c, but a bigger angle with b.

Watch this explained “Would they meet, further on?”, 1:25 into What "parallel" actually claims, and why it is hard to check · हिंदी में देखें

Open NCERT p. 119One way to think about it

1. Place one edge of the set square exactly along line l.
2. Hold a ruler firmly against another edge of the set square. From now on the ruler must not move.
3. Slide the set square along the ruler, keeping it pressed against the ruler and without turning it, until the edge that was on l passes through A.
4. Draw a line along that edge through A. This is the line parallel to l.
5. Why it works: the ruler cuts both l and the new line, so it is a transversal. The set square's edge made the same angle with the ruler in both positions, so the corresponding angles are equal, and equal corresponding angles mean the lines are parallel.

In shortYes. Put an edge of the set square on l, hold a ruler against another edge, slide the set square along the ruler until that edge reaches A, and draw along it. The ruler is a transversal, the corresponding angles at l and at the new line are equal, so the line through A is parallel to l.

Watch this explained “Through a point you are given”, 5:55 into Drawing a parallel line using the corresponding-angle test · हिंदी में देखें

Open NCERT p. 123Matches NCERT’s answer

1. (a) The two arrowed lines are parallel. 48° and a° both lie between the parallel lines, on opposite sides of the transversal, so they are alternate angles: a = 48.
2. (b) 52° (between the upper line going right and the transversal going down) and b° (between the transversal going up and the lower line going left) are alternate angles: b = 52.
3. (c) 81° (between the upper line going right and the transversal going down) and c° (between the transversal going up and the lower line going left) are alternate angles: c = 81. As a check, c and 99° are interior angles on the same side: 81 + 99 = 180.
4. (d) 99° (between the vertical going up and the lower line going right) and d° (between the upper line going left and the vertical going down) are alternate angles: d = 99.
5. (e) On the slanted transversal, 69° (between the transversal going up and the lower line going left) and e° (between the upper line going right and the transversal going down) are alternate angles: e = 69. The 97° and 83° belong to the other transversal and are not needed.
6. (f) 132° and f° are both between the parallel lines, on the same side (left) of the transversal, so they add to 180°: f = 180 − 132 = 48.
7. (g) The two vertical lines are parallel. At the right crossing, the 122° marked below-left lies between the slanted line going up-left and the vertical going down. g° is in exactly the same position at the left crossing, so they are corresponding angles: g = 122 (not 58 — the 58° is the angle above the slanted line).
8. (h) The right slanted line crosses both horizontal parallel lines. 75° (between the upper line going right and the slanted line going down) and h° (between the slanted line going up and the lower line going left) are alternate angles: h = 75. The 120° belongs to the other slanted line.
9. (i) The two arrowed lines are parallel and the middle line, from the top corner to the base point, cuts both. 54° (between that line going up and the right arrowed line going up) and i° (between the left arrowed line going down and that line going down) are alternate angles: i = 54.
10. (j) The three arrowed lines are parallel. The line from the 27° corner that passes the 97° mark reaches the third arrowed line at j°. 97° lies between the first arrowed line going down and this line going away from the corner; j° lies between the third arrowed line going up and this line going back towards the corner. Both arms point exactly the opposite way (a corresponding angle and then a vertically opposite one), so j = 97. Check: 27 + (180 − 124) + 97 = 180 in the triangle formed at the third line.

Answera = 48°, b = 52°, c = 81°, d = 99°, e = 69°, f = 48°, g = 122°, h = 75°, i = 54°, j = 97°

Watch this explained “The whole map”, 8:12 into Alternate angles, and interior angles that add to 180° · हिंदी में देखें

Open NCERT p. 124Matches NCERT’s answer

(i)

1. The two arrowed lines are parallel and the slanted line crosses both.
2. 42° is above the upper line, to the right of the slanted line. The corresponding angle at the lower line (above it, to the right of the slanted line) is also 42°.
3. a° is above the lower line on the left of the slanted line, so it forms a linear pair with that 42°: a = 180 − 42 = 138.
4. The 100° belongs to the other line and is not needed.

Answera = 138°

(ii)

1. The single-arrow lines are parallel, and the double-arrow lines are parallel.
2. 62° lies between the upper line going right and the right double-arrow line going down.
3. Corresponding angles carry it down the right double-arrow line to the lower line, and then along the lower line to the left double-arrow line: there, the angle between the lower line going right and the left double-arrow line going down is 62°.
4. a° lies between the lower line going left and the same double-arrow line going down, so it forms a linear pair with that 62°: a = 180 − 62 = 118.

Answera = 118°

(iii)

1. The three arrowed lines are parallel. The long line crosses all three; the shorter slanted line meets it exactly on the middle line.
2. 110° lies between the top line going right and the shorter line going up-left. At the middle line the corresponding angle is also 110°, so the angle between the middle line going left and the shorter line going up is 180 − 110 = 70°.
3. 35° is the angle between the shorter line and the long line, so the angle between the middle line going left and the long line going up is 70 + 35 = 105°.
4. a° is in the same position at the bottom line (between the bottom line going left and the long line going up), so it is the corresponding angle: a = 105.

