AP 7th Maths Unit 5 – Parallel and Intersecting Lines Solutions and Answers

AP SCERT Class 7 Maths Unit 5 – Parallel and Intersecting Lines | Solutions & Answers (Pages 258–304)

This post gives complete, page-by-page APSCERT Class 7 Maths Unit 5 – Parallel and Intersecting Lines solutions (also useful as Ganita Prakash Class 7 Chapter 5 answers) for pages 258 to 304. It covers intersecting lines, linear pairs, vertically opposite angles, perpendicular lines, parallel lines, transversals, corresponding angles, alternate angles, co-interior (interior) angles, and all “Figure it Out” exercises, in-text questions, and activities with clear step-by-step working, labelled diagrams and easy explanations suitable for CBSE/State Board Class 7 students preparing for exams, worksheets, and homework help on AP Teachers and TET aspirant resource pages.

📘 Pages 258–259: Unit Overview & Introduction

5.0 Introduction – Reflect & Answer: Take a piece of square paper, fold it in different ways, and observe the lines formed by the creases. Do they meet? Would they meet if extended beyond the paper?

Answer: Some pairs of crease-lines meet at a point within the paper – these are intersecting lines. Some other pairs never meet, however far we extend them – these are parallel lines. Whether two lines meet or not depends on their direction (slope); lines going in exactly the same direction never meet even if extended infinitely, while lines going in different directions will always meet at exactly one point somewhere (inside or outside the paper).

📘 Pages 260–263: 5.1 Intersecting Lines

Q1. How many angles do two intersecting lines form?

Answer: When two lines intersect, they form 4 angles (in Fig. 5.2, these are ∠a, ∠b, ∠c and ∠d).

Q2. Can two straight lines intersect at more than one point?

Answer: No. Two distinct straight lines can intersect at exactly one point only. (If they met at two points, they would have to be the very same line.)

🔵 Activity 1: Draw two intersecting lines, measure the four angles formed with a protractor. Draw four more such pairs and measure the angles at each intersection. What patterns do you observe?

Answer (Pattern observed):

• Angles that are opposite to each other (like ∠a & ∠c, or ∠b & ∠d) are always equal – these are called vertically opposite angles.
• Angles that are next to each other (adjacent), like ∠a & ∠b, always add up to 180° – these are called a linear pair.

Q3. In Fig. 5.2, if ∠a is 120°, find ∠b, ∠c and ∠d without measuring.

Answer:
∠a + ∠b = 180° (linear pair)  ⟹  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°.

Q4. Is it always true for any pair of intersecting lines that vertically opposite angles are equal?

Answer: Yes, this is always true. Since ∠a+∠b=180° and ∠a+∠d=180° (both are linear pairs on the straight line), we get ∠b = ∠d. Similarly, since ∠b+∠a=180° and ∠b+∠c=180°, we get ∠a = ∠c. This reasoning (without depending on any specific measurement) is called a proof in mathematics.

📗 Page 264: Figure it Out – 5.1

Q. List all the linear pairs and vertically opposite angles you observe in Fig. 5.3 (angles a, b, c, d formed by two intersecting lines l and m).

Linear Pairs (sum = 180°)	Vertically Opposite Angles
∠a & ∠b	∠a & ∠c
∠b & ∠c	∠b & ∠d
∠c & ∠d	–
∠d & ∠a	–

📘 Pages 266–267: 5.2 Perpendicular Lines & 5.3 Between Lines

Q1. Can you draw a pair of intersecting lines such that all four angles are equal? What will be the measure of each angle?

Answer: Yes. If all four angles formed are equal, and adjacent angles must add to 180° (linear pair), then each angle = 180° ÷ 2 = 90°. Such lines are called perpendicular lines – they intersect each other at right angles (90°). In Fig. 5.4, line l and line m are perpendicular to each other, written as l ⊥ m.

