💡 Print paper or share solutions as PDF via WhatsApp to students.
CHAPTER 5 — PARALLEL & INTERSECTING LINES
Class VII Mathematics — NCERT Ganita Prakash 2024
Preeti Kushwah Classes — Unit Test | Set 2
General Instructions:
1. All questions are compulsory.
2. Section A has 6 questions of 1 mark each.
3. Section B has 5 questions of 2 marks each.
4. Section C has 4 questions of 3 marks each.
5. Section D has 2 questions of 5 marks each.
6. There is 1 optional Bonus question worth 2 marks.
7. Show all working clearly. Use correct angle notation (∠) and properties.
1. All questions are compulsory.
2. Section A has 6 questions of 1 mark each.
3. Section B has 5 questions of 2 marks each.
4. Section C has 4 questions of 3 marks each.
5. Section D has 2 questions of 5 marks each.
6. There is 1 optional Bonus question worth 2 marks.
7. Show all working clearly. Use correct angle notation (∠) and properties.
Section A — (1 Mark Each) [6 × 1 = 6]
Q1.1
Define a transversal line in one sentence.
Q2.1
The sum of two angles forming a linear pair = ___°. Such angles are called _______.
Q3.1
State the Alternate Interior Angles property: When a transversal cuts two parallel lines, the alternate interior angles are _______.
Q4.1
At a single intersection point (where one transversal meets one line), how many distinct angle measures are there at most? ___
Q5.1
∠1 and ∠3 are vertically opposite angles at an intersection. If ∠1 = 74°, then ∠3 = ___°.
Q6.1
A transversal cuts two lines. At each intersection, there are 2 pairs of vertically opposite angles. How many VOA pairs are formed in total (at both intersections)? ___
Section B — (2 Marks Each) [5 × 2 = 10]
Q7.2
Two lines intersect at O. ∠AOD = 140°.
Find ∠BOC, ∠AOB, and ∠DOC. State the property used for each.
Find ∠BOC, ∠AOB, and ∠DOC. State the property used for each.
Q8.2
Lines p ∥ q. A transversal r cuts them. ∠4 = 55° at the upper intersection.
Find the corresponding angle ∠8 at the lower intersection. State the property.
Find the corresponding angle ∠8 at the lower intersection. State the property.
Q9.2
A transversal cuts two parallel lines. The two co-interior angles (same-side interior angles) are (5x + 10)° and (4x − 10)°.
Find x and both angles. Verify they sum to 180°.
Find x and both angles. Verify they sum to 180°.
Q10.2
A transversal cuts lines a and b. ∠1 = 70° at line a and ∠5 = 70° at line b (corresponding positions).
Are lines a and b parallel? Which property confirms this?
Are lines a and b parallel? Which property confirms this?
Q11.2
A transversal cuts two lines, forming angles ∠1 to ∠8 (∠1–∠4 at the upper intersection, ∠5–∠8 at the lower).
∠3 and ∠4 are interior angles (right and left of transversal, below upper line). ∠5 and ∠6 are interior angles (left and right of transversal, above lower line).
List all pairs of alternate interior angles.
∠3 and ∠4 are interior angles (right and left of transversal, below upper line). ∠5 and ∠6 are interior angles (left and right of transversal, above lower line).
List all pairs of alternate interior angles.
Section C — (3 Marks Each) [4 × 3 = 12]
Q12.3
Two lines intersect at a point. The angles ∠1 and ∠2 (which form a linear pair) are in the ratio 2 : 3.
(a) Find ∠1 and ∠2.
(b) Find ∠3 and ∠4 (the other two angles at the intersection).
(c) State the property used for ∠3 and ∠4.
(a) Find ∠1 and ∠2.
(b) Find ∠3 and ∠4 (the other two angles at the intersection).
(c) State the property used for ∠3 and ∠4.
Q13.3
A transversal cuts two parallel lines l ∥ m. At the upper intersection (P), ∠2 = 130° (upper-right angle).
