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Worksheet: Using Angle Relationships to Solve Algebraic Expressions

Q1:

Answer the following questions using the given diagram.

Form an equation that will allow you to calculate π‘₯ .

  • A 1 5 π‘₯ + 5 0 = 9 0 βˆ’ ( 5 π‘₯ + 6 0 )
  • B 1 5 π‘₯ + 5 0 = 1 8 0 βˆ’ ( 5 π‘₯ + 6 0 )
  • C 1 5 π‘₯ + 5 0 = 1 8 0 + 5 π‘₯ + 6 0
  • D 1 5 π‘₯ + 5 0 = 5 π‘₯ + 6 0
  • E 1 5 π‘₯ + 5 0 = 9 0 + 5 π‘₯ + 6 0

Find the value of π‘₯ .

  • A π‘₯ = 1
  • B π‘₯ = 1 9
  • C π‘₯ = 2
  • D π‘₯ = 1 7
  • E π‘₯ = 1 0

Q2:

If 𝐸 𝐢 is an altitude of β–³ 𝐴 𝐸 𝐷 , π‘š ∠ πœƒ = ( 5 π‘₯ + 2 ) ∘ , and π‘š ∠ πœ™ = ( 3 π‘₯ + 8 ) ∘ , find π‘š ∠ πœƒ .

Q3:

Find the values of π‘₯ and 𝑦 so that βƒ–     βƒ— 𝑃 𝑅 and βƒ–     βƒ— 𝑆 𝑄 are perpendicular.

  • A π‘₯ = 3 0 , 𝑦 = 2 9
  • B π‘₯ = 8 , 𝑦 = 2 9
  • C π‘₯ = 5 , 𝑦 = 6 5
  • D π‘₯ = 5 , 𝑦 = 2 9
  • E π‘₯ = 2 5 , 𝑦 = 6 5

Q4:

In the given diagram, 𝐴 𝐷 is a straight line, π‘š ∠ 𝐴 𝐡 𝐢 = ( 4 π‘₯ + 1 2 ) ∘ , and π‘š ∠ 𝐷 𝐡 𝐢 = ( 6 π‘₯ + 8 ) ∘ .

Form an equation that will allow you to calculate π‘₯ .

  • A 4 π‘₯ + 1 2 = 9 0
  • B 1 0 π‘₯ + 2 0 = 9 0
  • C 2 π‘₯ + 2 0 = 1 8 0
  • D 1 0 π‘₯ + 2 0 = 1 8 0
  • E 2 π‘₯ + 2 0 = 9 0

Solve for π‘₯ .

  • A π‘₯ = 1 6
  • B π‘₯ = 8
  • C π‘₯ = 1 9 . 5
  • D π‘₯ = 7
  • E π‘₯ = 3 5

Q5:

Answer the following questions using the given diagram.

Form an equation that will allow you to calculate π‘₯ .

  • A 3 π‘₯ + 4 6 = 9 0
  • B 3 π‘₯ + 4 4 = 9 0
  • C 3 π‘₯ + 2 = 4 6
  • D 3 π‘₯ + 4 8 = 9 0
  • E 3 π‘₯ + 4 8 = 4 6

Find the value of π‘₯ .

  • A π‘₯ = 1 4
  • B π‘₯ = 1 6
  • C π‘₯ = 2
  • D π‘₯ = 4 2
  • E π‘₯ = 3 0

Q6:

In the given diagram, ∠ 𝐴 𝐡 𝐢 is a right angle and π‘š ∠ 𝐴 𝐡 𝐷 is twice π‘š ∠ 𝐷 𝐡 𝐢 . Let π‘š ∠ 𝐷 𝐡 𝐢 = π‘₯ .

Form an equation that will allow you to calculate π‘₯ .

  • A 4 π‘₯ = 9 0 ∘
  • B 2 π‘₯ = 9 0 ∘
  • C 2 π‘₯ = 1 8 0 ∘
  • D 3 π‘₯ = 9 0 ∘
  • E 3 π‘₯ = 1 8 0 ∘

Solve for π‘₯ .

  • A π‘₯ = 3 0 ∘
  • B π‘₯ = 6 0 ∘
  • C π‘₯ = 9 0 ∘
  • D π‘₯ = 4 5 ∘
  • E π‘₯ = 2 2 . 5 ∘

Q7:

In the given diagram, 𝐴 𝐡 and 𝐢 𝐷 are straight lines. Answer the following questions.

Form an equation that will allow you to calculate π‘₯ .