Q2. Observe Fig. 5.5 and describe how the line segments meet or cross in each case:

Answer:

• Line segments FG and FH meet at the endpoint F at an angle of 115° (given example).
• Line segments AB and CD (crossing near X) intersect at a point X inside both segments.
• Line segments IJ and LM (crossing near Y) intersect at a point Y, which is close to the midpoint of both segments.
• Line segments ST and UV do not meet within the figure – they run close to each other without crossing.
• Line segments OP and QR also do not meet within the figure – they run in almost the same direction without crossing.

Q3. Are line segments ST and UV likely to meet if extended? Are OP and QR likely to meet if extended?

Answer: Yes – in both cases, the segments are not running in exactly the same direction (they are not truly parallel); they are slightly tilted towards each other. So, if extended far enough on the correct side, ST and UV would eventually meet, and OP and QR would eventually meet too. This is different from the piano-keys, bench, and window-blind pictures shown next – those lines are truly parallel and will never meet, however far they are extended.

Q4. Name some parallel lines you can spot in your classroom.

Answer (sample): The opposite edges of the blackboard, the opposite edges of a notebook page, the horizontal lines/rulings on notebook paper, the top and bottom edges of a door or window, the rungs of a ladder, and adjacent bars of a window grill are all examples of parallel lines.

Q5. Which pairs of lines appear to be parallel in Fig. 5.6?

Answer: The vertical line segments in the figure (running straight up-down, without any bend) appear parallel to each other, and the diagonal line segments that slant in the same direction (same tilt) appear parallel to each other. Lines that cross other lines or slant in a different direction are not parallel. (Students should verify by checking that the marked lines never get closer or farther apart along their length.)

📘 Pages 270–273: 5.4 Parallel & Perpendicular Lines in Paper Folding

🔵 Activity 2 – Take a square sheet of paper and answer:

1. How would you describe the opposite edges of the sheet? They are  parallel  to each other.

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

3. Fold the sheet horizontally in half. How many parallel lines do you see now?
Answer: 3 parallel lines (the top edge, the new crease, and the bottom edge). The new crease line is parallel to the top and bottom edges.

4. Make one more horizontal fold. How many parallel lines do you see now?
Answer: 5 parallel lines. Each new fold doubles the previous creases (in the unfolded sheet), adding 2 more lines each time.

5. What will happen if you do it once more? Is there a pattern?
Answer: The next fold gives 9 parallel lines. Yes, there is a pattern – number of lines = 3, 5, 9, 17 … i.e., after n folds, number of parallel lines = 2ⁿ + 1. The pattern extends further with every additional horizontal fold.

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

7. Fold the sheet along a diagonal. Can you find a fold that creates a line parallel to the diagonal?
Answer: Yes – folding each of the two triangular halves (formed by the diagonal crease) again in half, so that the fold runs at an equal distance from the diagonal on either side, creates a new crease that is parallel to the diagonal.

🔵 Triangle-fold Activity (Fig. 5.8): Are a, b, c parallel to p, q, r respectively? Why or why not?

Answer: Yes, they are parallel. When the paper is folded symmetrically (top-right corner and bottom-left corner folded onto the same crease lines), the two small triangles formed are mirror images of each other. This makes the corresponding sides equally inclined to the vertical crease lines, so the corresponding angles formed are equal – and equal corresponding angles mean the lines a||p, b||q, and c||r.

📗 Pages 274–277: Figure it Out – 5.2

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

Answer: The dashed coloured lines in the picture above are drawn perpendicular (at 90°) to each given black line, joining dot to dot on the grid, as required.

Q2. In Fig. 5.11, mark the parallel lines using arrow notation and mark right angles with a square symbol. (a) How did you spot the perpendicular lines? (b) How did you spot the parallel lines?

Answer:
(a) Perpendicular lines were spotted by checking where two line segments meet and form a square-corner (right angle, 90°) – these were marked with a small square symbol at the vertex.
(b) Parallel lines were spotted by checking pairs of segments that point in exactly the same direction and never get closer to each other – these were marked using matching arrow (>) symbols; a second, different pair of parallel lines was marked with double arrows (>>).

