NCERT Solutions Class 7 Maths Chapter 5 Parallel and Intersecting Lines

F R E E S T U D Y M AT E R I A L F O R E V E R Y S T U D E N T

C L A S S 7 · M AT H S

NCERT Solutions

Chapter 5: Parallel and
Intersecting Lines

NCERT Textbook — Ganita Prakash

BOOK PAGES SECTIONS QUESTIONS MEDIUM

106 – 126 18 55 English

Solutions, notes, sample papers & more at 80 pages

Class 7 Maths Chapter 5 Parallel and Intersecting Lines AglaSem · NCERT Solutions

CLASS 7 · MATHS · GANITA PRAKASH

NCERT Solutions — Chapter 5: Parallel and Intersecting
Lines
Chapter 5 of Ganita Prakash Part I studies how two lines on a flat surface can behave — they either intersect
at exactly one point, or they never meet at all and are parallel. Once a third line, a transversal, cuts a pair of
lines, eight angles appear, and their relationships let us decide, by reasoning alone, whether the two lines are
parallel.

TEXTBOOK BOOK PAGES

Ganita Prakash (Class 7) 106 – 126

SECTIONS QUESTIONS

18 55

MEDIUM

English

In-text Questions — Page 106
Section 5.1 Across the Line

MATH TALK

Q1 Take a piece of square paper and fold it in different ways. Now, on the creases
formed by the folds, draw lines using a pencil and a scale. You will notice different
lines on the paper. Take any pair of lines and observe their relationship with each
other. Do they meet? If they do not meet within the paper, do you think they would
meet if they were extended beyond the paper?

Some pairs of creases cross inside the paper — those lines meet. Other pairs do not cross inside
the paper, and for those there are two very different possibilities.

If the two creases are slanted differently, the gap between them keeps shrinking on one
side. Extend them beyond the paper and they will meet.
If the two creases keep exactly the same gap everywhere (for example, the two creases you
get by folding the sheet in half twice), they will never meet, however far you extend them.
Such lines are parallel.

Page 1 of 80

Class 7 Maths Chapter 5 Parallel and Intersecting Lines AglaSem · NCERT Solutions

Why it happens: Two straight lines in a plane can cross at most once. So either they
already cross, or they cross somewhere outside the paper, or they never cross at all.
The only way they never cross is if they lean at exactly the same angle — then the
distance between them stays the same for ever.

Check it yourself: Lay a ruler along one crease and slide it towards the other,
keeping it parallel. If the two creases stay the same distance apart all along the ruler,
they will never meet.

Q2 Let us observe what happens when two lines intersect. How many angles do they
form?

Four angles.

Two lines cross at 1 point

Each line is cut into 2 rays

2 rays + 2 rays = 4 rays from the point

4 rays around a point make 4 angles

Why it happens: Going once around the crossing point you pass the four rays one
after another, and between each ray and the next there is one angle. That is why the
count is 4, and why the four angles add up to 360°.

In Fig. 5.2, where line l intersects line m, these four angles are named ∠a, ∠b, ∠c and ∠d.

In-text Questions — Page 107

Page 2 of 80

Class 7 Maths Chapter 5 Parallel and Intersecting Lines AglaSem · NCERT Solutions

Section 5.1 Across the Line

MATH TALK TRY THIS

Q1 Can two straight lines intersect at more than one point?

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

Why it happens: Through any two points there is exactly one straight line. So if two
lines had two common points P and Q, both lines would have to be the same line
through P and Q — they would not be two different lines at all.

Tip: Try it with two rulers on your desk. Once the edges cross at one point, they
immediately start moving apart on both sides.

Q2 Activity 1: Draw two lines on a plain sheet of paper so that they intersect. Measure
the four angles formed with a protractor. Draw four such pairs of intersecting lines
and measure the angles formed at the points of intersection. What patterns do you
observe among these angles?

Whatever pair you draw, the same two patterns show up every time.

∠a = ∠c and ∠b = ∠d (the facing angles are equal)

∠a + ∠b = 180°, ∠b + ∠c = 180°

∠c + ∠d = 180°, ∠d + ∠a = 180°

∠a + ∠b + ∠c + ∠d = 360°

A sample set of readings from four different drawings:

Page 3 of 80

as e
Class 7 Maths Chapter 5 Parallel and Intersecting Lines
a g l AglaSem · NCERT Solutions

∠ A + ∠cBom
.
DRAWING ∠A ∠B ∠C ∠D
m sem
a180°
. co agl
1 120° 60° 120° 60°

e m
2
g l as
a
90° 90° 90° 90° 180°

3 35° 145° 35° 145° 180°

. com72° ag
4 72° 108°
em 108° 180°

g l as
a
Tip: If your readings are a degree or two off, it is the protractor and the thickness of
the pencil line — not the mathematics. Draw thin lines and read the protractor with
co m
e m.
as
your eye straight above the mark.
m l
m .co a g
l a se
g
a Q3 In Fig. 5.2, if ∠a is 120°, can you figure out the measurements of ∠b, ∠c and ∠d,
without drawing and measuring them?
m a s
m .co agl
l a se
a g

co m
m
m .
m as e
.co a g l
se m
g l a
a a b
l se m
com g l a
m . a
d gl ase
a c
co m
m .
m as e
.co a g l
se m
g l a Fig. 5.2, page 107 — line l crossing line m, forming the four angles ∠a, ∠b, ∠c and ∠d.
a c
m .
m a s e
e m . co agl
g l as
a
Yes — use linear pairs one after another.

co m
m .
m ase
.co


a g l Page 4 of 80

Class 7 Maths Chapter 5 Parallel and Intersecting Lines AglaSem · NCERT Solutions

∠a + ∠b = 180° (straight angle)

120° + ∠b = 180° → ∠b = 60°

∠b + ∠c = 180°
60° + ∠c = 180° → ∠c = 120°

∠c + ∠d = 180°

120° + ∠d = 180° → ∠d = 60°

So ∠a = ∠c = 120° and ∠b = ∠d = 60°. Check: 120 + 60 + 120 + 60 = 360° ✔

Why it happens: ∠a and ∠b sit side by side on a straight line, so together they make
a straight angle of 180°. The same is true of the next pair, and the next — that chain
of 180°s is what forces the facing angles to be equal.

Q4 Is this always true for any pair of intersecting lines? Check this for different
measures of ∠a. Using these measurements, can you reason whether this property
holds true for any measure of ∠a?

Yes, it is always true — and we can prove it without using any particular number.

