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CHAPTER 5 — PARALLEL & INTERSECTING LINES
Class VII Mathematics — NCERT Ganita Prakash 2024
Preeti Kushwah Classes — Unit Test | Set 1
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 (∠).
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 (∠).
Section A — (1 Mark Each) [6 × 1 = 6]
Q1.1
Two lines meeting at exactly one point are called _______ lines, and the meeting point is called the _______.
Q2.1
Vertically opposite angles are always _______. (equal / supplementary / complementary)
Q3.1
When two lines are perpendicular to each other, each of the four angles formed at the intersection = ___°.
Q4.1
Write the correct symbol notation: “Line l is parallel to line m” is written as ___________.
Q5.1
A transversal crossing two lines forms ___ angles in total (at both intersection points combined).
Q6.1
Corresponding angles formed by a transversal are equal only when the two lines are _______.
Section B — (2 Marks Each) [5 × 2 = 10]
Q7.2
Two lines intersect at a point. One of the angles formed is 65°.
Find the other three angles. Name the property used for each.
Find the other three angles. Name the property used for each.
Q8.2
Lines l ∥ m. A transversal t cuts them. At the upper intersection, ∠2 = 115°.
Find the corresponding angle ∠6 at the lower intersection. State the property used.
Find the corresponding angle ∠6 at the lower intersection. State the property used.
Q9.2
A transversal cuts two lines, forming angles ∠1 to ∠8 (∠1–∠4 at the upper intersection, ∠5–∠8 at the lower).
List all four pairs of corresponding angles.
List all four pairs of corresponding angles.
Q10.2
Two lines intersect at O. ∠AOB = (3x)° and its vertically opposite angle ∠COD = (x + 60)°.
Find the value of x and the measure of ∠AOB.
Find the value of x and the measure of ∠AOB.
Q11.2
Lines l ∥ m. A transversal cuts them. At the upper intersection, ∠3 = 72° (interior angle on the right of the transversal).
Find the alternate interior angle ∠5 at the lower intersection. State the property.
Find the alternate interior angle ∠5 at the lower intersection. State the property.
Section C — (3 Marks Each) [4 × 3 = 12]
Q12.3
Two lines intersect at a point, forming angles ∠1, ∠2, ∠3, ∠4 (going around). ∠1 = 125°.
(a) Find ∠2, ∠3, and ∠4.
(b) State the property used for each calculation.
(a) Find ∠2, ∠3, and ∠4.
(b) State the property used for each calculation.
Q13.3
A transversal cuts two parallel lines l ∥ m. At the upper intersection (P), ∠1 = 50°.
Find ∠2, ∠3, ∠4 at point P and ∠5 at the lower intersection (Q). State the property used for each.
(∠1 and ∠3 are vertically opposite; ∠1 and ∠2 are a linear pair; ∠1 and ∠5 are corresponding.)
Find ∠2, ∠3, ∠4 at point P and ∠5 at the lower intersection (Q). State the property used for each.
(∠1 and ∠3 are vertically opposite; ∠1 and ∠2 are a linear pair; ∠1 and ∠5 are corresponding.)
Q14.3
A transversal cuts two parallel lines. The two co-interior angles (same-side interior angles) are (3x + 15)° and (2x + 5)°.
(a) Find the value of x.
(b) Find both angles.
(c) Verify that they sum to 180°. State the property used.
(a) Find the value of x.
(b) Find both angles.
(c) Verify that they sum to 180°. State the property used.
Q15.3
Prove that vertically opposite angles are equal.
Hint: Use the linear pair property. Let two lines intersect at O forming ∠1, ∠2, ∠3, ∠4.
Hint: Use the linear pair property. Let two lines intersect at O forming ∠1, ∠2, ∠3, ∠4.
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.
The angles at P are labelled ∠1 (upper-left), ∠2 (upper-right), ∠3 (lower-right, interior), ∠4 (lower-left, interior).
The angles at Q are labelled ∠5 (upper-left, interior), ∠6 (upper-right, interior), ∠7 (lower-right), ∠8 (lower-left).