(a) Find all four angles at P: ∠1, ∠2, ∠3, ∠4.
(b) State the property to find ∠6, ∠7, and ∠8 at Q (lower intersection). Find each angle.
(∠1–∠4 at P, ∠5–∠8 at Q; corresponding pairs: ∠1&∠5, ∠2&∠6, ∠3&∠7, ∠4&∠8)
(a) Find all four angles at P: ∠1, ∠2, ∠3, ∠4.
(b) State the property to find ∠6, ∠7, and ∠8 at Q (lower intersection). Find each angle.
(∠1–∠4 at P, ∠5–∠8 at Q; corresponding pairs: ∠1&∠5, ∠2&∠6, ∠3&∠7, ∠4&∠8)
Q14.3
A transversal cuts two lines. The corresponding angles at the two intersections are (6x − 20)° and (4x + 10)°.
(a) Find the value of x. State which property you are using.
(b) Find the measure of each corresponding angle.
(c) State the converse: if corresponding angles are equal, what can you conclude?
(a) Find the value of x. State which property you are using.
(b) Find the measure of each corresponding angle.
(c) State the converse: if corresponding angles are equal, what can you conclude?
Q15.3
Describe the paper folding method to draw a line parallel to a given line l through a point P not on l.
Write clear step-by-step instructions (at least 4 steps). What angle property guarantees the lines are parallel?
Write clear step-by-step instructions (at least 4 steps). What angle property guarantees the lines are parallel?
Section D — (5 Marks Each) [2 × 5 = 10]
Q16.5
A transversal t cuts two parallel lines l (upper) and m (lower) at points P and Q respectively.
Angles at P: ∠1 (upper-left), ∠2 (upper-right), ∠3 (lower-right, interior), ∠4 (lower-left, interior).
Angles at Q: ∠5 (upper-left, interior), ∠6 (upper-right, interior), ∠7 (lower-right), ∠8 (lower-left).
At P, ∠3 = 65°.
(a) Find all four angles at P: ∠1, ∠2, ∠3, ∠4. State properties used. [2 marks]
(b) Find all four angles at Q. For each, name the property used (corresponding, VOA, or linear pair). [2 marks]
(c) Identify the two co-interior angle pairs. Verify each pair sums to 180°. [½ mark]
(d) If l and m were NOT parallel, would the alternate interior angles still be equal? Explain. [½ mark]
(a) Find all four angles at P: ∠1, ∠2, ∠3, ∠4. State properties used. [2 marks]
(b) Find all four angles at Q. For each, name the property used (corresponding, VOA, or linear pair). [2 marks]
(c) Identify the two co-interior angle pairs. Verify each pair sums to 180°. [½ mark]
(d) If l and m were NOT parallel, would the alternate interior angles still be equal? Explain. [½ mark]
Q17.5
(a) Prove that if a transversal cuts two lines such that the corresponding angles are equal, then the lines are parallel. [2 marks]
(b) Two railway lines (parallel) are crossed by a road. The road makes an angle of 75° with the first railway line. What angle does it make with the second railway line? Justify using the correct angle property. [1½ marks]
(c) Lines a ⊥ c and b ⊥ c. Prove that a ∥ b using angle properties of the transversal c. [1½ marks]
(b) Two railway lines (parallel) are crossed by a road. The road makes an angle of 75° with the first railway line. What angle does it make with the second railway line? Justify using the correct angle property. [1½ marks]
(c) Lines a ⊥ c and b ⊥ c. Prove that a ∥ b using angle properties of the transversal c. [1½ marks]
Bonus Question (Optional) [2 Marks]
Q18.2
★ A transversal cuts two parallel lines. The difference between the two co-interior angles is 40°. Find both co-interior angles. State the property used.
Preeti Kushwah Classes
Scan for more notes & papers
www.preetikushwahclasses.com
Scan for more notes & papers
www.preetikushwahclasses.com
Answer Key & Detailed Solutions — Set 2
Q1. [1 Mark]
A transversal is a line that crosses (intersects) two or more other lines at distinct points.