∠a + ∠b = 180° (straight angle on one line)

∠a + ∠d = 180° (straight angle on the other line)

So ∠a + ∠b = ∠a + ∠d

Take ∠a away from both sides → ∠b = ∠d

∠b + ∠a = 180° and ∠b + ∠c = 180°

So ∠b + ∠a = ∠b + ∠c → ∠a = ∠c

Why it happens: Both ∠b and ∠d are "what is left over" from 180° after removing
the same angle ∠a. Two numbers that fill the same gap must be equal. Notice that
no measurement was used — this is a proof, and it works for every value of ∠a.

Page 5 of 80

Class 7 Maths Chapter 5 Parallel and Intersecting Lines AglaSem · NCERT Solutions

Did you know? Angles like ∠a and ∠b are called a linear pair, and angles like ∠b
and ∠d are called vertically opposite angles. This little argument is one of the first
proofs in your geometry course.

Figure it Out — Page 108

Page 6 of 80

Class 7 Maths Chapter 5 Parallel and Intersecting Lines AglaSem · NCERT Solutions

Section 5.1 Across the Line

Q1 List all the linear pairs and vertically opposite angles you observe in Fig. 5.3: Linear
Pairs — ∠a and ∠b, … ; Pairs of Vertically Opposite Angles — ∠b and ∠d, …

Page 7 of 80

Class 7 Maths Chapter 5 Parallel and Intersecting Lines AglaSem · NCERT Solutions

b

a c

d
l
m

Page 8 of 80

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Class 7 Maths Chapter 5 Parallel and Intersecting Lines
a g l AglaSem · NCERT Solutions

Fig. 5.3, page 108 — lines l and m intersecting, with ∠a, ∠b, ∠c and ∠d around the
co m
e m.
crossing.

m l as
m .co a g
l a se
ag

In Fig. 5.3 the lines l and m cross once, giving ∠a on the left, ∠b at the top, ∠c on the right and

co m
. ag
∠d at the bottom.

e m
g l as
a
LINEAR PAIRS PAIRS OF VERTICALLY OPPOSITE ANGLES

∠a and ∠c
m
∠a and ∠b

co
m.
∠b and ∠c ∠b and ∠d
∠c and ∠d
m as e
∠d and ∠a
.co a g l
se m
g l a
a
So there are 4 linear pairs and 2 pairs of vertically opposite angles.

m a s
m .co agl
l a se m
a g

co m
b m .
m as e
.co a g l
se m
g l a
a
a c m
a se
.com a g l
m
ase
agl
d
co m
m .
m as e
.co a g l l
a sem
agl c
m .
Lines l and m crossing. ∠a and ∠c (blue) face each other, and so do ∠b and ∠d (red). Any two angles
m a s e
. co agl
side by side, such as ∠a and ∠b, form a linear pair.

e m
g l as
a

co m
m .
m ase
.co


a g l Page 9 of 80

Class 7 Maths Chapter 5 Parallel and Intersecting Lines AglaSem · NCERT Solutions

Why it happens: Walk once around the crossing point. Each neighbouring pair of
angles lies on one straight line, so each neighbouring pair is a linear pair — that
gives 4 of them. The angles that are not neighbours are the facing ones, and there
are only 2 such pairs.

Tip: Every vertically opposite pair here is equal: ∠a = ∠c and ∠b = ∠d. Every linear
pair adds to 180°.

In-text Questions — Page 108
Section 5.2 Perpendicular Lines

MATH TALK

Q1 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?

Yes, and each angle must be 90°.

∠a + ∠b + ∠c + ∠d = 360° (a full turn)

All four are equal, say each = x

4x = 360°

x = 360° ÷ 4 = 90°

Why it happens: The four angles fill up the complete turn around the crossing
point, which is 360°. Sharing 360° equally among four angles gives 90° each — a
right angle.

Lines that cross like this are called perpendicular lines. In Fig. 5.4, lines l and m are
perpendicular to each other. The right angle is shown by a small square, not by an arc.

Check it yourself: The corner of a page, the corner of your desk and the two edges
of a set square are all perpendicular pairs. Slide the corner of a page into the
crossing — if it fits exactly, the lines are perpendicular.

Page 10 of 80

Class 7 Maths Chapter 5 Parallel and Intersecting Lines AglaSem · NCERT Solutions

In-text Questions — Page 109
Section 5.3 Between Lines

MATH TALK

Q1 Observe Fig. 5.5 and describe the way the line segments meet or cross each other in
each case, with appropriate mathematical words (a point, an endpoint, the
midpoint, meet, intersect) and the degree measure of each angle. For example, line
segments FG and FH meet at the endpoint F at an angle 115.3°.

T
V O
Q
S
B
P
U
C

X R
D

A I
L
H
F

115.3° J

G
M

Fig. 5.5, page 109 — the pairs of line segments ST and UV, AB and CD, OP and QR, FG and
FH, and IJ and LM.

There are five pairs in Fig. 5.5. Here is a description of each, with the angle you should get on
measuring.

Page 11 of 80

Class 7 Maths Chapter 5 Parallel and Intersecting Lines AglaSem · NCERT Solutions

PAIR HOW THEY MEET ANGLE

FG and They meet at the endpoint F — F is an end of both segments 115.3°
FH (given example)

AB and They intersect at a point X. X is inside both segments, so it is about 60° (and 120° on the
CD an endpoint of neither other side)

IJ and They intersect at a point Y, which lies near the middle of about 79° (and 101° on the
LM each segment other side)

ST and They do not meet at all inside the figure — no common point no angle formed
UV

OP and They do not meet at all — they stay the same distance apart no angle formed
QR

Why it happens: The words describe where the meeting happens. If the common
point is an end of a segment we say the segments meet at an endpoint; if the
common point lies inside both, we say the segments intersect at that point.

Tip: Measure all four angles at X and at Y. You should find two equal pairs each time,
and each neighbouring pair adding to 180° — the very properties proved on page
107.

Q2 Are line segments ST and UV likely to meet if they are extended?

Yes. They look almost parallel, but they are not — they slowly come closer as you go to the right.

Gap between ST and UV at the left ends ≈ 12 units

Gap at the right ends ≈ 9 units

The gap is shrinking → the lines are converging

So if both segments are extended far enough towards the right, they will cross.

Page 12 of 80

Class 7 Maths Chapter 5 Parallel and Intersecting Lines AglaSem · NCERT Solutions

Why it happens: UV is tilted a little more steeply than ST (about 22° against about
20°). Two lines that are not tilted by exactly the same amount must meet
somewhere, however small the difference is — you may just have to extend them a
long way.

Tip: Do not trust the eye. Measure the perpendicular gap at two places far apart. If
the two gaps differ, the lines are not parallel.

Q3 Are line segments OP and QR likely to meet if they are extended?

