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1. Which geometric instruments are allowed in geometric constructions using Euclidean principles?
A. Protractor and Ruler
B. Graduated ruler and Compass
C. Un-graduated ruler (straightedge) and Compass
D. Set squares and Ruler
Answer: C. Un-graduated ruler (straightedge) and Compass
Explanation: Classical geometrical construction strictly uses an un-marked ruler (straightedge) to draw lines and a compass to draw arcs/circles.
2. The angle bisector divides a given angle into how many equal parts?
A. 3
B. 2
C. 4
D. 5
Answer: B. 2
Explanation: An angle bisector divides an angle into two equal parts.
3. Which of the following angles CANNOT be constructed using a ruler and compass only?
A. 60°
B. 45°
C. 85°
D. 105°
Answer: C. 85°
Explanation: Angles that are multiples of 7.5° (or 15°) can be constructed using ruler and compass. 85° is not a multiple of 7.5°.
4. Which of the following angles CAN be constructed using a compass and ruler?
A. 37.5°
B. 40°
C. 50°
D. 70°
Answer: A. 37.5°
Explanation: 37.5° is half of 75° (bisecting 75°), which is a standard constructible angle (multiple of 7.5°).
5. The perpendicular bisector of a line segment divides the line segment at an angle of:
A. 45°
B. 60°
C. 90°
D. 180°
Answer: C. 90°
Explanation: "Perpendicular" means the line forms right angles (90°) at the point of intersection.
6. To construct a perpendicular bisector of a line segment AB of length 8 cm, the compass radius taken should be:
A. Exactly 4 cm
B. Less than 4 cm
C. More than 4 cm
D. Any radius
Answer: C. More than 4 cm
Explanation: The compass radius must be greater than half the length of the line segment (> 4 cm) so that the arcs intersect on both sides.
7. To construct an angle of 60° at the initial point of a ray, the arc radius must be equal to:
A. The length of the ray
B. Half the length of the ray
C. Any convenient radius
D. 10 cm
Answer: C. Any convenient radius
Explanation: Any convenient radius can be used, provided the same radius is maintained when cutting the initial arc.
8. A triangle ABC can be constructed if its perimeter and ________ base angles are given.
A. One
B. Two
C. Three
D. Zero
Answer: B. Two
Explanation: A triangle can be constructed given its perimeter and two base angles.
9. To construct a triangle ABC, given BC = 6 cm, ∠B = 60°, which of the following conditions is required for the sum AB + AC?
A. AB + AC = BC
B. AB + AC < BC
C. AB + AC > BC
D. AB + AC = 0
Answer: C. AB + AC > BC
Explanation: By the triangle inequality theorem, the sum of any two sides must be strictly greater than the third side (AB + AC > 6 cm).
10. To construct a triangle given base BC, ∠B, and difference AB - AC, construction is possible only when AB - AC is:
A. Equal to BC
B. Greater than BC
C. Less than BC
D. Double of BC
Answer: C. Less than BC
Explanation: The difference between two sides of a triangle is always less than the third side (|AB - AC| < BC).
11. An angle of 22.5° can be constructed by bisecting an angle of:
A. 30°
B. 45°
C. 60°
D. 90°
Answer: B. 45°
Explanation: 22.5° is half of 45° (45° / 2 = 22.5°).
12. An angle of 135° can be obtained by adding 90° to:
A. 30°
B. 45°
C. 60°
D. 15°
Answer: B. 45°
Explanation: 90° + 45° = 135°. It can also be constructed by bisecting the region between 90° and 180°.
13. Bisecting an angle of 120° gives an angle of:
A. 30°
B. 45°
C. 60°
D. 90°
Answer: C. 60°
Explanation: 120° / 2 = 60°.
14. Which angle is formed by bisecting the angle between 60° and 120°?
A. 75°
B. 90°
C. 105°
D. 80°
Answer: B. 90°
Explanation: The midpoint between 60° and 120° is (60° + 120°) / 2 = 180° / 2 = 90°.
15. The angle 75° can be constructed by bisecting the angle between:
A. 60° and 90°
B. 0° and 90°
C. 90° and 120°
D. 60° and 120°
Answer: A. 60° and 90°
Explanation: The midpoint between 60° and 90° is (60° + 90°) / 2 = 150° / 2 = 75°.