The transversal forms angles at each point of intersection — 4 angles at each crossing, so 8 angles when cutting two lines.
The transversal forms angles at each point of intersection — 4 angles at each crossing, so 8 angles when cutting two lines.
Q2. [1 Mark]
The sum of two angles forming a linear pair = 180°.
Such angles are called a linear pair (they lie on a straight line on either side of a point).
Such angles are called a linear pair (they lie on a straight line on either side of a point).
Q3. [1 Mark]
When a transversal cuts two parallel lines, the alternate interior angles are equal.
Converse is also true: if alternate interior angles are equal, the lines are parallel.
Converse is also true: if alternate interior angles are equal, the lines are parallel.
Q4. [1 Mark]
At a single intersection point, at most 2 distinct angle measures exist.
The 4 angles form two pairs of vertically opposite angles — so there are only 2 different values: one angle and its supplement. (If the lines are perpendicular, there is only 1 distinct measure: 90°.)
The 4 angles form two pairs of vertically opposite angles — so there are only 2 different values: one angle and its supplement. (If the lines are perpendicular, there is only 1 distinct measure: 90°.)
Q5. [1 Mark]
∠1 = 74°, so ∠3 = 74°.
Property: Vertically Opposite Angles are equal.
Property: Vertically Opposite Angles are equal.
Q6. [1 Mark]
At each intersection there are 2 VOA pairs, and there are 2 intersections.
Total VOA pairs = 2 × 2 = 4 pairs.
Total VOA pairs = 2 × 2 = 4 pairs.
Q7. [2 Marks]
Given: ∠AOD = 140°
∠BOC = 140° — Vertically Opposite Angles (∠AOD and ∠BOC are VOA) [½ mark]
∠AOB = 40° — Linear Pair: ∠AOD + ∠AOB = 180°
140° + ∠AOB = 180° → ∠AOB = 40° [½ mark]
∠DOC = 40° — Vertically Opposite to ∠AOB [½ mark]
Summary: 140°, 40°, 140°, 40° [½ mark for summary]
∠BOC = 140° — Vertically Opposite Angles (∠AOD and ∠BOC are VOA) [½ mark]
∠AOB = 40° — Linear Pair: ∠AOD + ∠AOB = 180°
140° + ∠AOB = 180° → ∠AOB = 40° [½ mark]
∠DOC = 40° — Vertically Opposite to ∠AOB [½ mark]
Summary: 140°, 40°, 140°, 40° [½ mark for summary]
Q8. [2 Marks]
Given: p ∥ q, ∠4 = 55° at upper intersection.
∠8 is the corresponding angle to ∠4 (both are in the lower-left position at their respective intersections). [1 mark]
Property (Corresponding Angles): When a transversal cuts two parallel lines, corresponding angles are equal. [1 mark]
∴ ∠8 = 55°
∠8 is the corresponding angle to ∠4 (both are in the lower-left position at their respective intersections). [1 mark]
Property (Corresponding Angles): When a transversal cuts two parallel lines, corresponding angles are equal. [1 mark]
∴ ∠8 = 55°
Q9. [2 Marks]
Co-interior angles on parallel lines sum to 180°.
(5x + 10) + (4x − 10) = 180°
9x + 0 = 180°
9x = 180°
x = 20° [1 mark]
First angle = 5(20) + 10 = 100 + 10 = 110°
Second angle = 4(20) − 10 = 80 − 10 = 70° [½ mark]
Verification: 110° + 70° = 180° ✓ [½ mark]
(5x + 10) + (4x − 10) = 180°
9x + 0 = 180°
9x = 180°
x = 20° [1 mark]
First angle = 5(20) + 10 = 100 + 10 = 110°
Second angle = 4(20) − 10 = 80 − 10 = 70° [½ mark]
Verification: 110° + 70° = 180° ✓ [½ mark]
Q10. [2 Marks]
∠1 = 70° and ∠5 = 70° are corresponding angles (at the same position at each intersection).