No. OP and QR are parallel, so they will never meet however far they are extended.

Tilt of OP ≈ 33° below the horizontal

Tilt of QR ≈ 33° below the horizontal

Same tilt → the perpendicular gap stays the same everywhere

Why it happens: Two lines meet only when they lean differently. OP and QR lean by
the same amount, so the space between them never closes up.

Check it yourself: Measure the shortest (perpendicular) distance from O to line QR
and from P to line QR. Both give the same value for a parallel pair.

In-text Questions — Page 110
Section 5.3 Between Lines

MATH TALK

Q1 Name some parallel lines you can spot in your classroom.

Parallel lines are everywhere in a classroom:

The top and bottom edges of the blackboard, and its left and right edges.

Page 13 of 80

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Class 7 Maths Chapter 5 Parallel and Intersecting Lines
a g l AglaSem · NCERT Solutions

The ruled lines in your notebook.
co m
The two long edges of your scale (ruler).
e m.
m l as
.co a g
The opposite edges of a window pane, of the door, and of a floor tile.

a s em
The legs of the benches in a row, and the two rails of the ceiling fan's rod holder.

a l grooves of the roof sheets or the lines of bricks in a wall.
gThe

c om
Why it happens: All these pairs lie on the same flat surface and keep the same
. ag
s e
distance apart along their whole length,m so they can never meet.
a
agl
Tip: The edge of the board and a line drawn on the floor never meet either — but

co m
m.
they are not parallel, because they do not lie in the same plane.

m as e
.co a g l
a s em
a glQ2 Which pairs of lines appear to be parallel in Fig. 5.6 below?
m a s
m .co agl
l a se
a
d ag e

m
c

g
. co
e m
com l as
b

. a g
m
f

ase
agl
i h

se m
com g l a
.
Fig. 5.6, page 110 — nine line segments a to i drawn on dot paper.

m a
ase
agl

co m
Group the segments by the direction in which they point.
m .
m as e
.co a g l
em Vertical (straight up and down)
DIRECTION SEGMENTS THAT ARE PARALLEL

a s
agl a, i and h

.c
s e m
m a
Horizontal (straight across) c and g

e m . co agl
as
Slanting down to the right (45°) d and f

a
Slanting down to the left (45°) g l e and b

co m
m .
m ase
.co


a g l Page 14 of 80

Class 7 Maths Chapter 5 Parallel and Intersecting Lines AglaSem · NCERT Solutions

So the parallel sets are a, i, h; c and g; d and f; e and b.

Why it happens: On dot paper you can read the direction of a segment as "so many
dots across, so many dots up". Two segments with the same across-and-up step
point the same way, so they are parallel even though they have different lengths and
start at different dots.

Check it yourself: Count the steps for d and for f. Both go 1 dot right and 1 dot
down for every step — same direction, hence parallel.

Activity 2 — Page 111
Section 5.4 Parallel and Perpendicular Lines in Paper Folding

TRY THIS

Q1 Take a plain square sheet of paper (use a newspaper for this activity). How would
you describe the opposite edges of the sheet? They are _________________________ to each
other.

The opposite edges of the sheet are parallel to each other.

Why it happens: The left and right edges of a square sheet stay exactly the same
distance apart (the side length) all the way from top to bottom. Lines in a plane that
keep a constant gap can never meet, so they are parallel.

Q2 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.

The adjacent edges are perpendicular to each other.

Each corner of the sheet is a right angle
Angle between two adjacent edges = 90°

Page 15 of 80

Class 7 Maths Chapter 5 Parallel and Intersecting Lines AglaSem · NCERT Solutions

Why it happens: A square (or a rectangle) has four right-angled corners. Two edges
that meet at a corner cross at 90°, which is exactly the meaning of perpendicular.

Page 16 of 80

Class 7 Maths Chapter 5 Parallel and Intersecting Lines AglaSem · NCERT Solutions

Q3 Fold the sheet horizontally in half. A new line is formed (see Fig. 5.7). How many
parallel lines do you see now? How does the new line segment relate to the vertical
sides?

Page 17 of 80

Class 7 Maths Chapter 5 Parallel and Intersecting Lines AglaSem · NCERT Solutions

Page 18 of 80

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Class 7 Maths Chapter 5 Parallel and Intersecting Lines
a g l AglaSem · NCERT Solutions

co m
e m.
m l as
m .co a g
l a se
a g

co m
e m . ag
g l as
a

co m
em.
m l as
m .co a g
l a se
a g
m a s
m .co agl
l a se
a g

co m
m .
m as e
.co a g l
se m
g l a
a
se m
com g l a
m . a
ase
agl

co m
m .
m as e
.co a g l
se m
g l a
a c
m .
m a s e
e m . co agl
g l as
a

co m
m .
m ase
.co


a g l Page 19 of 80

Class 7 Maths Chapter 5 Parallel and Intersecting Lines AglaSem · NCERT Solutions

Fig. 5.7, page 111 — folding a square sheet horizontally in half, then again, then once
more, and finally a vertical fold.

You now see 3 parallel lines — the top edge, the new crease, and the bottom edge.

2 horizontal edges + 1 crease = 3 parallel lines

The new crease is perpendicular to the two vertical sides, and it cuts each of them exactly in
half.

Why it happens: Folding the top edge down onto the bottom edge makes the
crease run in exactly the same direction as those edges, so all three are parallel.
Since the vertical sides were already perpendicular to the top and bottom edges,
they are perpendicular to the crease as well.

Q4 Make one more horizontal fold in the folded sheet. How many parallel lines do you
see now?

5 parallel lines.

Folding the already-folded sheet in half makes 3 creases in all

3 creases + 2 horizontal edges = 5 parallel lines

Why it happens: The single crease you already had is copied on both halves of the
new fold, so one crease becomes three. Every one of them runs parallel to the top
and bottom edges.

Page 20 of 80

Class 7 Maths Chapter 5 Parallel and Intersecting Lines AglaSem · NCERT Solutions

Q5 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.

After a third fold you get 9 parallel lines, and after a fourth fold 17.

NUMBER OF FOLDS, N CREASES PARALLEL LINES

1 1 3

2 3 5

3 7 9

4 15 17

Creases after n folds = 2n – 1

Parallel lines = creases + 2 edges

= (2n – 1) + 2

= 2n + 1

Why it happens: Each new fold doubles the number of layers, so every existing
crease is copied once and one brand-new crease is added in the middle. That is why
the crease count goes 1 → 3 → 7 → 15, each time doubling and adding one.

Note: The pattern is 2n + 1 (2, raised to the power n), not 2 × n + 1. For n = 3 the
formula 2 × 3 + 1 would give 7, but folding really gives 9 lines — count them on your
own sheet.