At P, ∠1 = 80°.
(a) Find all four angles at P: ∠1, ∠2, ∠3, ∠4. State properties used. [2 marks]
(b) Find all four angles at Q: ∠5, ∠6, ∠7, ∠8. Use corresponding angles. [2 marks]
(c) Identify the alternate interior angle pairs. Are they equal? State the property. [½ mark]
(d) Name the co-interior angle pairs and verify each pair sums to 180°. [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: ∠5, ∠6, ∠7, ∠8. Use corresponding angles. [2 marks]
(c) Identify the alternate interior angle pairs. Are they equal? State the property. [½ mark]
(d) Name the co-interior angle pairs and verify each pair sums to 180°. [½ mark]
Q17.5
(a) Two roads cross at a point. One of the four angles formed is 50°. Find all four angles at the crossing. State properties used. [2 marks]
(b) A set of railway tracks (parallel lines) is crossed by a road. The road makes an angle of 65° with the first track. What angle does it make with the second track? Justify using angle properties. [2 marks]
(c) Describe the set square method to draw a line parallel to a given line through a point P above it. List at least 3 clear steps. [1 mark]
(b) A set of railway tracks (parallel lines) is crossed by a road. The road makes an angle of 65° with the first track. What angle does it make with the second track? Justify using angle properties. [2 marks]
(c) Describe the set square method to draw a line parallel to a given line through a point P above it. List at least 3 clear steps. [1 mark]
Bonus Question (Optional) [2 Marks]
Q18.2
★ Two parallel lines are cut by a transversal. One co-interior angle is 4 times the other. Find both angles. State the property that relates co-interior angles when lines are parallel.
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Answer Key & Detailed Solutions — Set 1
Q1. [1 Mark]
Two lines meeting at exactly one point are called intersecting lines.
The meeting point is called the point of intersection.
The meeting point is called the point of intersection.
Q2. [1 Mark]
Vertically opposite angles are always equal.
(They are formed on opposite sides of the intersection point when two lines cross.)
(They are formed on opposite sides of the intersection point when two lines cross.)
Q3. [1 Mark]
When two lines are perpendicular, each angle = 90°.
Notation: l ⊥ m means line l is perpendicular to line m.
Notation: l ⊥ m means line l is perpendicular to line m.
Q4. [1 Mark]
“Line l is parallel to line m” is written as: l ∥ m
Parallel lines are shown on diagrams with matching arrow marks (› or ») on each line.
Parallel lines are shown on diagrams with matching arrow marks (› or ») on each line.
Q5. [1 Mark]
A transversal crossing two lines forms 8 angles in total (4 at each intersection point).
Q6. [1 Mark]
Corresponding angles are equal only when the two lines are parallel.
Converse is also true: if corresponding angles are equal, the lines are parallel.
Converse is also true: if corresponding angles are equal, the lines are parallel.
Q7. [2 Marks]
Let the given angle = ∠1 = 65°.
∠3 = 65° (Vertically Opposite Angles — ∠1 and ∠3 are VOA) [½ mark]
∠2 = 115° (Linear Pair: ∠1 + ∠2 = 180° → 65 + ∠2 = 180°) [½ mark]
∠4 = 115° (Vertically Opposite to ∠2, or Linear Pair with ∠1) [½ mark]
Summary: Two pairs — 65°, 65°, 115°, 115°. [½ mark for correct summary]
∠3 = 65° (Vertically Opposite Angles — ∠1 and ∠3 are VOA) [½ mark]
∠2 = 115° (Linear Pair: ∠1 + ∠2 = 180° → 65 + ∠2 = 180°) [½ mark]
∠4 = 115° (Vertically Opposite to ∠2, or Linear Pair with ∠1) [½ mark]
Summary: Two pairs — 65°, 65°, 115°, 115°. [½ mark for correct summary]
Q8. [2 Marks]
Given: l ∥ m, ∠2 = 115° at the upper intersection.
