Worksheet: Parallel Lines and Transversals: Proportional Parts

In this worksheet, we will practice using parallelism of lines to find a missing length of a line segment in a transversal line cut by parallel lines.

Q1:

Using the information in the figure, determine the length of 𝐸𝐹.

Q2:

Given that 𝐴𝐢=7.5cm, 𝐡𝐷=14cm, πΉπ‘Œ=25.2cm, and 𝐹𝐾=42cm, determine the lengths of 𝐢𝑋 and 𝐷𝐹.

  • A36 cm, 22.5 cm
  • B36 cm, 42 cm
  • C67.2 cm, 22.5 cm
  • D37.5 cm, 42 cm

Q3:

If 𝐢𝐸=(π‘₯+2)cm, what is π‘₯?

Q4:

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

  • Aπ‘₯=3, 𝑦=6
  • Bπ‘₯=6, 𝑦=3.6
  • Cπ‘₯=12, 𝑦=18
  • Dπ‘₯=6, 𝑦=9
  • Eπ‘₯=2, 𝑦=9

Q5:

Given that 𝐴𝐷=π‘₯cm, 𝐷𝐡=30cm, 𝐡𝐸=(π‘₯+7)cm, and 𝐸𝐢=18cm, find the value of π‘₯.

Q6:

In the figure, lines 𝐿,𝐿,𝐿 , and 𝐿οŠͺ are all parallel. Given that 𝑋𝑍=12, 𝑍𝑁=8, 𝐴𝐡=10, and 𝐡𝐢=5, what is the length of 𝐢𝐷?

Q7:

Find the lengths of 𝐸𝐢 and 𝐷𝐡.

  • A𝐸𝐢=21cm, 𝐷𝐡=16cm
  • B𝐸𝐢=28cm, 𝐷𝐡=21cm
  • C𝐸𝐢=14cm, 𝐷𝐡=24cm
  • D𝐸𝐢=16cm, 𝐷𝐡=24cm

Q8:

Given that 𝐴𝐡=24cm, 𝐴𝐷=36cm, 𝐴𝐢=18cm, and πΈπ‘Œ=15cm, find the length of 𝐴𝐸 and 𝐷𝑋.

  • A𝐴𝐸=21cm, 𝐷𝑋=16cm
  • B𝐴𝐸=48cm, 𝐷𝑋=20cm
  • C𝐴𝐸=27cm, 𝐷𝑋=20cm
  • D𝐴𝐸=18cm, 𝐷𝑋=16cm

Q9:

Given the following figure, find the numerical values of π‘₯ and 𝑦.

  • Aπ‘₯=8, 𝑦=67
  • Bπ‘₯=73, 𝑦=67
  • Cπ‘₯=73, 𝑦=13
  • Dπ‘₯=8, 𝑦=13

Q10:

In the figure, 𝐴𝐡=3π‘₯, 𝐡𝐢=5π‘₯, 𝐷𝐸=(3π‘₯βˆ’6), and 𝐸𝐹=(4π‘₯βˆ’3). Find the value of π‘₯.

Q11:

In the following figure, 𝐴𝐡=(5π‘₯+4)cm, 𝐡𝐢=(6π‘₯βˆ’4)cm, 𝐷𝐸=(4π‘₯βˆ’3)cm, and 𝐸𝐹=(4π‘¦βˆ’7)cm. Find the values of π‘₯ and 𝑦.

  • Aπ‘₯=8, 𝑦=5
  • Bπ‘₯=0, 𝑦=5
  • Cπ‘₯=8, 𝑦=9
  • Dπ‘₯=0, 𝑦=8

Q12:

Given that 𝐴𝐡=(2π‘₯+4) cm, 𝐷𝐺=(3π‘₯+2) cm, 𝐴𝐢=(2𝑦+2) cm, 𝐴𝐺=4cm, and 𝐴𝐹=5cm, find the lengths of 𝐴𝐸 and 𝐷𝐺.

  • A𝐴𝐸=2cm, 𝐷𝐺=4cm
  • B𝐴𝐸=15cm, 𝐷𝐺=8cm
  • C𝐴𝐸=10cm, 𝐷𝐺=4cm
  • D𝐴𝐸=9cm, 𝐷𝐺=8cm

Q13:

Find the values of π‘₯ and 𝑦.

