Hard

A Tale of Three Intersecting Lines — Hard

15 questions 18 min PYQ-grade reasoning

Q1. Consider the lengths 10 cm, 15 cm and 30 cm. Which statements are correct? I. 10 + 15 < 30, so the inequality fails for the longest side. II. 10 + 30 > 15 and 15 + 30 > 10 hold, so a triangle is possible. III. A triangle cannot be formed with these three lengths. Which are correct?
- A. I and II only
- B. II and III only
- C. I and III only
- D. All three
Q2. Assertion (A): To check the triangle inequality for three lengths, it is enough to check whether the largest length is less than the sum of the other two. Reason (R): If the largest length is less than the sum of the other two, the other two inequalities automatically hold.
- A. Both A and R are true, and R is the correct explanation of A.
- B. Both A and R are true, but R is NOT the correct explanation of A.
- C. A is true, R is false.
- D. A is false, R is true.
Q3. On segment AB of length 8 cm, a circle of radius 3 cm is drawn at A and a circle of radius 4 cm is drawn at B. Which of the following best describes the two circles, and what does it mean for the triangle?
- A. The circles intersect at two points; a triangle with sides 8, 3, 4 exists.
- B. The circles touch at one point; the three lengths form a degenerate triangle.
- C. The circles do not intersect; no triangle with sides 8, 3, 4 exists.
- D. The circles are concentric; the triangle has sides 8, 3, 4 cm.
Q4. In the chapter's proof of the angle sum property, a line is drawn through vertex A parallel to BC. Consider these statements: I. The two angles at A on either side of the parallel line and the angle BAC together make a straight angle. II. Each of those two angles at A equals an interior angle of the triangle by alternate angles on a transversal. III. From I and II, ∠A + ∠B + ∠C = 180°. Which are correct?
- A. I and II only
- B. II and III only
- C. I and III only
- D. All three
Q5. In ΔPQR, side QR is extended to S. The exterior angle ∠PRS measures 130°, and ∠PQR = 70°. What is the measure of ∠QPR?
- A. 50°
- B. 60°
- C. 70°
- D. 110°
Q6. A student is asked to construct ΔABC with AB = 5 cm, AC = 4 cm and ∠A = 180°. What will happen?
- A. An obtuse-angled triangle of large area will form.
- B. A right-angled triangle will form with the right angle at A.
- C. No triangle will form — the two sides at A will lie along a straight line.
- D. An equilateral triangle of side 4 cm will form.
Q7. A triangle has to be constructed with two given angles 65° and 75° at the ends of a side of length 6 cm. What will be the measure of the third angle, and what type of triangle is formed?
- A. 40°, an acute-angled triangle
- B. 40°, a right-angled triangle
- C. 50°, an obtuse-angled triangle
- D. 140°, an obtuse-angled triangle
Q8. Assertion (A): Every equilateral triangle is also an isosceles triangle. Reason (R): An isosceles triangle is defined as a triangle with at least two sides equal.
- A. Both A and R are true, and R is the correct explanation of A.
- B. Both A and R are true, but R is NOT the correct explanation of A.
- C. A is true, R is false.
- D. A is false, R is true.
Q9. In right-angled ΔPQR, the right angle is at Q. Which of the following is the altitude from vertex P to the side QR?
- A. The segment PR itself
- B. The segment PQ itself
- C. A new perpendicular from P meeting QR somewhere between Q and R
- D. A perpendicular from P to the extension of QR beyond R
Q10. Consider these statements: I. A triangle cannot have two right angles. II. A triangle cannot have two obtuse angles. III. A triangle can have one right angle and one obtuse angle together. Which are correct?
- A. I and II only
- B. II and III only
- C. I and III only
- D. All three
Q11. While constructing ΔABC with three given sides using the arc method, the two arcs from A and B usually intersect at two points C₁ (above AB) and C₂ (below AB). Which of the following is the correct conclusion?
- A. Only C₁ gives a valid triangle; C₂ should be discarded.
- B. ΔABC₁ and ΔABC₂ are congruent triangles — mirror images across AB.
- C. ΔABC₁ and ΔABC₂ have different side lengths.
- D. Two different triangles with three different sets of sides can be built.
Q12. In ΔABC, a line ℓ is drawn through A parallel to BC. The angle ∠B = 65° appears at A as an alternate interior angle along the transversal AB. If ∠C = 55°, what is ∠BAC?
- A. 55°
- B. 60°
- C. 65°
- D. 120°
Q13. In ΔABC, side BC is extended to D so that B, C, D are collinear. The interior angle ∠ACB = 72°. Two students compute the exterior angle ∠ACD differently: Ravi: ∠ACD = 180° − 72° = 108°. Priya: ∠ACD = 90° − 72° = 18°. Who is correct, and why?
- A. Ravi, because ∠ACB and ∠ACD form a linear pair along BCD.
- B. Priya, because the exterior angle equals 90° minus the interior angle.
- C. Both, because both 108° and 18° are exterior angles.
- D. Neither, because the exterior angle equals 72°.
Q14. Consider these statements about why the triangle inequality must hold: I. The third vertex C is the intersection of two circles, one of radius b centred at A and one of radius a centred at B. II. Two circles meet at two points only when the distance between their centres is less than the sum of their radii. III. Hence side c = AB must be less than a + b for ΔABC to exist. Which are correct?
- A. I and II only
- B. II and III only
- C. I and III only
- D. All three
Q15. Assertion (A): In an obtuse-angled triangle, the altitude from the vertex of an acute angle to the opposite side may fall outside the triangle, on the extension of that side. Reason (R): The foot of the perpendicular from a vertex to the line of the opposite side need not lie between the two endpoints of that side.
- A. Both A and R are true, and R is the correct explanation of A.
- B. Both A and R are true, but R is NOT the correct explanation of A.
- C. A is true, R is false.
- D. A is false, R is true.

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