  • A 5 π‘₯ + 2 = 9 0
  • B 1 1 π‘₯ + 2 = 1 8 0
  • C 6 π‘₯ = 9 0
  • D 1 1 π‘₯ + 2 = 9 0
  • E 6 π‘₯ = 1 8 0

Find the value of π‘₯ .

  • A π‘₯ = 8
  • B π‘₯ = 1 5
  • C π‘₯ = 1 8
  • D π‘₯ = 1 6
  • E π‘₯ = 3 0

Find the value of 𝑦 .

  • A 𝑦 = 1 3 8
  • B 𝑦 = 4 2
  • C 𝑦 = 4 8
  • D 𝑦 = 3 0
  • E 𝑦 = 1 3 2

Find the value of 𝑧 .

  • A 𝑧 = 1 3 2
  • B 𝑧 = 1 3 8
  • C 𝑧 = 1 5 0
  • D 𝑧 = 4 8
  • E 𝑧 = 3 0

Q8:

Quadrilateral 𝐽 𝐾 𝐿 𝑀 is a rectangle. If π‘š ∠ 𝐾 𝐽 𝐿 = ( 4 π‘₯ + 7 ) ∘ and π‘š ∠ 𝐽 𝐿 𝐾 = ( 9 π‘₯ + 5 ) ∘ , find π‘₯ .

Q9:

In the given diagram, 𝐴 𝐡 is a straight line. Answer the following questions.

Form an equation that will allow you to calculate π‘₯ .

  • A 3 π‘₯ = 3 0
  • B 3 π‘₯ + 3 0 = 1 8 0
  • C 3 π‘₯ + 3 0 = 2 7 0
  • D 3 π‘₯ + 3 0 = 9 0
  • E 3 π‘₯ + 3 0 = 3 6 0

Find the value of π‘₯ .

  • A π‘₯ = 2 0
  • B π‘₯ = 8 0
  • C π‘₯ = 1 0
  • D π‘₯ = 5 0
  • E π‘₯ = 1 1 0

Q10:

In this figure, ∠ 𝑑 and ∠ 𝑒 form a linear pair. If π‘š ∠ π‘Ž = ( 2 π‘₯ ) ∘ , π‘š ∠ 𝑏 = ( 3 π‘₯ βˆ’ 2 9 ) ∘ , and π‘š ∠ 𝑐 = ( π‘₯ βˆ’ 1 ) ∘ , determine π‘š ∠ 𝑐 .

Q11:

Answer the following questions using the given diagram.

Form an equation that will allow you to calculate π‘₯ .

  • A 7 π‘₯ = 5 6
  • B 7 π‘₯ = 6 8
  • C 7 π‘₯ = 1 5 8
  • D 7 π‘₯ = 1 1 2
  • E 7 π‘₯ = 2 2

Find the value of π‘₯ .

  • A π‘₯ = 1 6
  • B π‘₯ = 3 5
  • C π‘₯ = 8
  • D π‘₯ = 1 0
  • E π‘₯ = 3

Q12:

The given figure shows two intersecting lines.

Find an expression for 𝑏 in terms of π‘Ž .

  • A 𝑏 = 9 0 + π‘Ž
  • B 𝑏 = 9 0 βˆ’ π‘Ž
  • C 𝑏 = 9 0 + 2 π‘Ž
  • D 𝑏 = 1 8 0 βˆ’ π‘Ž
  • E 𝑏 = 1 8 0 + 2 π‘Ž

Find an expression for 𝑑 in terms of π‘Ž .

  • A 𝑑 = 1 8 0 βˆ’ π‘Ž
  • B 𝑑 = 9 0 + 2 π‘Ž
  • C 𝑑 = 9 0 + π‘Ž
  • D 𝑑 = 9 0 βˆ’ π‘Ž
  • E 𝑑 = 1 8 0 + 2 π‘Ž

Hence, what is true of 𝑏 and 𝑑 ?

  • AThey are supplementary.
  • BThey are complementary.
  • CThey are equal.

We have worked out that 𝑏 = 1 8 0 βˆ’ π‘Ž . Use this to find an expression for 𝑐 in terms of π‘Ž .

  • A 𝑐 = π‘Ž
  • B π‘Ž βˆ’ 𝑐 = 9 0
  • C 𝑐 βˆ’ π‘Ž = + 2 7 0
  • D 𝑐 + π‘Ž = + 1 8 0
  • E 𝑐 + π‘Ž = βˆ’ 1 8 0

Q13:

In the given diagram, π‘š ∠ 𝐴 𝑋 𝐡 = ( 1 2 π‘₯ βˆ’ 4 ) ∘ and π‘š ∠ 𝐢 𝑋 𝐷 = ( 8 π‘₯ + 1 2 ) ∘ . Find the value of π‘₯ .