Q3. In the dot paper, draw different sets of parallel lines (with dots as endpoints).

Answer: Draw two or more segments joining dots so that each set has exactly the same slope/slant (e.g., 3 dots right & 1 dot up for every segment in a set) – all such segments will be parallel to one another.

Q4. Using your sense of how parallel lines look, draw lines parallel to the given segments in Fig. 5.12.

Answer: As shown above, a line is parallel to the given segment when it keeps the same slant/direction (moves the same number of dots across and up/down).
(a) Some segments (the steeply-slanted or unusually-angled ones) are more challenging to copy exactly using dots only.
(b) These are usually the diagonal segments that don’t move in a simple whole-number dot-to-dot pattern.
(c) Method: count how many dots the given segment moves horizontally and how many it moves vertically, then draw a new segment starting from a different dot but moving the same number of dots in the same directions.

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

Answer: Line a is parallel to line c. We decide this by comparing the direction (slant/inclination) of each line – line c tilts in exactly the same direction and by the same amount as line a, so the two never come closer or move apart; line b has a visibly different slant, so it is not parallel to a. (You can check by measuring the angle each line makes with a common horizontal/reference line using a protractor – equal angles confirm parallel lines.)

📘 Pages 278–279: 5.5 Transversals

Q1. Is it possible for all eight angles (formed when a transversal crosses two lines) to have different measurements? Why or why not?

Answer: No. The eight angles are formed at two points of intersection, and at each point, the two angles that are opposite each other (vertically opposite angles) are always equal. This means the 8 angles occur in 4 equal pairs, so there can be a maximum of only 4 different angle measures among all 8 angles – not 8 different values.

Q2. What about the four different angles – 6, 5, 3 and 2 (in Fig. 5.14)? Can these four all be different?

Answer: Yes. None of ∠2, ∠3, ∠5, ∠6 are vertically opposite to one another, so in general (when the two lines are not parallel and the transversal is not perpendicular to them) all four of these can have different measures – together with their vertically-opposite partners (∠1=∠3, ∠4=∠2, ∠5=∠7, ∠8=∠6) this gives the maximum of 4 distinct angle values referred to above.

📘 Pages 280–285: 5.6 Corresponding Angles

Quick reference (used throughout this section):

• Corresponding angles (same position at each intersection): ∠1&∠5, ∠2&∠6, ∠3&∠7, ∠4&∠8 – equal when lines are parallel.
• Alternate angles (opposite sides of transversal, between the lines): ∠4&∠6, ∠3&∠5 – equal when lines are parallel.
• Co-interior (interior) angles (same side of transversal, between the lines): ∠3&∠6, ∠4&∠5 – add up to 180° when lines are parallel.

🔵 Activity 3-4: Draw a transversal, make ∠a = 60° with line l, then draw a new line m through point Y making the corresponding angle ∠b = 60° too. What do you observe about lines l and m? Do they appear parallel?

Answer: Yes – lines l and m appear parallel to each other. This confirms the rule: when the corresponding angles formed by a transversal on a pair of lines are equal, the lines are parallel.

🔵 Activity 5: In Fig. 5.20, try to draw a transversal t to lines l and m such that one pair of corresponding angles is equal. Are you finding it hard to do this?

Answer: Yes, it is difficult (in fact, impossible) – because lines l and m in Fig. 5.20 are not parallel to each other. This demonstrates the converse rule: when a pair of lines are not parallel, the corresponding angles formed by any transversal on them can never be equal.

📘 Pages 286–289: 5.7 Drawing Parallel Lines

Q1. Using a ruler and a set-square, draw two lines perpendicular to line l (Fig. 5.21). Are these two lines parallel to each other? How are we sure? What angles are formed between these lines and line l?

Answer: Yes, the two lines are parallel to each other. We are sure because line l acts as a transversal, and it makes equal corresponding angles (both 90°) with the two new lines – and equal corresponding angles always mean the lines are parallel. The angle formed between line l and each of the new lines is 90° (a right angle).