  • Aπ‘₯=5, 𝑦=25
  • Bπ‘₯=6, 𝑦=21
  • Cπ‘₯=6, 𝑦=29
  • Dπ‘₯=5, 𝑦=17

Q14:

Given that 𝐴𝐡=(2π‘₯+1)cm, 𝐡𝐢=(π‘₯+3)cm, 𝐷𝐸=(3π‘₯βˆ’1)cm, 𝐸𝐹=𝑦cm, 𝐺𝐻=11cm, and 𝐻𝐼=10cm, find the values of π‘₯ and 𝑦.

  • Aπ‘₯=7.27, 𝑦=5.09
  • Bπ‘₯=5.05, 𝑦=5.51
  • Cπ‘₯=2.56, 𝑦=6.06
  • Dπ‘₯=6.11, 𝑦=6.06

Q15:

Given that 𝑀𝐢=15cm, 𝑀𝐷=10cm, and 𝐴𝑀=𝐷𝐹, determine the values of π‘₯ and 𝑦.

  • Aπ‘₯=8, 𝑦=3
  • Bπ‘₯=4, 𝑦=4
  • Cπ‘₯=6, 𝑦=28
  • Dπ‘₯=8, 𝑦=28

Q16:

In the diagram below, 𝐴𝐡=10, 𝐡𝐢=(π‘₯+1), 𝐢𝐷=20, 𝐸𝐹=10, and 𝐹𝐺=10. Find the value of π‘₯ and the length of 𝐺𝐻.

  • Aπ‘₯=10, 𝐺𝐻=10
  • Bπ‘₯=10, 𝐺𝐻=20
  • Cπ‘₯=9, 𝐺𝐻=20
  • Dπ‘₯=9, 𝐺𝐻=5

Q17:

In the figure, given that 𝐡𝐢=94, what is 𝐡𝐸?

Q18:

Suppose π‘‹π‘Œ and 𝑍𝐿 intersect at point 𝑀, and that 𝑋𝑍 and πΏπ‘Œ are parallel. If 𝑋𝑀=46, π‘Œπ‘€=64, and 𝑍𝐿=165, what is 𝑍𝑀?

Q19:

Suppose that, in the figure, 𝐴𝐢=30 and π‘šβˆ πΈπ·π΅=139∘. What is the length of 𝐴𝐸?

Q20:

Given that 𝑋𝐿=9cm, find the length of 𝑋𝑍.

Q21:

Given that 𝐸𝐷βˆ₯𝐢𝐡, find the value of π‘₯.

Q22:

Draw △𝐴𝐡𝐢 with 𝐴𝐡=8, π‘šβˆ π΄=39∘, and π‘šβˆ π΅=68∘. Bisect 𝐴𝐢 at 𝐷, and draw ⃖⃗𝐷𝐸 parallel to 𝐴𝐡 and meeting 𝐡𝐢 at 𝐸. Find the lengths 𝐡𝐸, 𝐢𝐸, and 𝐷𝐸 rounded to one decimal place.

  • Aπ΅πΈβ‰ˆ8.0, πΆπΈβ‰ˆ8.0, π·πΈβ‰ˆ2.6
  • Bπ΅πΈβ‰ˆ4.0, πΆπΈβ‰ˆ4.0, π·πΈβ‰ˆ2.6
  • Cπ΅πΈβ‰ˆ5.2, πΆπΈβ‰ˆ5.2, π·πΈβ‰ˆ4.0
  • Dπ΅πΈβ‰ˆ2.6, πΆπΈβ‰ˆ2.6, π·πΈβ‰ˆ4.0

Q23:

Given that π΄π‘Œ=16cm, π·π‘Œ=21cm, and 𝐴𝐡=88.53cm, find the perimeter of β–³π΄π·π‘Œ. Round your answer to two decimal places.

Q24:

𝑋,π‘Œ, and 𝑍 are three parallel planes intersected by the two coplanar straight lines 𝐿 and 𝐿. If 𝐴𝐡𝐡𝐢=23, 𝐴𝐷=15cm, and 𝐢𝐹=13cm, find the length of 𝐡𝐸.

Q25:

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

  • Aπ‘₯=4, 𝑦=1.8
  • Bπ‘₯=8, 𝑦=1.5
  • Cπ‘₯=16, 𝑦=9
  • Dπ‘₯=8, 𝑦=9

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