  • A π‘₯ = 2
  • B π‘₯ = 8
  • C π‘₯ = 1 6
  • D π‘₯ = 4
  • E π‘₯ = 7

Q14:

Find .

  • A
  • B
  • C
  • D

Q15:

In the given diagram, 𝐴 𝐡 and 𝐢 𝐷 are straight lines. Answer the following questions.

Form an equation that will allow you to calculate π‘₯ .

  • A 1 0 π‘₯ + 2 0 = 6 0
  • B 1 0 π‘₯ + 2 0 = 1 8 0
  • C 1 0 π‘₯ + 1 0 = 9 0
  • D 1 0 π‘₯ + 2 0 = 9 0
  • E 1 0 π‘₯ + 1 0 = 1 8 0

Find the value of π‘₯ .

  • A π‘₯ = 7
  • B π‘₯ = 8
  • C π‘₯ = 4
  • D π‘₯ = 1 6
  • E π‘₯ = 1 7

Find the value of 𝑦 .

  • A 𝑦 = 4 5
  • B 𝑦 = 6 0
  • C 𝑦 = 3 0
  • D 𝑦 = 3 8
  • E 𝑦 = 5 2

Find the value of 𝑧 .

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

Q16:

In the given figure, οƒͺ 𝐾 𝐽 and οƒͺ 𝐾 𝐿 are opposite rays, and  𝐾 𝑁 bisects ∠ 𝐿 𝐾 𝑀 . If π‘š ∠ 𝑁 𝐾 𝐿 = ( 2 π‘₯ βˆ’ 8 ) ∘ and π‘š ∠ 𝐽 𝐾 𝑀 = ( 2 π‘₯ + 7 ) ∘ , find π‘š ∠ 𝐽 𝐾 𝑁 .

Q17:

If π‘š ∠ 𝐽 𝑁 𝑃 = ( 4 π‘₯ + 2 ) ∘ , π‘š ∠ 𝐾 𝑁 𝐿 = ( 7 π‘₯ βˆ’ 2 8 ) ∘ , and π‘š ∠ 𝐾 𝑁 𝐽 = ( 6 π‘₯ + 1 9 ) ∘ , find the measure of each angle.

  • A π‘š ∠ 𝐽 𝑁 𝑃 = 4 6 ∘ , π‘š ∠ 𝐾 𝑁 𝐿 = 4 4 ∘ , π‘š ∠ 𝐾 𝑁 𝐽 = 9 0 ∘
  • B π‘š ∠ 𝐽 𝑁 𝑃 = 7 6 ∘ , π‘š ∠ 𝐾 𝑁 𝐿 = 1 0 3 ∘ , π‘š ∠ 𝐾 𝑁 𝐽 = 1 3 1 ∘
  • C π‘š ∠ 𝐽 𝑁 𝑃 = 4 9 ∘ , π‘š ∠ 𝐾 𝑁 𝐿 = 5 4 ∘ , π‘š ∠ 𝐾 𝑁 𝐽 = 9 0 ∘
  • D π‘š ∠ 𝐽 𝑁 𝑃 = 4 6 ∘ , π‘š ∠ 𝐾 𝑁 𝐿 = 4 9 ∘ , π‘š ∠ 𝐾 𝑁 𝐽 = 8 5 ∘
  • E π‘š ∠ 𝐽 𝑁 𝑃 = 4 2 ∘ , π‘š ∠ 𝐾 𝑁 𝐿 = 1 0 5 ∘ , π‘š ∠ 𝐾 𝑁 𝐽 = 4 7 ∘

Q18:

In the given figure, find the values of π‘₯ and 𝑦 .

  • A π‘₯ = 1 4 , 𝑦 = 3 8
  • B π‘₯ = 1 5 , 𝑦 = 3 7
  • C π‘₯ = 8 9 , 𝑦 = 9 1
  • D π‘₯ = 1 5 , 𝑦 = 3 8
  • E π‘₯ = 8 9 , 𝑦 = 1

Q19:

From the intersecting lines in the figure, determine the value of π‘₯ .

Q20:

In the figure below, οƒͺ 𝐾 𝐽 and  𝐾 𝑀 are opposite rays, and  𝐾 𝑁 bisects ∠ 𝐽 𝐾 𝐿 . If π‘š ∠ 𝐽 𝐾 𝑁 = ( 5 π‘₯ βˆ’ 7 ) ∘ and π‘š ∠ 𝑁 𝐾 𝐿 = ( 4 π‘₯ + 4 ) ∘ , find π‘š ∠ 𝐽 𝐾 𝑁 .