Worksheet: Angle between Two Straight Lines in the Coordinate Plane

In this worksheet, we will practice finding the measure of an acute angle between two straight lines in the coordinate plane.

Q1:

Determine the measure of the acute angle between the straight line π‘₯βˆ’π‘¦+4=0 and the straight line passing through the points (3,βˆ’2) and (βˆ’2,4) to the nearest second.

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

Q2:

A line passing through (8,2) makes an angle πœƒ with the line 6π‘₯+4𝑦+9=0, and tanπœƒ=1513. What is the equation of this line?

  • A 7 1 π‘₯ βˆ’ 6 9 𝑦 + 4 1 0 = 0 , βˆ’ 1 9 π‘₯ βˆ’ 9 𝑦 + 1 1 0 = 0
  • B βˆ’ 7 1 π‘₯ βˆ’ 9 𝑦 + 2 1 4 = 0 , βˆ’ 1 9 π‘₯ + 6 9 𝑦 + 5 1 4 = 0
  • C βˆ’ 6 9 π‘₯ + 7 1 𝑦 + 4 1 0 = 0 , βˆ’ 9 π‘₯ βˆ’ 1 9 𝑦 + 1 1 0 = 0
  • D βˆ’ 9 π‘₯ βˆ’ 7 1 𝑦 + 2 1 4 = 0 , βˆ’ 6 9 π‘₯ + 1 9 𝑦 + 5 1 4 = 0

Q3:

Determine, to the nearest second, the measure of the acute angle between two straight lines having slopes of 5 and 14.

  • A 7 5 1 5 β€² 2 3 β€² β€² ∘
  • B 6 6 4 8 β€² 5 β€² β€² ∘
  • C 6 4 3 9 β€² 1 4 β€² β€² ∘
  • D 7 6 3 6 β€² 2 7 β€² β€² ∘

Q4:

Find, to the nearest second, the measure of the acute angle included between the straight line 6π‘₯βˆ’7𝑦+40=0 and the straight line whose slope is 73.

  • A 4 6 4 5 4 5 ∘   
  • B 5 7 5 5 4 ∘   
  • C 2 6 1 2 0 ∘   
  • D 3 6 2 5 5 1 ∘   

Q5:

Find the measure of the acute angle between the two straight lines whose equations are 11π‘₯+10π‘¦βˆ’28=0 and 2π‘₯+𝑦+15=0 to the nearest second.

  • A 4 4 5 β€² 2 6 β€² β€² ∘
  • B 1 5 4 2 β€² 3 1 β€² β€² ∘
  • C 5 4 3 8 β€² 1 5 β€² β€² ∘
  • D 2 2 1 4 β€² 5 6 β€² β€² ∘

Q6:

Determine the measure of the acute angle between the two straight lines 𝐿∢=βŸ¨βˆ’4,βˆ’3⟩+𝐾⟨4,βˆ’9⟩r and 𝐿∢7π‘₯βˆ’3𝑦+17=0 to the nearest second.

  • A 4 1 7 β€² 1 7 β€² β€² ∘
  • B 4 7 9 β€² 4 0 β€² β€² ∘
  • C 1 7 β€² 2 4 β€² β€² ∘
  • D 0 5 4 β€² 3 4 β€² β€² ∘

Q7:

Find the measure of the acute angle that lies between the straight line whose direction vector is r=⟨1,βˆ’3⟩, and the straight line whose equation is βˆ’2π‘₯βˆ’5𝑦+1=0 in degrees, minutes, and the nearest second.

  • A 4 9 4 5 β€² 4 9 β€² β€² ∘
  • B 8 6 3 8 β€² 1 β€² β€² ∘
  • C 2 6 3 3 β€² 5 4 β€² β€² ∘
  • D 4 0 1 4 β€² 1 1 β€² β€² ∘

Q8:

Find the measure of the acute angle between the following pair of straight lines: (βˆ’9,βˆ’3)+𝑠(βˆ’1,βˆ’6) and (7,βˆ’7)+𝑠(4,βˆ’12).

  • A 2 7 5 3 β€² 5 0 . 2 β€² β€² ∘
  • B 9 2 7 β€² 4 4 . 4 β€² β€² ∘
  • C 2 6 3 3 β€² 5 4 . 2 β€² β€² ∘
  • D 1 0 0 β€² 2 8 . 7 β€² β€² ∘

Q9:

If the acute angle between the straight lines whose equations are π‘˜π‘¦βˆ’2π‘₯+19=0 and 9π‘₯βˆ’7π‘¦βˆ’8=0 is πœ‹4, find all the possible values of π‘˜.