Since corresponding angles are equal (both 70°), lines a and b are parallel. [1 mark]
Property (Converse of Corresponding Angles): If a transversal cuts two lines such that the corresponding angles are equal, then the lines are parallel. [1 mark]
Since corresponding angles are equal (both 70°), lines a and b are parallel. [1 mark]
Property (Converse of Corresponding Angles): If a transversal cuts two lines such that the corresponding angles are equal, then the lines are parallel. [1 mark]
Q11. [2 Marks]
Interior angles between the two lines: ∠3 and ∠4 (at upper intersection P); ∠5 and ∠6 (at lower intersection Q).
Alternate interior angles are interior angles on opposite sides of the transversal:
Pair 1: ∠3 and ∠5 (∠3 is right-side at P; ∠5 is left-side at Q — opposite sides) [1 mark]
Pair 2: ∠4 and ∠6 (∠4 is left-side at P; ∠6 is right-side at Q — opposite sides) [1 mark]
Alternate interior angles are interior angles on opposite sides of the transversal:
Pair 1: ∠3 and ∠5 (∠3 is right-side at P; ∠5 is left-side at Q — opposite sides) [1 mark]
Pair 2: ∠4 and ∠6 (∠4 is left-side at P; ∠6 is right-side at Q — opposite sides) [1 mark]
Q12. [3 Marks]
∠1 : ∠2 = 2 : 3 and ∠1 + ∠2 = 180° (Linear Pair).
Let ∠1 = 2k and ∠2 = 3k.
2k + 3k = 180°
5k = 180°
k = 36°
(a) ∠1 = 72° and ∠2 = 108° [1 mark]
(b) ∠3 = 72° (VOA of ∠1) and ∠4 = 108° (VOA of ∠2) [1 mark]
(c) Property: ∠3 is vertically opposite to ∠1, so ∠3 = ∠1 = 72°. ∠4 is vertically opposite to ∠2, so ∠4 = ∠2 = 108°. (Vertically Opposite Angles are equal.) [1 mark]
Let ∠1 = 2k and ∠2 = 3k.
2k + 3k = 180°
5k = 180°
k = 36°
(a) ∠1 = 72° and ∠2 = 108° [1 mark]
(b) ∠3 = 72° (VOA of ∠1) and ∠4 = 108° (VOA of ∠2) [1 mark]
(c) Property: ∠3 is vertically opposite to ∠1, so ∠3 = ∠1 = 72°. ∠4 is vertically opposite to ∠2, so ∠4 = ∠2 = 108°. (Vertically Opposite Angles are equal.) [1 mark]
Q13. [3 Marks]
Given: l ∥ m, ∠2 = 130° at P.
(a) Angles at P: [1 mark]
∠2 = 130° (given)
∠4 = 130° (VOA of ∠2)
∠1 = 50° (Linear Pair: 180° − 130°)
∠3 = 50° (VOA of ∠1)
(b) Angles at Q (using Corresponding Angles, l ∥ m): [2 marks, ½ each]
∠6 = 130° (corresponds to ∠2; Property: Corresponding Angles equal when l ∥ m)
∠8 = 130° (corresponds to ∠4)
∠7 = 50° (corresponds to ∠3)
(∠5 = 50° corresponds to ∠1 — not asked but worth noting.)
(a) Angles at P: [1 mark]
∠2 = 130° (given)
∠4 = 130° (VOA of ∠2)
∠1 = 50° (Linear Pair: 180° − 130°)
∠3 = 50° (VOA of ∠1)
(b) Angles at Q (using Corresponding Angles, l ∥ m): [2 marks, ½ each]
∠6 = 130° (corresponds to ∠2; Property: Corresponding Angles equal when l ∥ m)
∠8 = 130° (corresponds to ∠4)
∠7 = 50° (corresponds to ∠3)
(∠5 = 50° corresponds to ∠1 — not asked but worth noting.)