16. The angle 105° is constructed by bisecting the angle between:
A. 60° and 90°
B. 90° and 120°
C. 90° and 180°
D. 120° and 150°
Answer: B. 90° and 120°
Explanation: The midpoint between 90° and 120° is (90° + 120°) / 2 = 210° / 2 = 105°.
17. Is it possible to construct a triangle with sides 3 cm, 4 cm, and 8 cm?
A. Yes
B. No
C. Depends on the angles
D. Data insufficient
Answer: B. No
Explanation: Here, 3 + 4 = 7 cm, which is NOT greater than 8 cm. A triangle cannot be formed.
18. Every point on the perpendicular bisector of a line segment is ________ from the endpoints of the segment.
A. Unequal
B. Equidistant
C. Double distance
D. Half distance
Answer: B. Equidistant
Explanation: By property of perpendicular bisectors, any point on it is equidistant from both endpoints of the segment.
19. To construct an equilateral triangle of side 5 cm, we need to construct base of 5 cm and base angles of:
A. 45° each
B. 60° each
C. 90° each
D. 30° each
Answer: B. 60° each
Explanation: All internal angles of an equilateral triangle are equal to 60°.
20. The angle bisector of an angle of 150° gives two angles of:
A. 75° each
B. 60° each
C. 50° each
D. 90° each
Answer: A. 75° each
Explanation: Bisecting 150° yields 150° / 2 = 75° on each side.
21. Is it possible to construct a triangle ABC where BC = 5 cm, ∠B = 45° and AB + AC = 4 cm?
A. Yes
B. No
C. Only if ∠C = 90°
D. Cannot be determined
Answer: B. No
Explanation: Sum of two sides (AB + AC = 4 cm) must be greater than third side (BC = 5 cm). Since 4 < 5, construction is impossible.
22. To construct a right triangle whose base is 4 cm and sum of its hypotenuse and other side is 8 cm, the base angle at the right vertex is:
A. 45°
B. 60°
C. 90°
D. 30°
Answer: C. 90°
Explanation: In a right triangle, the angle between the base and the vertical leg is 90°.
23. Construction of a triangle ABC with AB = 4 cm, BC = 5 cm is NOT possible if AC is equal to:
A. 6 cm
B. 7 cm
C. 8 cm
D. 9 cm
Answer: D. 9 cm
Explanation: AB + BC = 4 + 5 = 9 cm. For a valid triangle, AC must be strictly LESS than 9 cm. So AC = 9 cm is not possible.
24. Bisecting a 90° angle yields an angle of:
A. 30°
B. 45°
C. 60°
D. 15°
Answer: B. 45°
Explanation: 90° / 2 = 45°.
25. Bisecting a 45° angle gives an angle of:
A. 22.5°
B. 15°
C. 30°
D. 12.5°
Answer: A. 22.5°
Explanation: 45° / 2 = 22.5° (or 22½°).
26. Which angle is constructed by bisecting the angle between 0° and 60°?
A. 15°
B. 30°
C. 45°
D. 20°
Answer: B. 30°
Explanation: The midpoint between 0° and 60° is 60° / 2 = 30°.
27. Which angle is constructed by bisecting the angle between 0° and 30°?
A. 15°
B. 10°
C. 20°
D. 25°
Answer: A. 15°
Explanation: 30° / 2 = 15°.
28. Which of the following angle measures CANNOT be constructed with compass and ruler?
A. 15°
B. 30°
C. 35°
D. 45°
Answer: C. 35°
Explanation: 35° is not a multiple of 7.5° and cannot be obtained by bisecting standard constructible angles.
29. The locus of points equidistant from two fixed points A and B is the:
A. Angle bisector
B. Circle with diameter AB
C. Perpendicular bisector of AB
D. Parallel line to AB
Answer: C. Perpendicular bisector of AB
Explanation: The perpendicular bisector contains all points that are equal distances from point A and point B.
30. When constructing a triangle ABC given base BC, base angle ∠B, and difference of sides (AC - AB) where AC > AB, the ray BX is drawn and segment cut on the ray is in direction:
A. Same direction of ray BX
B. Opposite direction of ray BX (extended downwards)
C. Perpendicular to BX
D. Parallel to BC
Answer: B. Opposite direction of ray BX (extended downwards)
Explanation: When AC > AB (i.e. AB < AC), the line segment equal to (AC - AB) is cut on the extension of ray BX in the opposite direction.
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