Q6 Make a vertical fold in the square sheet. This new vertical line is ___________ to the
previous horizontal lines.

This new vertical line is perpendicular to the previous horizontal lines.

Page 21 of 80

Class 7 Maths Chapter 5 Parallel and Intersecting Lines AglaSem · NCERT Solutions

Vertical crease meets each horizontal crease at 90°

Why it happens: The vertical fold brings the left edge onto the right edge, so the
crease runs in the direction of the vertical sides. Those sides are perpendicular to the
horizontal creases, so the new crease is perpendicular to all of them.

Tip: All the horizontal creases are parallel to one another, so a single line
perpendicular to one of them is automatically perpendicular to all of them.

Q7 Fold the sheet along a diagonal. Can you find a fold that creates a line parallel to
the diagonal line?

Yes. Here is one easy way for a square sheet ABCD with the diagonal crease AC.

1. Fold along both diagonals once to mark the centre O of the square.
2. Now fold corner B so that it lands exactly on O.
3. Unfold. The new crease is parallel to the diagonal AC.

Why it happens: B is carried to O along the line BD, which is perpendicular to AC. A
fold moves a point straight across the crease, so the crease must be perpendicular
to BD — and any line perpendicular to BD runs in the same direction as AC. Same
direction means parallel.

Try This: Fold B onto the midpoint of BO instead. You get another crease parallel to
AC, but closer to B. In fact every fold that takes B to a point of BD gives a line parallel
to AC.

In-text Questions — Page 112

Page 22 of 80

Class 7 Maths Chapter 5 Parallel and Intersecting Lines AglaSem · NCERT Solutions

Section 5.4 Paper Folding, and Notations

TRY THIS

Q1 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?

c a c
a

b b

q q

r p r p

Page 23 of 80

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Class 7 Maths Chapter 5 Parallel and Intersecting Lines
a g l AglaSem · NCERT Solutions

Fig. 5.8, page 112 — the square sheet creased into quarters, with the top right and
co m
e m.
bottom left corners folded into the triangles a, b, c and p, q, r.

m l as
m .co a g
l a se
ag

Yes. a ∥ p, b ∥ q and c ∥ r.

. comWHY THEY ARE PARALLEL ag
e m
as
PAIR WHAT THE TWO SIDES ARE

a g l
a and The vertical sides of the two triangles, All the creases run parallel to the vertical edges of the
lying along crease lines
m
p square

co
e m.
as
b and The horizontal sides of the two triangles Both are perpendicular to those vertical creases, so

m l
.co g
q they point the same way

m The two fold lines (the slanting sides) a
l a se
g
c and r Each is the diagonal of an equal square, so each makes

a the same 45° with the edges

m a s
m .co
Why it happens: The two corner folds are made in exactly the same way at opposite agl
l a se
corners. So the whole picture looks unchanged if you turn the sheet through half a
a g
turn (180°) about its centre — the top-right triangle lands exactly on the bottom-left
one. A half-turn always sends a line to a line pointing in the same direction, that is,
co m
to a parallel line. That single fact explains all three pairs at once.
m .
m as e
.co a g l
s e m it yourself: Trace triangle abc on tracing paper, put a pin through the centre
Check

agla of the square, and rotate the tracing by 180°. It falls exactly on triangle pqr.
se m
com g l a
m. a
ase
agl
Figure it Out — Pages 113–114

co m
m .
m as e
.co a g l
se m
g l a
a c
m .
m a s e
e m . co agl
g l as
a

co m
m .
m as e
.co


a g l Page 24 of 80

Class 7 Maths Chapter 5 Parallel and Intersecting Lines AglaSem · NCERT Solutions

Section 5.4 Parallel and Perpendicular Lines in Paper Folding

Q1 Draw some lines perpendicular to the lines given on the dot paper in Fig. 5.10.

Fig. 5.10, page 113 — four line segments drawn on dot paper.

Read each given segment as a step: "so many dots right, so many dots down". To turn it, swap
the two numbers and change one sign.

GIVEN SEGMENT ITS STEP A PERPENDICULAR SEGMENT

Horizontal one, top row 2 right, 0 down 0 right, 1 down — a vertical segment

Slanting one, bottom left 2 right, 2 up 2 right, 2 down — the other 45° slant

Vertical one, middle 0 right, 2 down 2 right, 0 down — a horizontal segment

Steep one, right side 2 right, 3 down 3 right, 2 up

Page 25 of 80

Class 7 Maths Chapter 5 Parallel and Intersecting Lines AglaSem · NCERT Solutions

Grey = the four given lines; blue = the perpendiculars drawn on the dots. Each blue segment crosses
its grey partner at 90°, and every endpoint sits on a dot.

Why it happens: If one segment goes p right and q down, a segment that goes q
right and p up turns the direction by exactly a quarter turn. The dot grid keeps both
endpoints on dots, so you never need a protractor.

Check it yourself: Put the corner of a page at each crossing. It should fit snugly with
no gap.

Page 26 of 80

Class 7 Maths Chapter 5 Parallel and Intersecting Lines AglaSem · NCERT Solutions

Q2 In Fig. 5.11, mark the parallel lines using the notation given 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?

Fig. 5.11, page 113 — five shapes drawn on squared paper.

Work shape by shape on the grid.

Mark every vertical edge with a single arrow head (>), and every horizontal edge with a
double arrow head (>>).
The slanting edges that run along the grid diagonals fall into two families — down-to-the-
right and up-to-the-right. Mark one family with three arrow heads and the other with four.

Page 27 of 80

Class 7 Maths Chapter 5 Parallel and Intersecting Lines AglaSem · NCERT Solutions

Wherever a vertical edge meets a horizontal edge, draw the small square symbol for 90°.

(a) How did you spot the perpendicular lines? The vertical and horizontal lines of the grid
paper meet at a right angle (90°), so wherever one edge follows a grid column and the other
follows a grid row, the two edges are perpendicular. The corner of a page or a set square
confirms it.
(b) How did you spot the parallel lines? By finding lines that always stay the same distance
apart. On grid paper this is easy: two edges are parallel when they take the same step — for
example both go 1 square right and 1 square down.

Why it happens: A grid is itself made of two families of parallel lines that cut each
other at 90°. Any edge drawn along the grid inherits these relationships, so you can
judge parallel and perpendicular by counting squares instead of measuring.

Q3 In the dot paper following, draw different sets of parallel lines. The line segments
can be of different lengths but should have dots as endpoints.

Choose a step and repeat it starting from different dots. Some easy sets are given below.