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

Q10:

Let πœƒ be the angle between two lines that pass through (4,βˆ’2). If tanπœƒ=121 and the slopes of the lines are π‘š and 45π‘š, with π‘š>0, find the equations of these lines.

  • A ( π‘₯ βˆ’ 4 𝑦 βˆ’ 1 2 = 0 π‘₯ βˆ’ 5 𝑦 βˆ’ 1 4 = 0 ) ( βˆ’ 5 π‘₯ + 𝑦 + 2 2 = 0 4 π‘₯ βˆ’ 𝑦 βˆ’ 1 8 = 0 ) a n d o r a n d
  • B ( π‘₯ βˆ’ 5 𝑦 βˆ’ 1 4 = 0 4 π‘₯ βˆ’ 𝑦 βˆ’ 1 8 = 0 ) ( π‘₯ βˆ’ 4 𝑦 βˆ’ 1 2 = 0 βˆ’ 5 π‘₯ + 𝑦 + 2 2 = 0 ) a n d o r a n d
  • C ( βˆ’ 5 π‘₯ + 𝑦 + 2 2 = 0 π‘₯ βˆ’ 5 𝑦 βˆ’ 1 4 = 0 ) ( 4 π‘₯ βˆ’ 𝑦 βˆ’ 1 8 = 0 π‘₯ βˆ’ 4 𝑦 βˆ’ 1 2 = 0 ) a n d o r a n d

Q11:

If πœƒ is the measure of the acute angle between the two straight lines whose equations are π‘Žπ‘₯βˆ’3π‘¦βˆ’8=0 and βˆ’π‘₯+3𝑦+10=0 and tanπœƒ=1, find all the possible values of π‘Ž.

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

Q12:

Determine the measure of the positive angle that the straight line 𝐿 makes with the positive direction of the π‘₯-axis approximated to the nearest second, given that 𝐿 passes through points 𝐴(βˆ’1,βˆ’4) and 𝐡(3,βˆ’5).

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

Q13:

Given that 𝐴𝐡𝐢 is a right-angled triangle at 𝐴, the equation of ⃖⃗𝐡𝐢 is r=⟨1,4⟩+πΎβŸ¨βˆ’6,βˆ’4⟩, and the equation of ⃖⃗𝐴𝐡 is r=⟨4,βˆ’9⟩+𝐾⟨1,βˆ’8⟩, find π‘šβˆ π΄πΆπ΅ approximated to the nearest minute.

  • A 5 9 2 5 β€² ∘
  • B 3 0 3 5 β€² ∘
  • C 2 6 3 4 β€² ∘
  • D 6 3 2 6 β€² ∘

Q14:

Determine, to the nearest second, the measure of the positive angle that the straight line 14π‘₯+12𝑦=βˆ’27 makes with the positive π‘₯-axis.

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

Q15:

Let 𝑀 be the line on points (0,βˆ’8) and (βˆ’4,10), and 𝐿 the perpendicular to 𝑀 that passes through the origin (0,0). What is the measure of the positive angle that 𝐿 makes with the positive π‘₯-axis? Give your answer to the nearest second.

  • A 1 0 2 3 1 β€² 4 4 β€² β€² ∘
  • B 1 6 7 2 8 β€² 1 6 β€² β€² ∘
  • C 1 2 3 1 β€² 4 4 β€² β€² ∘
  • D 7 7 2 8 β€² 1 6 β€² β€² ∘

Q16:

Determine, to the nearest second, the measure of the positive angle that the straight line 𝐿∢π‘₯=3+10𝑠, 𝑦=βˆ’15βˆ’21𝑠 makes with the positive π‘₯-axis.

  • A 6 4 3 2 β€² 1 2 β€² β€² ∘
  • B 2 5 2 7 β€² 4 8 β€² β€² ∘
  • C 1 5 4 3 2 β€² 1 2 β€² β€² ∘
  • D 1 1 5 2 7 β€² 4 8 β€² β€² ∘

Q17:

Determine, to the nearest second, the measure of the positive angle made with the positive π‘₯-axis by the perpendicular straight line to the straight line r=⟨8,βˆ’9⟩+𝐾⟨10,7⟩.

  • A 3 4 5 9 β€² 3 1 β€² β€² ∘
  • B 1 2 4 5 9 β€² 3 1 β€² β€² ∘
  • C 5 5 0 β€² 2 9 β€² β€² ∘
  • D 1 4 5 0 β€² 2 9 β€² β€² ∘

Q18:

Find the measure of the acute angle between ⃖⃗𝐴𝐢 and ⃖⃗𝐡𝐢 approximated to the nearest second.