Answera = 105°

(iv)

1. The two arrowed lines are parallel. The vertical line meets the base at a right angle (square mark).
2. At the foot, the left arrowed line makes 67° with the base, so it makes 90 − 67 = 23° with the vertical line.
3. The vertical line cuts both arrowed lines. a° (at the top, between the vertical going down and the right arrowed line going down) and that 23° are alternate angles: a = 23.

Answera = 23°

Watch this explained “The whole map”, 8:12 into Alternate angles, and interior angles that add to 180° · हिंदी में देखें

Open NCERT p. 124Matches NCERT’s answer

First figure

1. The two arrowed lines are parallel. The square mark shows the vertical line is perpendicular to the upper line, so it is perpendicular to the lower line too.
2. At the lower crossing, 65° lies between the vertical going up and the slanted line going up-right, so the slanted line makes 90 − 65 = 25° with the lower line going right.
3. x° is vertically opposite that 25° angle, so x = 25.
4. At the upper crossing, the corresponding angle between the upper line going right and the slanted line going up-right is also 25°.
5. y° lies below the upper line, between the upper line going right and the slanted line going down-left, so it forms a linear pair with that 25°: y = 180 − 25 = 155.

Answerx = 25°, y = 155°

Second figure

1. The two arrowed lines are parallel, and the two slanted lines meet at a point on the upper line.
2. At the lower line, one slanted line makes 53° and the other makes 78° with the line going right. By corresponding angles they make the same 53° and 78° with the upper line going right, at their meeting point.
3. x° is the angle between the two slanted lines above the upper line, so x = 78 − 53 = 25.

Answerx = 25°

Watch this explained “The whole map”, 8:12 into Alternate angles, and interior angles that add to 180° · हिंदी में देखें

Open NCERT p. 124Matches NCERT’s answer

1. IKBA and GED are the two arrowed lines, so they are parallel. C, B, E, H lie on one straight line, and J, K, E, F lie on another.
2. Line CBEH cuts the parallel lines at B and E. ∠ABC (between BA going right and BC going up) corresponds to ∠DEB (between ED going right and EB going up), so ∠DEB = 45°.
3. ∠GEH is vertically opposite ∠DEB, so ∠GEH = 45°.
4. Line JKEF cuts the parallel lines at K and E. ∠IKJ (between KI going left and KJ going up) corresponds to ∠GEK (between EG going left and EK going up), so ∠GEK = 78°.
5. ∠FED is vertically opposite ∠GEK, so ∠FED = 78°.
6. G, E and D lie on one straight line, so ∠GEH + ∠HEF + ∠FED = 180°: ∠HEF = 180 − 45 − 78 = 57°.

Answer∠GEH = 45°, ∠HEF = 57°, ∠FED = 78°

Watch this explained “Two hops, and no looking up”, 2:05 into Alternate angles, and interior angles that add to 180° · हिंदी में देखें

Open NCERT p. 125Matches NCERT’s answer

1. AB, CD and EF are parallel. B, D and E lie on one straight line, which cuts all three.
2. ∠BEF = 55° is the angle at E between ED (towards D and B) and EF (going down).
3. At D, y° is the angle between DE (towards E) and CD going down. y° and ∠DEF are both between the parallel lines CD and EF, on the same side of the line BDE, so they add to 180°: y = 180 − 55 = 125.
4. At B, x° is the angle between BD (towards D) and AB going down. It is in the same position as y° at D, so x° and y° are corresponding angles (AB is parallel to CD): x = 125.
5. The fact that EA is perpendicular to AB is not needed for x and y.

Answerx = 125° and y = 125°

Watch this explained “The same walk, a different second step”, 6:23 into Alternate angles, and interior angles that add to 180° · हिंदी में देखें

Open NCERT p. 125Matches NCERT’s answer

1. LM and PQ are both marked with an upward arrow, so they are parallel; M, N, O, P are joined in a zig-zag between them.
2. Draw a line through N parallel to LM (and so also to PQ).
3. By alternate angles on transversal MN, the part of ∠MNO next to NM equals ∠LMN = 40°, leaving 96 − 40 = 56° next to NO.
4. Draw a second line through O parallel to LM (and PQ) as well, so it is parallel to the line already drawn through N.
5. By alternate angles on transversal NO, the 56° found above reappears as the part of ∠NOP next to ON.
6. By alternate angles on transversal OP, ∠OPQ = 52° reappears as the other part of ∠NOP, next to OP.
7. Adding the two parts: ∠NOP = 56 + 52 = 108°.

Answer∠NOP = 108°

Watch this explained “Two hops, and no looking up”, 2:05 into Alternate angles, and interior angles that add to 180° · हिंदी में देखें