SET STEP USED EXAMPLE SEGMENTS (DOTS JOINED)

1 3 right, 0 down Three horizontal segments of lengths 2, 3 and 4 units in different rows

2 0 right, 2 down Vertical segments in different columns

3 1 right, 1 down 45° segments starting from different dots

4 2 right, 1 up Gently rising segments — a harder but neat set

Why it happens: Two segments are parallel exactly when they have the same
direction. On dot paper the direction is completely fixed by the step, so same step =
parallel, whatever the length and wherever you start.

Tip: Make the segments different lengths on purpose. It reminds you that being
parallel is about direction, not size.

Page 28 of 80

as e
Class 7 Maths Chapter 5 Parallel and Intersecting Lines
a g l AglaSem · NCERT Solutions

co m
m.
Using your sense of how parallel lines look, try to draw lines parallel to the line
e
Q4

comones? (c) How did you do it?
segments on this dot paper. (a) Did you find it challenging to draw some of them?
g l as
. a
em
(b) Which

a s
a gl

m
For each segment in Fig. 5.12, count its step and repeat that same step from a different dot.
. co ag
e m
(a) Did you find it challenging? Yes — some were much harder than others.

g l as
a
(b) Which ones? The segments e, f, g and h. These do not run along a row, a column or a 45°
diagonal, so the eye cannot judge them easily. The horizontal, vertical and 45° ones (a, b, c, d)

m
were simple.
co
e m.
(c) How did you do it? Two reliable methods:

m l as
.co a g
1. Count the step. If the segment goes, say, 3 dots right and 1 dot down, start at any other dot

a s emgo 3 right and 1 down. Join the two dots.
and

a gl2. Keep the distance equal. Mark two points at the same perpendicular distance from the
given segment, one near each end, and join them.

m a s
e
. c o
m like "3 right, 1 down" has no landmark on the agl
s
Why it happens: A slanted direction
awrong. Counting dots turns the direction into two
agl
page to copy, so guessing goes
whole numbers, which are easy to repeat exactly.

. com
m is right, the
Check it yourself: Draw any line cutting both segments. If your edrawing
s
o m
cangles a
gl will be equal.
m
two. a
it makes in the same position at the two crossings

ase
agl
se m
com g l a
m . a
ase
agl

co m
m .
m as e
.co a g l
se m
g l a
a c
m .
m a s e
e m . co agl
g l as
a

com
m .
m ase
.co


a g l Page 29 of 80

Class 7 Maths Chapter 5 Parallel and Intersecting Lines AglaSem · NCERT Solutions

Q5 In Fig. 5.13, which line is parallel to line a — line b or line c? How do you decide this?

b
c

a

Fig. 5.13, page 114 — line a, and lines b and c drawn from one common point.

Line c is parallel to line a.
Lines b and c start from the same point, and in the figure there is already a segment joining that
point to the end of line a. Use it as a transversal and compare the angles it makes.

Angle between the joining line and line a ≈ 29°

Angle between the joining line and line c ≈ 29° → equal ✔

Angle between the joining line and line b ≈ 32° → not equal ✘

Corresponding angles are equal only for c and a, so c ∥ a. Line b leans a little more, so b and a
would meet if extended far enough to the right.

Why it happens: Eyes are poor at judging small differences in slope — 29° and 32°
look alike. A transversal converts the question into a comparison of two angles,
which a protractor (or tracing paper) settles exactly.

Page 30 of 80

Class 7 Maths Chapter 5 Parallel and Intersecting Lines AglaSem · NCERT Solutions

Tip: Another quick check — measure the perpendicular gap between a and c near
both ends. For a parallel pair the two readings match.

In-text Questions — Page 115
Sections 5.5 Transversals and 5.6 Corresponding Angles

MATH TALK

Q1 Is it possible for all the eight angles to have different measurements? Why, why
not?

No, it is not possible. The eight angles can show at most four different sizes.

At line l: ∠1 = ∠3 and ∠2 = ∠4 (vertically opposite)

At line m: ∠5 = ∠7 and ∠6 = ∠8 (vertically opposite)

8 angles → grouped into 4 equal pairs
So at most 4 different measures

Page 31 of 80

Class 7 Maths Chapter 5 Parallel and Intersecting Lines AglaSem · NCERT Solutions

1 2 l
4 3

5 6 m
8 7

t
Transversal t cuts lines l and m, making eight angles. ∠1, ∠2, ∠3, ∠4 at the first crossing; ∠5, ∠6, ∠7,
∠8 at the second.

Why it happens: At each crossing the four angles are not independent — the facing
ones must be equal and neighbours must add to 180°. So each crossing offers only
two different sizes, and two crossings give at most 2 + 2 = 4.

Did you know? If lines l and m happen to be parallel, even those four sizes collapse
to two: one acute and one obtuse, adding to 180°.

Q2 What about five different angles — 6, 5, 4, 3 and 2?

Five is not possible either. Nor is three or six.

Page 32 of 80

Class 7 Maths Chapter 5 Parallel and Intersecting Lines AglaSem · NCERT Solutions

Possible numbers of different measures = 1, 2 or 4

4 → the general case (l and m not parallel)

2 → l ∥ m, transversal slanting

1 → t perpendicular to both l and m (all eight angles 90°)

To get five different values you would need at least one crossing to show three different angles
— impossible, because ∠2 = ∠4 and ∠1 = ∠3 always.

Why it happens: Angles come in forced pairs. Once you fix one angle at a crossing,
the other three are decided — one equals it, and the other two are its supplement.
So the count of different sizes can only be 1, 2 or 4.

Page 33 of 80

as e
Class 7 Maths Chapter 5 Parallel and Intersecting Lines
a g l AglaSem · NCERT Solutions

co m
m.
In Fig. 5.14, since ∠1 and ∠3 are vertically opposite angles, they are equal. Are there
e
Q3

m l as
.co
other pairs of vertically opposite angles?

a g
se m
g l a
a

co m
e m . ag
g l as
a
1 2
lco m
em.
m l as
.co 4 3 a g
a sem
agl m
5 6
m a s
em8
.co agl
a s
a gl 7

co m
m .
m
t
as e
.co a g l
se m
g l a
a
m
Fig. 5.14, page 115 — transversal t crossing lines l and m, forming the eight angles ∠1 to

a se
com l
∠8.
. a g
m
ase
agl

co m
.
Yes — four pairs in all, two at each crossing.

em
m l as
.co a g
CROSSING VERTICALLY OPPOSITE PAIRS RELATION

a s em t with line l
agl
∠1 and ∠3 equal

.c
t with line l ∠2 and ∠4 equal
s e m
m a
m . co agl
se
t with line m ∠5 and ∠7 equal

l a
t with line m
a∠6g and ∠8 equal

co m
m .
m ase
.co


a g l Page 34 of 80

Class 7 Maths Chapter 5 Parallel and Intersecting Lines AglaSem · NCERT Solutions

Why it happens: Each crossing of two lines is a complete little picture of its own, and
every such crossing has exactly two pairs of facing angles. Two crossings therefore
give 2 × 2 = 4 pairs.