Q2. Draw two more parallel lines using the long side of the set-square (Fig. 5.22). How do you know these two lines are parallel? Can you check if the corresponding angles are equal?

Answer: These two lines are parallel because the set-square is kept at the same fixed angle against the ruler (line l) both times, so the corresponding angle made with line l is exactly the same in both cases. This can be checked/verified by tracing one angle onto tracing paper and placing it over the other angle – they will match exactly – or by measuring both angles with a protractor.

📗 Page 288: Figure it Out – 5.3

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

Answer – Method (using ruler and set-square):

1. Draw line l with the ruler.
2. Using the set-square and ruler, draw a line t passing through point A, perpendicular to line l (making a 90° angle at the point where t meets l).
3. Now, using the set-square again on line t, draw a new line m through point A that is perpendicular to t (i.e., making another 90° angle, this time at A).
4. Since both line l and line m are perpendicular to the same line t, the corresponding angles (both 90°) made with transversal t are equal, so line m is parallel to line l and passes through A, as required.

🔵 Paper-folding construction (Pages 288–289): Fold a perpendicular to line l through point A (call this crease t). Then fold a perpendicular to t through A again (call this line m). Why are lines l and m parallel to each other?

Answer: Because both line l and line m are perpendicular to the very same line t. When two lines are each perpendicular to a common third line, the corresponding angles they make with that common line are both 90° (equal) – and equal corresponding angles always mean the two lines are parallel to each other.

📘 Pages 290–293: 5.8 Alternate Angles

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

Answer:
∠b = ∠f = 120° (corresponding angles, since l || m)
∠d = ∠b = 120° (vertically opposite angles)
So, ∠d = 120°. In general, ∠f = ∠d always, because ∠f = ∠b (corresponding angles) and ∠b = ∠d (vertically opposite angles) – regardless of the actual measure of ∠f. This proves: alternate angles formed by a transversal on a pair of parallel lines are always equal.

📘 Pages 292–297: Solved Examples Recap (Examples 1–4)

Example 1: Parallel lines l, m cut by transversal t; ∠6 = 135°. Then ∠2 = ∠4 = ∠6 = ∠8 = 135°, and ∠1 = ∠3 = ∠5 = ∠7 = 45° (linear pairs with 135°).

Example 2: ∠a = 120°, ∠f = 70°. Since ∠b = 60° (linear pair with ∠a) should equal the corresponding angle ∠f for the lines to be parallel, but 60° ≠ 70°, lines l and m are NOT parallel.

Example 3: Parallel lines l, m; ∠3 = 50°. Then ∠2 = 130° (linear pair), and since ∠2 = ∠6 (corresponding angles), ∠6 = 130°. (∠3 and ∠6 are called co-interior angles; they add to 180°.)

Example 4: AB||CD, AD||BC, ∠DAC = 65°, ∠ADC = 60°. Working: ∠DAB = 180−60 = 120°, so ∠CAB = 120−65 = 55°; ∠BCD = 180−60 = 120°; and ∠ABC = 60°.

📗 Pages 298–302: Figure it Out – 5.4 (Important Exercise)

Q1. Find the angles marked below (Fig. 5.30, a to j):

Angle	Answer	Reasoning (rule applied)
a	48°	Vertically opposite to the given 48° angle
b	52°	Alternate angle to the given 52° (lines parallel)
c	81°	Corresponding angle to 81° (99°+81°=180° confirms straight line)
d	99°	Corresponding/vertically opposite to 99°
e	111°	97°+83°=180° (straight line); e is linear pair with 69° → e=180−69
f	48°	Co-interior with 132° (lines parallel): f = 180−132
g	58°	Corresponding angle to the marked 58° (parallel vertical rays)
h	45°	Base angle = 180−120=60° (linear pair); triangle angle sum: 180−75−60
i	110°	Linear pair with 70°: i = 180−70
j	56°	Using linear pair & triangle/parallel-line angle relations

Q2. Find the angle represented by a (Fig. 5.31):

Given angles	a = ?
100°, 42° (parallel lines)	58°
62° (parallel lines, corresponding)	62°
110°, 35° (parallel lines)	75°
67°, with a right angle	23°

Method used: Using linear pair (angles on a straight line = 180°), vertically opposite angles, corresponding/alternate angle equality on parallel lines, and (where a right angle is marked) the fact that a right angle = 90° along with triangle angle-sum = 180°.

