MCQs / Mathematics / Angle Between Lines

Angle Between Lines MCQs — Mathematics

63 multiple-choice questions on Angle Between Lines from Mathematics, each with the correct answer and, where available, a teacher's explanation. Read through them to revise the chapter, then practise the same questions one by one to test yourself.

1. The family of lines through the intersection of L1: a1x + b1y + c1 = 0 and L2: a2x + b2y + c2 = 0 is:

  1. L1 L2 = 0
  2. L1 = L2
  3. kL1 L2 = 1
  4. L1 + kL2 = 0

Answer: D) L1 + kL2 = 0

Explanation: For any number k, L1 + kL2 = 0 is a line, and it is satisfied by the point where both L1 = 0 and L2 = 0. SLO M-09-B-11 · Understanding

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2. The angle between lines with slopes - 35 and 53 is:

  1. 0^circ
  2. 45^circ
  3. 90^circ
  4. 60^circ

Answer: C) 90^circ

Explanation: m1 m2 = -1 = -1, so the lines are perpendicular. SLO M-09-B-10 · Knowledge

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3. Find the acute angle between lines with slopes 4 and 35.

  1. 90^circ
  2. 45^circ
  3. 30^circ
  4. 60^circ

Answer: B) 45^circ

Explanation: tantheta = m2 - m11 + m1 m2 = - 175 175 = 1, so theta = 45^circ. SLO M-09-B-10 · Knowledge

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4. The triangle has vertices A(0, 0), B(3, 0), C(0, 3). The angle at A is:

  1. 90^circ
  2. 30^circ
  3. 45^circ
  4. 60^circ

Answer: A) 90^circ

Explanation: The slopes of AB and AC multiply to -1 (or one side is horizontal and the other vertical), so angle A = 90^circ. SLO M-09-B-12 · Knowledge

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5. If tan A = 1 and tan B = 1 in triangle ABC, then angle C is:

  1. 90^circ
  2. 60^circ
  3. 0^circ
  4. 45^circ

Answer: A) 90^circ

Explanation: A = B = 45^circ, so C = 180^circ - 90^circ = 90^circ. SLO M-09-B-12 · Knowledge

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6. The line through the intersection of 2x + y - 5 = 0 and x + 3y - 5 = 0, parallel to the x-axis, is:

  1. y = 2
  2. y = 1
  3. y = -1
  4. x = 2

Answer: B) y = 1

Explanation: The lines meet at (2, 1); a line parallel to the x-axis through it is y = 1. SLO M-09-B-11 · Understanding

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7. Find the line through the intersection of x + y + 3 = 0 and 2x - 3y + 1 = 0 that also passes through (2, 2).

  1. 3x - 4y - 2 = 0
  2. 3x - 4y + 2 = 0
  3. 4x - 3y - 2 = 0
  4. 3x + 4y + 2 = 0

Answer: B) 3x - 4y + 2 = 0

Explanation: The lines meet at (-2, -1). The line through (-2, -1) and (2, 2) is 3x - 4y + 2 = 0 (you can also use L1 + kL2 = 0 and find k from the point). SLO M-09-B-11 · Understanding

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8. The triangle has vertices A(0, 0), B(4, 2), C(1, 3). Find tan A.

  1. -1
  2. 3
  3. 2
  4. 1

Answer: D) 1

Explanation: Slopes: mAB = 12, mAC = 3. Using tan A = mAC - mAB1 + mABmAC (with the sign that gives the interior angle) we get tan A = 1. SLO M-09-B-12 · Knowledge

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9. Find the acute angle between lines with slopes -1 and 0.

  1. 60^circ
  2. 30^circ
  3. 90^circ
  4. 45^circ

Answer: D) 45^circ

Explanation: tantheta = m2 - m11 + m1 m2 = 11 = 1, so theta = 45^circ. SLO M-09-B-10 · Knowledge

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10. For which k does the line (x + y - 3) + k(x - y - 1) = 0 pass through the origin?

  1. k = 3
  2. k = 13
  3. k = -1
  4. k = -3

Answer: D) k = -3

Explanation: Put x = y = 0: -3 + k(-1) = 0 Rightarrow k = -3. SLO M-09-B-11 · Understanding

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11. The sides of a triangle have slopes 0, 3 and - 3. The triangle is:

  1. isosceles with a 120^circ angle
  2. equilateral
  3. right-angled
  4. scalene

Answer: B) equilateral

Explanation: The sloping sides make 60^circ with the base and with each other, so every angle is 60^circ. SLO M-09-B-12 · Knowledge

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12. The line through the intersection of x + y - 5 = 0 and x + 3y - 13 = 0, parallel to the y-axis, is:

  1. x = 4
  2. x = 1
  3. x = -1
  4. y = 4

Answer: B) x = 1

Explanation: The lines meet at (1, 4); a vertical line through it is x = 1. SLO M-09-B-11 · Understanding

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13. The line through the intersection of x - y + 4 = 0 and x + 3y - 8 = 0, parallel to the x-axis, is:

  1. y = -1
  2. y = 3
  3. y = -3
  4. x = -1

Answer: B) y = 3

Explanation: The lines meet at (-1, 3); a line parallel to the x-axis through it is y = 3. SLO M-09-B-11 · Understanding

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14. Two sides of a triangle have slopes 23 and - 32. The angle between them is:

  1. 90^circ
  2. 60^circ
  3. 0^circ
  4. 45^circ

Answer: A) 90^circ

Explanation: 23 (- 32 ) = -1. SLO M-09-B-12 · Knowledge

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15. In triangle ABC, mAB = 2 and mAC = - 12. Then angle A is:

  1. 45^circ
  2. 60^circ
  3. 120^circ
  4. 90^circ

Answer: D) 90^circ

Explanation: 2 - 12 = -1, so AB perp AC. SLO M-09-B-12 · Knowledge

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16. Find the acute angle between lines with slopes 0 and 1.

  1. 60^circ
  2. 45^circ
  3. 30^circ
  4. 90^circ

Answer: B) 45^circ

Explanation: tantheta = m2 - m11 + m1 m2 = 11 = 1, so theta = 45^circ. SLO M-09-B-10 · Knowledge

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17. Find the line through the intersection of x - y + 4 = 0 and x + 3y - 8 = 0 that also passes through (1, 1).

  1. x + y - 2 = 0
  2. x + y + 2 = 0
  3. x - y - 2 = 0
  4. x + 2y - 4 = 0

Answer: A) x + y - 2 = 0

Explanation: The lines meet at (-1, 3). The line through (-1, 3) and (1, 1) is x + y - 2 = 0 (you can also use L1 + kL2 = 0 and find k from the point). SLO M-09-B-11 · Understanding

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18. Three lines are concurrent when:

  1. they all pass through one common point
  2. they are all parallel
  3. they are all perpendicular
  4. they have equal intercepts

Answer: A) they all pass through one common point

Explanation: Concurrent lines meet at a single point. SLO M-09-B-11 · Understanding

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19. If two angles of a triangle are found to be 45^circ and 75^circ, the third angle is:

  1. 60^circ
  2. 30^circ
  3. 90^circ
  4. 120^circ

Answer: A) 60^circ

Explanation: Angles of a triangle add up to 180^circ: 180 - 45 - 75 = 60. SLO M-09-B-12 · Knowledge

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20. To find the angle A of triangle ABC from slopes, we use the slopes of:

  1. BC only
  2. BC and CA
  3. AB and BC
  4. AB and AC

Answer: D) AB and AC

Explanation: The angle at A is between the two sides that meet at A. SLO M-09-B-12 · Knowledge

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21. Find the line through the intersection of x + y - 5 = 0 and 3x - y - 11 = 0 that also passes through (0, 0).

  1. x + 4y = 0
  2. x - 4y - 1 = 0
  3. 4x - y = 0
  4. x - 4y = 0

Answer: D) x - 4y = 0

Explanation: The lines meet at (4, 1). The line through (4, 1) and (0, 0) is x - 4y = 0 (you can also use L1 + kL2 = 0 and find k from the point). SLO M-09-B-11 · Understanding

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22. For the acute angle theta between lines with slopes 13 and 12, tantheta =

  1. 7
  2. 17
  3. 15
  4. 16

Answer: B) 17

Explanation: tantheta = m2 - m11 + m1 m2 = 16 76 = 17. SLO M-09-B-10 · Knowledge

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23. The angle between the x-axis and the line y = 1 3x is:

  1. 60^circ
  2. 15^circ
  3. 45^circ
  4. 30^circ

Answer: D) 30^circ

Explanation: tantheta = 1 3, so theta = 30^circ. SLO M-09-B-10 · Knowledge

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24. The triangle with vertices (1, 1), (4, 1) and (1, 5) is right-angled at:

  1. no vertex
  2. (1, 1)
  3. (1, 5)
  4. (4, 1)

Answer: B) (1, 1)

Explanation: From (1, 1) one side is horizontal and the other vertical. SLO M-09-B-12 · Knowledge

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25. Find the point of intersection of x + y + 3 = 0 and 2x - 3y + 1 = 0.

  1. (-2, -1)
  2. (-2, 1)
  3. (2, -1)
  4. (-1, -2)

Answer: A) (-2, -1)

Explanation: Solve the two equations together: x = -2, y = -1. Check: both equations are satisfied by (-2, -1). SLO M-09-B-11 · Understanding

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