Tip: Do not confuse these with corresponding angles — ∠1 and ∠5, ∠2 and ∠6, ∠3
and ∠7, ∠4 and ∠8 — which sit in the same position at the two different crossings.

Activity 3 — Page 116
Section 5.6 Corresponding Angles

TRY THIS

Q1 Activity 3: Draw a pair of lines and a transversal such that they form two distinct
angles. Step 1: Draw a line l and a transversal t intersecting it at point X. Step 2:
Measure ∠a formed by lines l and t (let us say it is 60°). Step 3: Mark a point Y on line
t. Step 4: Draw a line m through point Y that forms a 60° angle to line t.

Follow the four steps with a ruler and a protractor.

1. Draw line l, then draw t cutting it at X.
2. Measure ∠a between l and t. Suppose it comes to 60°.
3. Mark any point Y further along t.
4. At Y, set off 60° from t — on the same side and in the same position as ∠a — and draw line
m.

∠a = 60° (at X, between l and t)

∠b = 60° (at Y, between m and t)

∠a and ∠b are corresponding angles

Only two sizes appear anywhere: 60° and 120°

Why it happens: Copying the angle at Y makes line m lean exactly like line l. Every
angle at Y is then a copy of the matching angle at X, so no new sizes can appear.

Page 35 of 80

Class 7 Maths Chapter 5 Parallel and Intersecting Lines AglaSem · NCERT Solutions

Tip: Instead of a protractor you may trace ∠a on tracing paper and slide the tracing
along t up to Y. Copying is more accurate than measuring.

Q2 How many distinct angles have formed now?

Only two distinct angles: 60° and 120°.

At X: ∠a = 60°
Linear pair → 180° – 60° = 120°

Vertically opposite → 60° and 120° again

At Y the same two values repeat

Total distinct measures = 2

Why it happens: One angle fixes all four at a crossing — the angle itself, its
supplement, and their two facing copies. Since we deliberately copied 60° at Y, the
second crossing repeats the very same two values.

Q3 What do you observe about lines l and m? Do they appear to be parallel to each
other?

Yes, l and m appear to be parallel — and they really are.

Corresponding angles ∠a = ∠b = 60°

Equal corresponding angles ⟹ l ∥ m

Why it happens: The angle a line makes with the transversal tells you how much
that line leans. Since l and m make the same 60° with t, they lean by the same
amount, so the gap between them never changes and they can never meet.

Page 36 of 80

Class 7 Maths Chapter 5 Parallel and Intersecting Lines AglaSem · NCERT Solutions

Did you know? This is the important rule of the chapter: when the corresponding
angles made by a transversal on a pair of lines are equal, the lines are parallel.
It is a way of testing for parallelism without extending the lines at all.

Activities 4 and 5 — Page 117

Page 37 of 80

Class 7 Maths Chapter 5 Parallel and Intersecting Lines AglaSem · NCERT Solutions

Section 5.6 Corresponding Angles

TRY THIS

Q1 Activity 4: Fig. 5.19 has a pair of parallel lines l and m (what is the notation used in
the figure to indicate they are parallel?). Line t is the transversal across these two
lines.

l a

m b

t

Fig. 5.19, page 117 — parallel lines l and m (marked with matching arrow heads) cut by
transversal t; ∠a and ∠b are corresponding angles.

The notation is a pair of matching arrow heads (>) — one drawn on line l and one on line m.

Page 38 of 80

as e
Class 7 Maths Chapter 5 Parallel and Intersecting Lines
a g l AglaSem · NCERT Solutions

co m
m.
Why it happens: A picture cannot be extended for ever, so we cannot see that two

m l a
lines never meet. The arrow marks are a promise from the person who drew the
o se
.cthose lines are parallel. a g
m
figure that

l a se
g
aTip: If a figure has a second set of parallel lines, that set is marked with double

co m
ag
arrow heads (>>), a third set with triple arrow heads, and so on. Perpendicular lines
.
s em
are marked with a small square instead.
a
agl

co m
em.
m l as
m .co a g
l a se
a g
m a s
m.co agl
l a se
a g

co m
m .
m as e
.co a g l
se m
g l a
a
se m
com g l a
m . a
ase
agl

co m
m .
m as e
.co a g l
se m
g l a
a c
m .
m a s e
e m . co agl
g l as
a

co m
m .
m ase
.co


a g l Page 39 of 80

Class 7 Maths Chapter 5 Parallel and Intersecting Lines AglaSem · NCERT Solutions

Q2 Take a tracing paper and trace ∠a on it. Now place this tracing paper over ∠b and
see if the angles align exactly. Check the other corresponding angles in the figure
using a protractor. Are all the corresponding angles equal to each other?

l a

m b

t

Fig. 5.19, page 117 — parallel lines l and m (marked with matching arrow heads) cut by
transversal t; ∠a and ∠b are corresponding angles.

Yes. The tracing of ∠a falls exactly on ∠b, and every other corresponding pair matches too.

CORRESPONDING PAIR WHAT YOU FIND

∠a and ∠b equal

the other three pairs at the two crossings equal as well

Page 40 of 80

Class 7 Maths Chapter 5 Parallel and Intersecting Lines AglaSem · NCERT Solutions

l ∥ m and t is a transversal

⟹ every pair of corresponding angles is equal

Why it happens: Sliding the tracing paper straight along the transversal from the
first crossing to the second is a translation. Because l and m point in the same
direction, the whole first crossing slides exactly onto the second — so every angle
lands on its corresponding partner.

Check it yourself: Once one corresponding pair is known to be equal, the other
three follow at once, using linear pairs and vertically opposite angles.

Q3 Activity 5: In Fig. 5.20, draw a transversal t to the lines l and m such that one pair of
corresponding angles is equal. You can measure the angles with a protractor. Are
you finding it hard to draw a transversal such that the corresponding angles are
equal?

l

m

Fig. 5.20, page 117 — lines l and m, drawn without any transversal.

Yes — and it is impossible, because the lines l and m in Fig. 5.20 are not parallel.