Q14. [3 Marks]
(a) If the lines are parallel, corresponding angles are equal. [½ mark]
6x − 20 = 4x + 10
6x − 4x = 10 + 20
2x = 30
x = 15° [1 mark]
(b) Each corresponding angle = 6(15) − 20 = 90 − 20 = 70°
Check: 4(15) + 10 = 60 + 10 = 70° ✓ [1 mark]
(c) Converse Property: If a transversal cuts two lines such that the corresponding angles are equal, then the two lines are parallel. [½ mark]
6x − 20 = 4x + 10
6x − 4x = 10 + 20
2x = 30
x = 15° [1 mark]
(b) Each corresponding angle = 6(15) − 20 = 90 − 20 = 70°
Check: 4(15) + 10 = 60 + 10 = 70° ✓ [1 mark]
(c) Converse Property: If a transversal cuts two lines such that the corresponding angles are equal, then the two lines are parallel. [½ mark]
Q15. [3 Marks]
Paper Folding Method to draw a parallel line:
Step 1: Draw line l on a sheet of paper and mark a point P above (or below) the line. [½ mark]
Step 2: Fold the paper along a line through P so that line l folds onto itself. This creates a fold-crease that is perpendicular to line l. Call this crease PQ (where Q is on line l). [1 mark]
Step 3: Without unfolding, fold the paper again along a new crease through P, this time making line PQ fold onto itself. This new fold-crease passes through P and is perpendicular to PQ. [1 mark]
Step 4: Open the paper. The second fold-crease through P is the required line parallel to l. [½ mark]
Angle Property: Both lines are perpendicular to PQ (they each make a 90° angle with PQ). Since the corresponding angles (both 90°) are equal, the two lines are parallel.
Step 1: Draw line l on a sheet of paper and mark a point P above (or below) the line. [½ mark]
Step 2: Fold the paper along a line through P so that line l folds onto itself. This creates a fold-crease that is perpendicular to line l. Call this crease PQ (where Q is on line l). [1 mark]
Step 3: Without unfolding, fold the paper again along a new crease through P, this time making line PQ fold onto itself. This new fold-crease passes through P and is perpendicular to PQ. [1 mark]
Step 4: Open the paper. The second fold-crease through P is the required line parallel to l. [½ mark]
Angle Property: Both lines are perpendicular to PQ (they each make a 90° angle with PQ). Since the corresponding angles (both 90°) are equal, the two lines are parallel.
Q16. [5 Marks]
Given: l ∥ m, ∠3 = 65° at P (lower-right interior).
(a) Angles at P: [2 marks]
∠3 = 65° (given)
∠1 = 65° (VOA of ∠3 — Vertically Opposite Angles)
∠2 = 115° (Linear Pair: 180° − 65°)
∠4 = 115° (VOA of ∠2, or Linear Pair with ∠3)
(b) Angles at Q: [2 marks]
∠7 = 65° (corresponds to ∠3, l ∥ m → Corresponding Angles)
∠5 = 65° (VOA of ∠7)
∠6 = 115° (corresponds to ∠2, or Linear Pair with ∠5)
∠8 = 115° (corresponds to ∠4, or VOA of ∠6)
(c) Co-interior Angle Pairs: [½ mark]
Pair 1: (∠3, ∠6) → 65° + 115° = 180° ✓
Pair 2: (∠4, ∠5) → 115° + 65° = 180° ✓
(d) Non-parallel lines: [½ mark]
No. If l and m were NOT parallel, the alternate interior angles would not be equal. The equality of alternate interior angles is a property that holds only when the lines are parallel. (Conversely, if alternate angles are equal, the lines must be parallel.)
