Chapter 5: Lines and Angles - Board Exam Practice Questions

Section A: Very Short Answer Questions (1 Mark Each)

1. What is the measure of a straight angle?
2. If two angles are complementary and one angle is 35°, find the other angle.
3. What is the sum of all angles around a point?
4. If two parallel lines are cut by a transversal, what can you say about corresponding angles?
5. Define vertically opposite angles.
6. What is the measure of each angle of a linear pair if they are equal?
7. If ∠AOB = 45° and ∠BOC = 55°, find ∠AOC when rays OA, OB, and OC are in the same plane.
8. Name the angle which is equal to its supplement.
9. What is the sum of interior angles on the same side of a transversal when two parallel lines are cut by it?
10. If two angles are in the ratio 2:3 and they are supplementary, find the smaller angle.

Section B: Short Answer Questions (2 Marks Each)

11. Prove that vertically opposite angles are equal.
12. Two supplementary angles are in the ratio 7:5. Find both angles.
13. If AB || CD and EF is a transversal intersecting them at P and Q respectively. If ∠APE = 55°, find all other angles.
14. The difference between two complementary angles is 20°. Find both angles.
15. In the given figure, if AB || CD, find the value of x. [Assume a figure where alternate angles or corresponding angles are marked]
16. If two angles of a triangle are 50° and 70°, find the third angle. What type of triangle is it?
17. Prove that the sum of angles in a linear pair is 180°.
18. If ∠A and ∠B are supplementary angles and ∠A = (2x + 10)° and ∠B = (3x - 5)°, find x and both angles.
19. In the figure, l₁ || l₂. If ∠1 = 65°, find ∠2, ∠3, and ∠4.
20. Two angles are complementary. The larger angle is 15° more than twice the smaller angle. Find both angles.

Section C: Long Answer Questions (3-4 Marks Each)

21. Theorem Proof: State and prove that if two parallel lines are cut by a transversal, then each pair of alternate interior angles are equal.
22. In triangle ABC, if the exterior angle at vertex A is 110° and ∠B = 45°, find ∠C and ∠A.
23. In the given figure, AB || DE. Find the values of x, y, and z. [Consider a complex figure with multiple angles marked]
24. Case Study: A ladder is placed against a wall making an angle of 60° with the ground. The ladder, ground, and wall form a triangle.
    • What is the angle between the ladder and the wall?
    • What is the angle between the wall and ground?
    • Justify your answers using properties of triangles and parallel lines.
25. Prove that: "If two lines intersect each other, then the vertically opposite angles are equal." Also, find the measure of all angles if one angle is (2x + 10)° and its vertically opposite angle is (3x - 15)°.
26. In triangle PQR, the exterior angles at P, Q, and R are in the ratio 3:4:5. Find:
    • All exterior angles
    • All interior angles
    • What type of triangle is PQR?

Section D: Application-Based Questions (4-5 Marks Each)

27. Architecture Application: A roof truss is designed such that two parallel beams are connected by cross-beams. If one cross-beam makes an angle of 55° with the first parallel beam:
    • Find all angles formed at the intersection points
    • Justify using properties of parallel lines and transversals
    • Draw a neat diagram
28. Real-life Problem: Two straight roads intersect each other. The angle between them is 70°. A third road runs parallel to one of the original roads. Find all angles formed when this third road intersects with the other original road. Support your answer with a diagram and proper reasoning.
29. Multi-step Problem: In the figure, AB || CD || EF and PQ is a transversal cutting these parallel lines at points X, Y, and Z respectively. If ∠PXA = 50° and ∠QYC = 130°, find:
    • ∠AXY
    • ∠XYD
    • ∠YZF
    • ∠EZY Justify each step with appropriate reasons.
30. Proof-based Application: State and prove: "The sum of the three interior angles of a triangle is 180°." Using this theorem, solve: In triangle ABC, ∠A = x°, ∠B = (2x + 10)°, and ∠C = (x + 20)°. Find all three angles and classify the triangle.

Section E: Higher Order Thinking Questions (5 Marks Each)

31. Challenge Problem: Three parallel lines l₁, l₂, and l₃ are cut by two transversals m and n. At the intersection points, various angles are formed. If the angle between the transversals is 45°, and one of the angles formed by l₁ and m is 65°, find all possible angles in the figure. Draw the complete diagram and show all calculations.
32. Reasoning Question: "If two lines are cut by a transversal such that the interior angles on the same side of the transversal are supplementary, then the lines are parallel."
    • State this as a theorem
    • Provide a complete proof
    • Give a practical application of this theorem
33. Investigation Problem: In a triangle ABC, the bisector of exterior angle at A meets BC extended at D. If ∠B = 40° and ∠C = 60°:
    • Find ∠BAC
    • Find the exterior angle at A
    • Find ∠BAD and ∠CAD
    • Prove that ∠BAD - ∠CAD = ∠C - ∠B

Answer Key Summary

Section A (1-10): 180°, 55°, 360°, equal, angles opposite each other when two lines intersect, 90° each, 100°, 90°, 180°, 72°

Section B: Focus on step-by-step solutions using angle properties

Section C & D: Require detailed proofs, diagrams, and multi-step reasoning

Section E: Advanced applications requiring deep understanding of concepts

Important Theorems to Remember:

1. Linear Pair Axiom
2. Vertically Opposite Angles Theorem
3. Parallel Lines and Transversal Properties
4. Angle Sum Property of Triangle
5. Exterior Angle Property of Triangle

Marking Scheme Guidelines:

• Always draw neat, labeled diagrams (1 mark)
• State given information clearly (0.5 marks)
• Show step-by-step working (2-3 marks)
• State final answers clearly (0.5 marks)
• Use proper geometric reasoning and theorems (1 mark)