Page 41 of 80

Class 7 Maths Chapter 5 Parallel and Intersecting Lines AglaSem · NCERT Solutions

Try any transversal you like

Corresponding angles always differ

The difference stays the same for every transversal
= the angle at which l and m would meet

Why it happens: Equal corresponding angles would force l and m to lean identically
— that is, to be parallel. Since they are not parallel, no transversal can ever make a
corresponding pair equal.

Did you know? This gives the rule the other way round too: when a pair of lines is
not parallel, the corresponding angles formed by any transversal can never be
equal. Equality of corresponding angles is therefore both necessary and sufficient
for two lines to be parallel.

In-text Questions — Pages 118–119

Page 42 of 80

Class 7 Maths Chapter 5 Parallel and Intersecting Lines AglaSem · NCERT Solutions

Section 5.7 Drawing Parallel Lines

MATH TALK

Q1 Can you draw a pair of parallel lines using a ruler and a set square? Fig. 5.21 shows
how you can do it. Draw a line l with a scale. By sliding your set square you can
make two lines perpendicular to line l.

l

Page 43 of 80

as e
Class 7 Maths Chapter 5 Parallel and Intersecting Lines
a g l AglaSem · NCERT Solutions

Fig. 5.21, page 118 — a ruler laid along line l, and a set square slid along it to draw two
co m
em.
lines perpendicular to l.

m l as
m .co a g
l a se
ag

Yes. Here is the method shown in Fig. 5.21.

com
1. Draw line l with the scale.
e m . ag
l as
2. Place one edge of the set square along l and draw a line along the edge that stands at 90° to
g
it. a
3. Slide the set square along l to a new place and draw a second line along that same 90° edge.

co m
e m.
m l as
.co g
Both new lines make 90° with l

m a
l a se
These 90° angles are corresponding angles

a g Equal corresponding angles ⟹ the two new lines are parallel

om a s
e
. c
m does not turn it, so the drawing edge keeps agl
s
Why it happens: Sliding the set square
a
a l lines therefore lean identically and can never meet.
pointing the same way. Bothgnew

co m
m .
o m l a se
Are these two lines parallel to each other? How are we sure that they are parallel to
ag and line l?
.ceach other? What angles are formed between these lines
Q2

se m
g l a
a ANSWER

se m
com
Yes, they are parallel. The angles between each of them and line l are 90°.
g l a
m . a
e
s square)
a(set
a g l
Angle of first line with l = 90°

Angle of second line with l = 90°

co m
Treat l as a transversal of the two new lines
m .
m as e
.co
Corresponding angles = 90° = 90° → equal
a g l
se m ⟹ the two lines are parallel
g l a
a c
m .
s e
. c om
Why it happens: We are not relying on how the picture looks. We are using the rule
a g la
s e m guarantee parallelism — reasoning, not eyesight.
that equal corresponding angles
a
agl

co m
m .
m as e
.co


a g l Page 44 of 80

Class 7 Maths Chapter 5 Parallel and Intersecting Lines AglaSem · NCERT Solutions

Did you know? A useful consequence: two lines that are both perpendicular to the
same line are parallel to each other.

Page 45 of 80

Class 7 Maths Chapter 5 Parallel and Intersecting Lines AglaSem · NCERT Solutions

Q3 Draw two more parallel lines using the long side of the set square as shown in Fig.
5.22.

Page 46 of 80

Class 7 Maths Chapter 5 Parallel and Intersecting Lines AglaSem · NCERT Solutions

Page 47 of 80

Class 7 Maths Chapter 5 Parallel and Intersecting Lines AglaSem · NCERT Solutions

l

Page 48 of 80

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Class 7 Maths Chapter 5 Parallel and Intersecting Lines
a g l AglaSem · NCERT Solutions

Fig. 5.22, page 118 — the long side of the set square slid along the ruler to draw two
co m
e m.
more parallel lines.

m l as
m .co a g
l a se
ag

This time the drawing edge is not at 90° to l, but the idea is exactly the same.

co m
1. Keep the scale fixed on the paper as a guide rail.
e m . ag
l as
2. Hold the long side of the set square against the scale and draw a line along the opposite
g
edge. a
3. Slide the set square along the scale — without turning it — and draw a second line along the

co m
m.
same edge.

as e
. com a g l
e m
Both lines
s
make the same angle with the scale's edge

aglaThat edge acts as a transversal
s
Corresponding angles equal ⟹ the lines are parallel
m a
em
.co agl
a s
a gl the set square from rotating. Only sliding is
Why it happens: The scale stops
allowed, and sliding preserves direction.

co m
m .
o m l a se
Q4 .cHow do you know these two lines are parallel? Can you
a g if the corresponding
m
e angles are equal?
check

aglas
se m
m a

o
Draw any line cutting both of them and.ccompare the two angles that sit in the same position.
a g l
m
se at each crossing on the same side of the transversal and
By protractor: measure thela
agtwo readings agree.
angle
in the same position. The

m
By tracing paper: trace one angle, slide the tracing along the transversal to the other
. co
m
crossing. It fits exactly.

m as e
.co a g l
s e m If the two readings are equal → the lines are parallel
agla
.c
If they differ by even 2° or 3° → the lines are not parallel

s e m
m a
co agl
and they will meet somewhere far away

m .
as e
a g l

com
m .
m ase
.co


a g l Page 49 of 80

Class 7 Maths Chapter 5 Parallel and Intersecting Lines AglaSem · NCERT Solutions

Why it happens: Equality of corresponding angles is both necessary and sufficient
for a pair of lines to be parallel. So this single check settles the question completely.

Tip: Small measuring errors are normal. If your two readings differ by less than a
degree, treat the lines as parallel and blame the protractor and the thickness of the
pencil line.

Figure it Out — Page 119
Section 5.7 Drawing Parallel Lines

Q1 Can you draw a line parallel to l, that goes through point A? How will you do it with
the tools from your geometry box? Describe your method.

Yes. Tools needed: a ruler (scale), a set square and a pencil.

1. Place one edge of the set square exactly along line l.
2. Hold the ruler firmly against another edge of the set square, so the ruler cannot move — it is
now a rail.
3. Slide the set square along the ruler until the edge that was on l reaches point A.
4. Draw a line along that edge through A.
5. This new line passes through A and is parallel to l.

The set square only slides, never turns

So the drawing edge keeps the direction of l

⟹ new line ∥ l, and it passes through A

Alternative method with the protractor: draw any line through A that cuts l at a point P.
Measure the angle it makes with l at P. Copy that same angle at A, in the corresponding position,
and draw the second arm. Equal corresponding angles make the new line parallel to l.
Alternative method with paper folding: fold a crease t through A perpendicular to l; then fold
a crease m through A perpendicular to t. Then m ∥ l.

