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Worksheet: Angle between Two Straight Lines in Two Dimensions

Q1:

Determine the size of the acute angle between the straight line and the straight line passing through the points and to the nearest second.

  • A
  • B
  • C
  • D

Q2:

Determine the measure of the positive angle that the straight line 𝐿 makes with the positive direction of the 𝑥 -axis approximated to the nearest second, given that 𝐿 passes through points 𝐴 ( 1 , 4 ) and 𝐵 ( 3 , 5 ) .

  • A 1 4 2 1 0
  • B 1 0 4 2 1 0
  • C 6 9 2 6 3 8
  • D 1 6 5 5 7 5 0
  • E 2 0 3 3 2 2

Q3:

If the acute angle between the straight lines whose equations are 𝑘 𝑦 2 𝑥 + 1 9 = 0 and 9 𝑥 7 𝑦 8 = 0 is 𝜋 4 , find all the possible values of 𝑘 .

  • A 7 , 7
  • B 1 6 , 1 4
  • C 1 8 , 1
  • D 1 4 , 16

Q4:

Given that is a right triangle at , the equation of is , and the equation of is , find approximated to the nearest minute.

  • A
  • B
  • C
  • D

Q5:

Determine, to the nearest second, the measure of the positive angle that the straight line 1 4 𝑥 + 1 2 𝑦 = 2 7 makes with the positive 𝑥 -axis.

  • A 4 0 3 6 5
  • B 4 9 2 3 5 5
  • C 1 3 9 2 3 5 5
  • D 1 3 0 3 6 5

Q6:

A line passing through ( 8 , 2 ) makes an angle 𝜃 with the line 6 𝑥 + 4 𝑦 + 9 = 0 , and t a n 𝜃 = 1 5 1 3 . What is the equation of this line?

  • A 6 9 𝑥 + 7 1 𝑦 + 4 1 0 = 0 , 9 𝑥 1 9 𝑦 + 1 1 0 = 0
  • B 7 1 𝑥 9 𝑦 + 2 1 4 = 0 , 1 9 𝑥 + 6 9 𝑦 + 5 1 4 = 0
  • C 7 1 𝑥 6 9 𝑦 + 4 1 0 = 0 , 1 9 𝑥 9 𝑦 + 1 1 0 = 0
  • D 9 𝑥 7 1 𝑦 + 2 1 4 = 0 , 6 9 𝑥 + 1 9 𝑦 + 5 1 4 = 0

Q7:

Determine, to the nearest second, the measure of the acute angle between two straight lines having gradients of 5 and .

  • A
  • B
  • C
  • D

Q8:

Determine, to the nearest second, the measure of the acute angle between two straight lines having gradients of 7 and .

  • A
  • B
  • C
  • D

Q9:

Determine, to the nearest second, the measure of the acute angle between two straight lines having gradients of and .

  • A
  • B
  • C
  • D

Q10:

Find the size of the acute angle between the two straight lines whose equations are and to the nearest second.

  • A
  • B
  • C
  • D

Q11:

Find the size of the acute angle between the two straight lines whose equations are and to the nearest second.

  • A
  • B
  • C
  • D

Q12:

Determine the size of the acute angle between the two straight lines and to the nearest second.

  • A
  • B
  • C
  • D

Q13:

Determine the size of the acute angle between the two straight lines and to the nearest second.

  • A
  • B
  • C
  • D

Q14:

Let be the angle between two lines that pass through . If and the gradients of the lines are and , with , find the equations of these lines.

  • A
  • B
  • C

Q15:

Find, to the nearest second, the measure of the acute angle included between the straight line and the straight line whose gradient is .

  • A
  • B
  • C
  • D

Q16:

Let be the line on points and , and the perpendicular to that passes through the origin . What is the size of the positive angle that makes with the positive -axis? Give your answer to the nearest second.

  • A
  • B
  • C
  • D

Q17:

Determine, to the nearest second, the size of the positive angle that the straight line , makes with the positive -axis.

  • A
  • B
  • C
  • D

Q18:

Determine, to the nearest second, the size of the positive angle made with the positive -axis by the perpendicular straight line to the straight line .

  • A
  • B
  • C
  • D

Q19:

Find the size of the acute angle between and approximated to the nearest second.

  • A
  • B
  • C
  • D

Q20:

Find, to the nearest second, the size of the angle between the line and the positive -axis.

  • A
  • B
  • C
  • D

Q21:

Find the size of the acute angle that lies between the straight line whose direction vector is , and the straight line whose equation is in degrees, minutes, and the nearest second.

  • A
  • B
  • C
  • D

Q22:

Find the size of the acute angle between and the -axis approximated to the nearest second.

  • A
  • B
  • C
  • D

Q23:

If 𝜃 is the measure of the acute angle between the two straight lines whose equations are 𝑎 𝑥 3 𝑦 8 = 0 and 𝑥 + 3 𝑦 + 1 0 = 0 and t a n 𝜃 = 1 , find all the possible values of 𝑎 .

  • A 6 , 1
  • B 6 , 3 2
  • C 1 2 , 2
  • D 3 2 , 6

Q24:

If points , , and form a right triangle at , find the value of , and then determine the measures of the other two angles to the nearest second.

  • A , ,
  • B , ,
  • C , ,
  • D , ,

Q25:

Find the size of the acute angle between the following pair of straight lines: and .

  • A
  • B
  • C
  • D