Q3. In the figures below (Fig. 5.32), what angles do x and y stand for?

Figure (i): Right angle, 65° and y° at one point; x° at the base.
y = 180−65 = 115° (linear pair); x = 65° (corresponding angle, parallel lines).

Figure (ii): 53° and 78° at one intersection; x° is the angle where the two transversals meet.
By exterior angle property: x = 53°+78° = 131°.

Q4. In Fig. 5.33, ∠ABC = 45° and ∠IKJ = 78°. Find ∠GEH, ∠HEF, ∠FED.

Answer (using corresponding & linear-pair relationships, since line IJ || line GD):
∠GEH = 180° − ∠IKJ = 180−78 = 102°
∠FED = 180° − ∠ABC = 180−45 = 135°
∠HEF = ∠GEH − (angle contributed by the ABC side) = 57° (found using the angles around point E, since all angles around E on the lower side together make 180°: 45°+57°+78° = 180°)

Q5. In Fig. 5.34, AB||CD||EF, and EA⊥AB. If ∠BEF = 55°, find x and y.

Answer:
Since AB || EF, and BE is a transversal, ∠ABE(x°) and ∠BEF(55°) are alternate interior angles.
So, x = 55°.
Since AB || CD, and the same line BDE is the transversal, ∠BDC(y°) corresponds to ∠ABE(x°).
So, y = 55°.

Q6. What is the measure of angle ∠NOP in Fig. 5.35? [Hint: Draw lines parallel to LM and PQ through points N and O.]

Answer:
Draw a line through N parallel to LM, and a line through O parallel to PQ (these two new lines are also parallel to each other, since both are parallel to the same LM/PQ direction).
At N: this splits ∠MNO(96°) into 40° (alternate to ∠LMN=40°) and (96−40)=56° remaining.
At P: alternate angle to ∠OPQ(52°) gives 52° at O (on the same auxiliary line).
By alternate angles between the two auxiliary parallel lines (transversal NO): the 56° portion at N corresponds to an equal 56° portion at O.
So, ∠NOP = 56° + 52° = 108°.

📘 Pages 302–303: 5.9 Parallel Illusions

Q. The pictures show lines that do not seem to be parallel – or are they? What causes these illusions?

Answer: In each picture, the long straight lines actually are parallel (or straight) – they are perfectly straight and evenly spaced. The illusion happens because of the shorter background lines/patterns crossing them at different angles. Our brain uses the angle at which lines cross as a clue to judge direction, and the crossing pattern tricks our visual system into perceiving the straight lines as bent, curved, or converging, even though they are not. This is a well-known type of optical illusion called the café-wall / radiating-line illusion.

📘 Page 304: Summary & Fun Puzzle

• When two lines intersect, they form 4 angles. Vertically opposite angles are equal; linear pairs add up to 180°.
• When all four angles formed are 90°, the lines are perpendicular.
• Lines on the same plane that never meet, however far extended, are parallel lines.
• A line crossing a pair of lines is called a transversal; it forms 8 angles (2 sets of 4).
• If lines are parallel: corresponding angles are equal, alternate angles are equal, and co-interior (same-side) angles add up to 180°.
• If corresponding angles formed by a transversal are equal, the two lines must be parallel (and vice-versa).

🧩 Puzzle: How many triangles?

Answer: Counting triangles of every size (small unit triangles as well as the bigger triangles formed by combining them) in the given triangular grid gives a total of 27 triangles.