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Lesson: Angle between Two Straight Lines in Two Dimensions

Sample Question Videos

Worksheet • 25 Questions • 1 Video

Q1:

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

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

Q2:

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

Q3:

If the acute angle between the straight lines whose equations are π‘˜ 𝑦 βˆ’ 2 π‘₯ + 1 9 = 0 and 9 π‘₯ βˆ’ 7 𝑦 βˆ’ 8 = 0 is πœ‹ 4 , find all the possible values of π‘˜ .

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

Q4:

Given that 𝐴 𝐡 𝐢 is a right-angled triangle at 𝐴 , the equation of βƒ–     βƒ— 𝐡 𝐢 is r = ⟨ 1 , 4 ⟩ + 𝐾 ⟨ βˆ’ 6 , βˆ’ 4 ⟩ , and the equation of βƒ–     βƒ— 𝐴 𝐡 is r = ⟨ 4 , βˆ’ 9 ⟩ + 𝐾 ⟨ 1 , βˆ’ 8 ⟩ , find π‘š ∠ 𝐴 𝐢 𝐡 approximated to the nearest minute.

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

Q5:

Determine, to the nearest second, the measure of the positive angle that the straight line 1 4 π‘₯ + 1 2 𝑦 = βˆ’ 2 7 makes with the positive π‘₯ -axis.

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

Q6:

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

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

Q7:

Determine, to the nearest second, the measure of the acute angle between two straight lines having slopes of 5 and 1 4 .

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

Q8:

Determine, to the nearest second, the measure of the acute angle between two straight lines having slopes of 7 and βˆ’ 8 7 .

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

Q9:

Determine, to the nearest second, the measure of the acute angle between two straight lines having slopes of βˆ’ 1 3 and βˆ’ 7 .

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

Q10:

Find the measure of the acute angle between the two straight lines whose equations are 1 1 π‘₯ + 1 0 𝑦 βˆ’ 2 8 = 0 and 2 π‘₯ + 𝑦 + 1 5 = 0 to the nearest second.

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

Q11:

Find the measure of the acute angle between the two straight lines whose equations are π‘₯ βˆ’ 2 𝑦 + 2 = 0 and 7 π‘₯ βˆ’ 1 0 𝑦 + 4 5 = 0 to the nearest second.

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

Q12:

Determine the measure of the acute angle between the two straight lines 𝐿 ∢ ⃑ π‘Ÿ = ( βˆ’ 4 , βˆ’ 3 ) + 𝐾 ( 4 , βˆ’ 9 ) 1 and 𝐿 ∢ 7 π‘₯ βˆ’ 3 𝑦 + 1 7 = 0 2 to the nearest second.

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

Q13:

Determine the measure of the acute angle between the two straight lines 𝐿 ∢ ⃑ π‘Ÿ = ( βˆ’ 7 , 0 ) + 𝐾 ( 1 , βˆ’ 4 ) 1 and 𝐿 ∢ 1 6 π‘₯ + 7 𝑦 βˆ’ 2 4 = 0 2 to the nearest second.

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

Q14:

Let πœƒ be the angle between two lines that pass through ( 4 , βˆ’ 2 ) . If t a n πœƒ = 1 2 1 and the slopes of the lines are π‘š and 4 5 π‘š , with π‘š > 0 , find the equations of these lines.

  • A ( 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
  • B ( 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
  • C ( 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

Q15:

Find, to the nearest second, the measure of the acute angle included between the straight line 6 π‘₯ βˆ’ 7 𝑦 + 4 0 = 0 and the straight line whose slope is 7 3 .

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

Q16:

Let 𝑀 be the line on points ( 0 , βˆ’ 8 ) and ( βˆ’ 4 , 1 0 ) , 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 2 3 1 β€² 4 4 β€² β€² ∘
  • B 1 0 2 3 1 β€² 4 4 β€² β€² ∘
  • C 7 7 2 8 β€² 1 6 β€² β€² ∘
  • D 1 6 7 2 8 β€² 1 6 β€² β€² ∘

Q17:

Determine, to the nearest second, the measure of the positive angle that the straight line 𝐿 ∢ π‘₯ = 3 + 1 0 𝑠 , 𝑦 = βˆ’ 1 5 βˆ’ 2 1 𝑠 makes with the positive π‘₯ -axis.

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

Q18:

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 ⃑ π‘Ÿ = ( 8 , βˆ’ 9 ) + 𝐾 ( 1 0 , 7 ) .

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

Q19:

Find the measure of the acute angle between βƒ–     βƒ— 𝐴 𝐢 and βƒ–     βƒ— 𝐡 𝐢 approximated to the nearest second.

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

Q20:

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

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

Q21:

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’ 2 π‘₯ βˆ’ 5 𝑦 + 1 = 0 in degrees, minutes, and the nearest second.

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

Q22:

Find the measure of the acute angle between βƒ–     βƒ— 𝐴 𝐡 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 ∘

Q23:

If πœƒ is the measure of the acute angle between the two straight lines whose equations are π‘Ž π‘₯ βˆ’ 3 𝑦 βˆ’ 8 = 0 and βˆ’ π‘₯ + 3 𝑦 + 1 0 = 0 and t a n πœƒ = 1 , find all the possible values of π‘Ž .

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

Q24:

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 = 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 ∘
  • B 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 ∘
  • 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 = 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 ∘

Q25:

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

  • A
  • B
  • C
  • D
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