Why it happens: All three methods do the same thing in different clothing — they
force the new line to make the same angle with some transversal as l does. Equal
corresponding angles guarantee parallelism.

Page 50 of 80

Class 7 Maths Chapter 5 Parallel and Intersecting Lines AglaSem · NCERT Solutions

Did you know? Through a point not on a line there is exactly one line parallel to it.
So whichever method you use, everyone in the class gets the same line.

In-text Questions — Page 120
Section 5.8 Alternate Angles

MATH TALK TRY THIS

Q1 Why are lines l and m parallel to each other? (Making Parallel Lines through Paper
Folding, Fig. 5.24)

A

l l

t t
A A
m

l l

Fig. 5.24, page 119 — making a crease m through point A parallel to crease l, using two
perpendicular folds.

Because both of them are perpendicular to the same crease t.

Page 51 of 80

Class 7 Maths Chapter 5 Parallel and Intersecting Lines AglaSem · NCERT Solutions

Fold 1: crease t ⟂ l, passing through A

Fold 2: crease m ⟂ t, passing through A

Treat t as a transversal of l and m
Corresponding angles = 90° and 90° → equal

⟹l∥m

Why it happens: A fold that lays a line exactly on top of itself must be perpendicular
to that line. So each fold creates a true right angle — no measuring needed. Two
lines that make the same 90° with a common transversal lean the same way, hence
they are parallel.

Tip: This is the folding version of the set-square method on page 118. Both rest on
the same rule: two lines perpendicular to one line are parallel to each other.

Page 52 of 80

Class 7 Maths Chapter 5 Parallel and Intersecting Lines AglaSem · NCERT Solutions

Q2 Activity 6: In Fig. 5.25, if ∠f is 120° what is the measure of its alternate angle ∠d?

l
a b

d c

m
e f

h g

t

Fig. 5.25, page 120 — parallel lines l and m cut by transversal t; the eight angles ∠a to
∠h.

∠d = 120°.

∠f = 120° (given)

∠b = ∠f = 120° (corresponding angles, l ∥ m)

∠d = ∠b = 120° (vertically opposite angles)

So ∠d = 120°

The same chain works for any value of ∠f:

Page 53 of 80

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Class 7 Maths Chapter 5 Parallel and Intersecting Lines
a g l AglaSem · NCERT Solutions

co m
e m.
∠f = ∠b (corresponding)

com g l as
.
∠b = ∠d (vertically opposite)
a
a s em ∠f = ∠d, always
agl
Therefore

. com
Why it happens: To find the alternate angle of ∠f, first step across to its
ag
m then turn to the facing angle ∠d. Each
corresponding angle ∠b at the othereline,
s
a
agl ends of the chain must be equal — without any
step preserves the size, so the two
measurement.

co m
se m.
o m l a
Did you know? This proves the rule: alternate angles formed by a transversal
g5.25, ∠c and ∠e are
m .c a
se other alternate pair.
intersecting a pair of parallel lines are always equal. In Fig.

g l a
the
a
m a s
m.co agl
Examples — Pages 120–122
l a se
a g

co m
m .
m as e
.co a g l
se m
g l a
a
se m
com g l a
m . a
ase
agl

co m
m .
m as e
.co a g l
se m
g l a
a c
m .
m a s e
e m . co agl
g l as
a

co m
m .
m ase
.co


a g l Page 54 of 80

Class 7 Maths Chapter 5 Parallel and Intersecting Lines AglaSem · NCERT Solutions

Section 5.8 Alternate Angles

MATH TALK

Q1 Example 1: In Fig. 5.26, parallel lines l and m are intersected by the transversal t. If
∠6 is 135°, what are the measures of the other angles?

l

2
135°
1 3 m
6
4
5 7
8
t

Fig. 5.26, page 121 — parallel lines l and m cut by transversal t, with ∠6 = 135°.

Start from ∠6 = 135° and travel through the figure.

∠2 = ∠6 = 135° (corresponding angles, l ∥ m)

∠8 = ∠6 = 135° (vertically opposite)

∠4 = ∠8 = 135° (corresponding)

∠2 = ∠4 = 135° (vertically opposite) ✔

∠5 + ∠6 = 180° (linear pair)

∠5 = 180° – 135° = 45°

Similarly ∠1 = ∠3 = ∠7 = 45°

Page 55 of 80

Class 7 Maths Chapter 5 Parallel and Intersecting Lines AglaSem · NCERT Solutions

ANGLE MEASURE

∠2, ∠4, ∠6, ∠8 135°

∠1, ∠3, ∠5, ∠7 45°

Why it happens: Only two sizes can appear when a transversal crosses parallel lines,
and they must add to 180°. Here they are 135° and 45°, and 135 + 45 = 180 ✔

Page 56 of 80

Class 7 Maths Chapter 5 Parallel and Intersecting Lines AglaSem · NCERT Solutions

Q2 Example 2: In Fig. 5.27, lines l and m are intersected by the transversal t. If ∠a is
120° and ∠f is 70°, are lines l and m parallel to each other?

120°

a b l

d c

70°
e f m

h g

t

Fig. 5.27, page 121 — lines l and m cut by transversal t, with ∠a = 120° and ∠f = 70°.

No, l and m are not parallel.

Page 57 of 80

Class 7 Maths Chapter 5 Parallel and Intersecting Lines AglaSem · NCERT Solutions

∠a + ∠b = 180° (linear pair)

120° + ∠b = 180°

∠b = 60°

∠b and ∠f are corresponding angles

For l ∥ m we would need ∠b = ∠f

But 60° ≠ 70° ✘

The corresponding angles are unequal, so the lines are not parallel.

Why it happens: ∠f is 10° larger than ∠b, which means line m leans 10° more than
line l. Lines leaning differently must meet somewhere — here they would meet on
the side where the gap closes.

Tip: Never compare 120° with 70° directly. First bring both angles into the same
position at the two crossings, then compare.

Page 58 of 80

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Class 7 Maths Chapter 5 Parallel and Intersecting Lines
a g l AglaSem · NCERT Solutions

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Example 3: In Fig. 5.28, parallel lines l and m are intersected by the transversal t. If
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∠3 is 50°, what is the measure of ∠6?
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a Fig. 5.28, page 121 — parallel lines l and m cut by transversal t, with ∠3 = 50°.

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∠6 = 130°.

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∠2 + ∠3 = 180° (linear pair)
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∠2 = 180° – 50° = 130°
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So ∠6 = 130° .c
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Notice the by-product: ∠3 + ∠6 =m
a s e 50° + 130° = 180°. Angles ∠3 and ∠6 are called interior angles

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on the same side of the transversal.

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a g l Page 59 of 80