∠6 is the corresponding angle to ∠2 (both are in the upper-right position at their respective intersections). [1 mark]
Property (Corresponding Angles): When a transversal cuts two parallel lines, the corresponding angles are equal. [1 mark]
∴ ∠6 = 115°
∠6 is the corresponding angle to ∠2 (both are in the upper-right position at their respective intersections). [1 mark]
Property (Corresponding Angles): When a transversal cuts two parallel lines, the corresponding angles are equal. [1 mark]
∴ ∠6 = 115°
Q9. [2 Marks]
The four pairs of corresponding angles are: [½ mark each]
1. ∠1 and ∠5 (both upper-left at their intersections)
2. ∠2 and ∠6 (both upper-right at their intersections)
3. ∠3 and ∠7 (both lower-right at their intersections)
4. ∠4 and ∠8 (both lower-left at their intersections)
Corresponding angles occupy the same position (same corner) at each intersection point.
1. ∠1 and ∠5 (both upper-left at their intersections)
2. ∠2 and ∠6 (both upper-right at their intersections)
3. ∠3 and ∠7 (both lower-right at their intersections)
4. ∠4 and ∠8 (both lower-left at their intersections)
Corresponding angles occupy the same position (same corner) at each intersection point.
Q10. [2 Marks]
∠AOB and ∠COD are vertically opposite angles, so they are equal. [½ mark]
∴ 3x = x + 60
3x − x = 60
2x = 60
x = 30° [1 mark]
∠AOB = 3 × 30 = 90° [½ mark]
(The lines are perpendicular since the angle is 90°!)
∴ 3x = x + 60
3x − x = 60
2x = 60
x = 30° [1 mark]
∠AOB = 3 × 30 = 90° [½ mark]
(The lines are perpendicular since the angle is 90°!)
Q11. [2 Marks]
Given: l ∥ m, ∠3 = 72° (interior angle, lower-right at P).
∠5 is the alternate interior angle to ∠3 (∠5 is interior at Q, on the left of the transversal — the opposite side). [1 mark]
Property (Alternate Interior Angles): When a transversal cuts two parallel lines, the alternate interior angles are equal. [1 mark]
∴ ∠5 = 72°
∠5 is the alternate interior angle to ∠3 (∠5 is interior at Q, on the left of the transversal — the opposite side). [1 mark]
Property (Alternate Interior Angles): When a transversal cuts two parallel lines, the alternate interior angles are equal. [1 mark]
∴ ∠5 = 72°
Q12. [3 Marks]
Given: ∠1 = 125°
(a) ∠3 = 125°
Property: Vertically Opposite Angles — ∠1 and ∠3 are formed on opposite sides of the intersection. [1 mark]
(b) ∠2 = 55°
Property: Linear Pair — ∠1 + ∠2 = 180° (angles on a straight line)
125° + ∠2 = 180° → ∠2 = 55° [1 mark]
(c) ∠4 = 55°
Property: Vertically Opposite Angles — ∠4 is VOA of ∠2 (or Linear Pair with ∠3). [1 mark]
Summary: ∠1 = ∠3 = 125°; ∠2 = ∠4 = 55°
(a) ∠3 = 125°
Property: Vertically Opposite Angles — ∠1 and ∠3 are formed on opposite sides of the intersection. [1 mark]
(b) ∠2 = 55°
Property: Linear Pair — ∠1 + ∠2 = 180° (angles on a straight line)
125° + ∠2 = 180° → ∠2 = 55° [1 mark]
(c) ∠4 = 55°
Property: Vertically Opposite Angles — ∠4 is VOA of ∠2 (or Linear Pair with ∠3). [1 mark]
Summary: ∠1 = ∠3 = 125°; ∠2 = ∠4 = 55°
Q13. [3 Marks]
Given: l ∥ m, ∠1 = 50° at P.