  • A 5 9 2 β€² 1 0 β€² β€² ∘
  • B 4 7 1 3 β€² 5 2 β€² β€² ∘
  • C 7 1 5 9 β€² 4 5 β€² β€² ∘
  • D 2 8 2 3 β€² 3 5 β€² β€² ∘

Q19:

Find, to the nearest second, the measure of the angle between the line βˆ’3π‘₯+4π‘¦βˆ’2=0 and the positive π‘₯-axis.

  • A 1 4 3 7 β€² 4 8 β€² β€² ∘
  • B 5 3 7 β€² 4 8 β€² β€² ∘
  • C 1 2 6 5 2 β€² 1 2 β€² β€² ∘
  • D 3 6 5 2 β€² 1 2 β€² β€² ∘

Q20:

Find the measure of the acute angle between ⃖⃗𝐴𝐡 and the π‘₯-axis approximated to the nearest second.

  • A 3 5 3 2 β€² 1 6 β€² β€² ∘
  • B 5 4 2 7 β€² 4 4 β€² β€² ∘
  • C 5 3 7 β€² 4 8 β€² β€² ∘
  • D 5 9 2 β€² 1 0 β€² β€² ∘

Q21:

If points 𝐴(βˆ’6,2), 𝐡(βˆ’2,βˆ’8), and 𝐢(3,𝑦) form a right-angled triangle at 𝐡, find the value of 𝑦, and then determine the measures of the other two angles to the nearest second.

  • A 𝑦 = 4 , π‘š ∠ 𝐴 = 5 0 2 1 β€² 3 1 β€² β€² ∘ , π‘š ∠ 𝐢 = 3 9 3 8 β€² 2 9 β€² β€² ∘
  • B 𝑦 = 6 , π‘š ∠ 𝐴 = 5 4 4 β€² 3 7 β€² β€² ∘ , π‘š ∠ 𝐢 = 3 5 5 5 β€² 2 3 β€² β€² ∘
  • C 𝑦 = βˆ’ 4 , π‘š ∠ 𝐴 = 3 0 4 3 β€² 5 6 β€² β€² ∘ , π‘š ∠ 𝐢 = 5 9 1 6 β€² 4 β€² β€² ∘
  • D 𝑦 = βˆ’ 6 , π‘š ∠ 𝐴 = 2 6 3 3 β€² 5 4 β€² β€² ∘ , π‘š ∠ 𝐢 = 6 3 2 6 β€² 6 β€² β€² ∘

Q22:

Find the measure of the acute angle included between the two straight line 𝐿 and 𝐿 whose equations are r=⟨2,βˆ’7⟩+πΎβŸ¨βˆ’1,8⟩ and π‘₯=3+12𝑑, 𝑦=4π‘‘βˆ’5, respectively, in terms of degrees, minutes, and seconds, to the nearest second.

  • A 7 2 1 5 β€² 1 9 β€² β€² ∘
  • B 7 0 4 9 β€² 1 6 β€² β€² ∘
  • C 7 8 4 1 β€² 2 4 β€² β€² ∘
  • D 7 7 4 4 β€² 7 β€² β€² ∘

Q23:

Determine, to the nearest second, the measure of the positive angle that a straight line makes with the positive π‘₯-axis while passing through the two points (2,7) and (βˆ’5,βˆ’5).

  • A 1 2 0 1 5 β€² 2 3 β€² β€² ∘
  • B 5 9 4 4 β€² 3 7 β€² β€² ∘
  • C 1 4 9 4 4 β€² 3 7 β€² β€² ∘
  • D 3 0 1 5 β€² 2 3 β€² β€² ∘

Q24:

A line meets the π‘₯-axis at 14 and the 𝑦-axis at βˆ’19. Determine, to the nearest second, the measure of the positive angle that the line makes with the positive π‘₯-axis.

  • A 3 6 2 3 β€² 4 β€² β€² ∘
  • B 1 4 3 3 6 β€² 5 6 β€² β€² ∘
  • C 5 3 3 6 β€² 5 6 β€² β€² ∘
  • D 1 2 6 2 3 β€² 4 β€² β€² ∘

Q25:

A line has slope 0.492. What is the measure of the angle that this line makes with the positive π‘₯-axis? Give your answer to the nearest second.

  • A 2 6 1 1 β€² 5 0 β€² β€² ∘
  • B 1 1 6 1 1 β€² 5 0 β€² β€² ∘
  • C 6 3 4 8 β€² 1 0 β€² β€² ∘
  • D 4 4 3 2 β€² 1 7 β€² β€² ∘

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