(a) Angles at P: [2 marks]
∠3 = 65° (given)
∠1 = 65° (VOA of ∠3 — Vertically Opposite Angles)
∠2 = 115° (Linear Pair: 180° − 65°)
∠4 = 115° (VOA of ∠2, or Linear Pair with ∠3)
(b) Angles at Q: [2 marks]
∠7 = 65° (corresponds to ∠3, l ∥ m → Corresponding Angles)
∠5 = 65° (VOA of ∠7)
∠6 = 115° (corresponds to ∠2, or Linear Pair with ∠5)
∠8 = 115° (corresponds to ∠4, or VOA of ∠6)
(c) Co-interior Angle Pairs: [½ mark]
Pair 1: (∠3, ∠6) → 65° + 115° = 180° ✓
Pair 2: (∠4, ∠5) → 115° + 65° = 180° ✓
(d) Non-parallel lines: [½ mark]
No. If l and m were NOT parallel, the alternate interior angles would not be equal. The equality of alternate interior angles is a property that holds only when the lines are parallel. (Conversely, if alternate angles are equal, the lines must be parallel.)
Q17. [5 Marks]
(a) Proof (Converse of Corresponding Angles): [2 marks]
Given: Transversal t cuts lines l and m such that corresponding angles are equal (∠1 = ∠5).
To prove: l ∥ m.
∠1 = ∠5 (given — corresponding angles equal) …(i)
∠1 = ∠3 (VOA at intersection with l) …(ii)
From (i) and (ii): ∠3 = ∠5
But ∠3 and ∠5 are alternate interior angles. If alternate interior angles are equal, the lines are parallel.
∴ l ∥ m ■
(b) Railway tracks: [1½ marks]
The road acts as a transversal cutting two parallel railway lines. Angle at first track = 75°.
By the Corresponding Angles Property (transversal cutting parallel lines), the corresponding angle at the second track is also 75°.
∴ The road makes a 75° angle with the second railway line. [1 mark for answer + ½ for property]
(c) Proof that a ∥ b when a ⊥ c and b ⊥ c: [1½ marks]
Since a ⊥ c: the transversal c makes a 90° angle with line a.
Since b ⊥ c: the transversal c makes a 90° angle with line b.
The corresponding angles that c makes with a and b are both 90° → they are equal.
By the Converse of Corresponding Angles: since corresponding angles are equal, a ∥ b. ■
Given: Transversal t cuts lines l and m such that corresponding angles are equal (∠1 = ∠5).
To prove: l ∥ m.
∠1 = ∠5 (given — corresponding angles equal) …(i)
∠1 = ∠3 (VOA at intersection with l) …(ii)
From (i) and (ii): ∠3 = ∠5
But ∠3 and ∠5 are alternate interior angles. If alternate interior angles are equal, the lines are parallel.
∴ l ∥ m ■
(b) Railway tracks: [1½ marks]
The road acts as a transversal cutting two parallel railway lines. Angle at first track = 75°.
By the Corresponding Angles Property (transversal cutting parallel lines), the corresponding angle at the second track is also 75°.
∴ The road makes a 75° angle with the second railway line. [1 mark for answer + ½ for property]
(c) Proof that a ∥ b when a ⊥ c and b ⊥ c: [1½ marks]
Since a ⊥ c: the transversal c makes a 90° angle with line a.
Since b ⊥ c: the transversal c makes a 90° angle with line b.
The corresponding angles that c makes with a and b are both 90° → they are equal.
By the Converse of Corresponding Angles: since corresponding angles are equal, a ∥ b. ■
Q18. Bonus [2 Marks]
Let the two co-interior angles be x and y.
Property: Co-interior angles on parallel lines are supplementary: x + y = 180° …(i) [½ mark]
Given: Difference = 40° → x − y = 40° …(ii) [½ mark]
Adding (i) and (ii):
2x = 220° → x = 110°
From (i): y = 180° − 110° = 70° [1 mark]
Verification: 110° + 70° = 180° ✓; 110° − 70° = 40° ✓
Property: Co-interior angles on parallel lines are supplementary: x + y = 180° …(i) [½ mark]
Given: Difference = 40° → x − y = 40° …(ii) [½ mark]
Adding (i) and (ii):
2x = 220° → x = 110°
From (i): y = 180° − 110° = 70° [1 mark]
Verification: 110° + 70° = 180° ✓; 110° − 70° = 40° ✓