At point P:
∠3 = 50° — Vertically Opposite Angles (∠1 and ∠3 are VOA) [½ mark]
∠2 = 130° — Linear Pair: 180° − 50° = 130° [½ mark]
∠4 = 130° — VOA of ∠2 (or Linear Pair with ∠1) [½ mark]
At point Q:
∠5 = 50° — Corresponding Angles: ∠5 corresponds to ∠1; l ∥ m → ∠5 = ∠1 = 50° [1½ marks]
At point P:
∠3 = 50° — Vertically Opposite Angles (∠1 and ∠3 are VOA) [½ mark]
∠2 = 130° — Linear Pair: 180° − 50° = 130° [½ mark]
∠4 = 130° — VOA of ∠2 (or Linear Pair with ∠1) [½ mark]
At point Q:
∠5 = 50° — Corresponding Angles: ∠5 corresponds to ∠1; l ∥ m → ∠5 = ∠1 = 50° [1½ marks]
Q14. [3 Marks]
Co-interior angles (same-side interior angles) on parallel lines sum to 180°.
(a) (3x + 15) + (2x + 5) = 180°
5x + 20 = 180°
5x = 160°
x = 32° [1 mark]
(b) First angle = 3(32) + 15 = 96 + 15 = 111° [½ mark]
Second angle = 2(32) + 5 = 64 + 5 = 69° [½ mark]
(c) Verification: 111° + 69° = 180° ✓ [½ mark]
Property: Co-interior angles (also called co-interior or same-side interior angles) formed by a transversal on parallel lines are supplementary (sum = 180°). [½ mark]
(a) (3x + 15) + (2x + 5) = 180°
5x + 20 = 180°
5x = 160°
x = 32° [1 mark]
(b) First angle = 3(32) + 15 = 96 + 15 = 111° [½ mark]
Second angle = 2(32) + 5 = 64 + 5 = 69° [½ mark]
(c) Verification: 111° + 69° = 180° ✓ [½ mark]
Property: Co-interior angles (also called co-interior or same-side interior angles) formed by a transversal on parallel lines are supplementary (sum = 180°). [½ mark]
Q15. [3 Marks]
To prove: Vertically opposite angles are equal.
Given: Two lines AB and CD intersect at O, forming ∠1, ∠2, ∠3, ∠4 (in order around O). [1 mark for setup]
Proof:
∠1 and ∠2 lie on a straight line AB → ∠1 + ∠2 = 180° …(i) [Linear Pair] [1 mark]
∠2 and ∠3 lie on a straight line CD → ∠2 + ∠3 = 180° …(ii) [Linear Pair]
From (i) and (ii):
∠1 + ∠2 = ∠2 + ∠3
∠1 = ∠3 ∴ Vertically opposite angles are equal. ■ [1 mark]
(Similarly ∠2 = ∠4 can be proved.)
Given: Two lines AB and CD intersect at O, forming ∠1, ∠2, ∠3, ∠4 (in order around O). [1 mark for setup]
Proof:
∠1 and ∠2 lie on a straight line AB → ∠1 + ∠2 = 180° …(i) [Linear Pair] [1 mark]
∠2 and ∠3 lie on a straight line CD → ∠2 + ∠3 = 180° …(ii) [Linear Pair]
From (i) and (ii):
∠1 + ∠2 = ∠2 + ∠3
∠1 = ∠3 ∴ Vertically opposite angles are equal. ■ [1 mark]
(Similarly ∠2 = ∠4 can be proved.)
Q16. [5 Marks]
Given: l ∥ m, ∠1 = 80° at P.
(a) Angles at P: [2 marks]
∠1 = 80° (given)
∠3 = 80° (Vertically Opposite Angles: ∠1 and ∠3 are VOA)
∠2 = 100° (Linear Pair: 180° − 80°)
∠4 = 100° (Vertically Opposite to ∠2)
(b) Angles at Q (using Corresponding Angles, l ∥ m): [2 marks]
∠5 = 80° (corresponds to ∠1)
∠6 = 100° (corresponds to ∠2)
∠7 = 80° (corresponds to ∠3)
∠8 = 100° (corresponds to ∠4)
(c) Alternate Interior Angles: [½ mark]
Pairs: (∠3, ∠5) and (∠4, ∠6)
∠3 = ∠5 = 80° ✓ ∠4 = ∠6 = 100° ✓
Property: Alternate interior angles are equal when lines are parallel.
(d) Co-interior Angle Pairs: [½ mark]
Pairs: (∠3, ∠6) and (∠4, ∠5)
∠3 + ∠6 = 80° + 100° = 180° ✓
∠4 + ∠5 = 100° + 80° = 180° ✓
(a) Angles at P: [2 marks]
∠1 = 80° (given)
∠3 = 80° (Vertically Opposite Angles: ∠1 and ∠3 are VOA)
∠2 = 100° (Linear Pair: 180° − 80°)
∠4 = 100° (Vertically Opposite to ∠2)
(b) Angles at Q (using Corresponding Angles, l ∥ m): [2 marks]
∠5 = 80° (corresponds to ∠1)
∠6 = 100° (corresponds to ∠2)
∠7 = 80° (corresponds to ∠3)
∠8 = 100° (corresponds to ∠4)
(c) Alternate Interior Angles: [½ mark]
Pairs: (∠3, ∠5) and (∠4, ∠6)
∠3 = ∠5 = 80° ✓ ∠4 = ∠6 = 100° ✓
Property: Alternate interior angles are equal when lines are parallel.
(d) Co-interior Angle Pairs: [½ mark]
Pairs: (∠3, ∠6) and (∠4, ∠5)
∠3 + ∠6 = 80° + 100° = 180° ✓
∠4 + ∠5 = 100° + 80° = 180° ✓
Q17. [5 Marks]
(a) Two roads crossing: [2 marks]
Let the first angle = 50°.
Vertically opposite angle = 50° (VOA property)
Linear pair angle = 180° − 50° = 130°
The fourth angle = 130° (VOA of 130°)
Four angles: 50°, 130°, 50°, 130°
(b) Railway tracks (parallel lines): [2 marks]
The road acts as a transversal. The angle at the first track = 65°.
The corresponding angle at the second (parallel) track = 65°.
Property: Corresponding Angles — equal when lines are parallel.
∴ The road makes a 65° angle with the second track.
(c) Set Square Method (parallel line through point P): [1 mark]
Step 1: Place the set square so one of its right-angle sides lies along the given line.
Step 2: Place a ruler along the other side (hypotenuse) of the set square.
Step 3: Holding the ruler fixed, slide the set square along the ruler until its edge reaches point P.
Step 4: Draw the line along the set square’s edge through P. This line is parallel to the original.
Let the first angle = 50°.
Vertically opposite angle = 50° (VOA property)
Linear pair angle = 180° − 50° = 130°
The fourth angle = 130° (VOA of 130°)
Four angles: 50°, 130°, 50°, 130°
(b) Railway tracks (parallel lines): [2 marks]
The road acts as a transversal. The angle at the first track = 65°.
The corresponding angle at the second (parallel) track = 65°.
Property: Corresponding Angles — equal when lines are parallel.
∴ The road makes a 65° angle with the second track.
(c) Set Square Method (parallel line through point P): [1 mark]
Step 1: Place the set square so one of its right-angle sides lies along the given line.
Step 2: Place a ruler along the other side (hypotenuse) of the set square.
Step 3: Holding the ruler fixed, slide the set square along the ruler until its edge reaches point P.
Step 4: Draw the line along the set square’s edge through P. This line is parallel to the original.
Q18. Bonus [2 Marks]
Let the two co-interior angles be x and 4x.
Property: Co-interior angles on parallel lines are supplementary (sum = 180°). [½ mark]
x + 4x = 180°
5x = 180°
x = 36° [½ mark]
Smaller angle = 36°; Larger angle = 4 × 36° = 144° [1 mark]
Verification: 36° + 144° = 180° ✓
Property: Co-interior angles on parallel lines are supplementary (sum = 180°). [½ mark]
x + 4x = 180°
5x = 180°
x = 36° [½ mark]
Smaller angle = 36°; Larger angle = 4 × 36° = 144° [1 mark]
Verification: 36° + 144° = 180° ✓