Worksheet: Inequalities in Two Triangles

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In this worksheet, we will practice using the hinge theorem and its converse in triangles to find the possible range of a side length or angle in two triangles.

Q1:

Consider the following two triangles.

Length π‘Ž is congruent to 𝑑 and length 𝑏 is congruent to 𝑒. Given that πœ™ is greater than πœƒ, what does the hinge theorem tell us about the lengths 𝑐 and 𝑓?

  • A 𝑓 and 𝑐 are equal.
  • BThere is no relation between 𝑓 and 𝑐.
  • C 𝑓 is less than 𝑐.
  • D 𝑓 is greater than 𝑐.

Q2:

Given that π‘šβˆ π΄πΆπ·=(6π‘₯βˆ’12)∘, use the hinge theorem to find the range of all possible values of π‘₯ in the figure.

  • A 2 ≀ π‘₯ ≀ 1 1
  • B 2 < π‘₯ < 7
  • C βˆ’ 2 < π‘₯ < 7
  • D βˆ’ 2 ≀ π‘₯ ≀ 7
  • E 2 < π‘₯ < 1 1

Q3:

Consider triangles 𝐴𝐡𝐢 and 𝐷𝐸𝐹 in the figure.

Without completing any calculations, use the hinge theorem to determine whether 𝐷𝐹 is greater than, less than, or equal to 𝐴𝐢.

  • A 𝐷 𝐹 is greater than 𝐴𝐢.
  • B 𝐷 𝐹 is less than 𝐴𝐢.
  • C 𝐷 𝐹 and 𝐴𝐢 are equal.

Q4:

Given that 𝐢𝐸=5π‘₯βˆ’11 and 𝐢𝐡=9, find the range of all possible values of π‘₯ using the Hinge theorem.

  • A βˆ’ 1 1 5 < π‘₯ < 4
  • B 1 1 5 < π‘₯ < 4
  • C 1 1 5 ≀ π‘₯ ≀ 4
  • D 1 1 5 ≀ π‘₯ < 4
  • E βˆ’ 1 1 5 ≀ π‘₯ ≀ 4

Q5:

In the figure, π‘šβˆ π‘‹π‘π‘Š=(π‘Ž+20)∘. Use the hinge theorem to find the range of all possible values of π‘Ž.

  • A 2 2 < π‘Ž < 1 6 0
  • B 2 2 < π‘Ž < 2 0 0
  • C 6 2 < π‘Ž < 1 6 0
  • D 2 2 < π‘Ž < 1 6 0
  • E 6 2 < π‘Ž < 2 0 0

Q6:

In the triangles, using the converse of the hinge theorem, determine whether πœƒ is greater than, less than, or equal to πœ™.

  • A πœƒ is larger than πœ™.
  • B πœƒ is equal to πœ™.
  • C πœƒ is less than πœ™.

Q7:

Consider the two triangles in the diagram.

Length π‘Ž is equal to length 𝑑 and length 𝑏 is equal to length 𝑒. Given that the perimeter of triangle 1 is less than the perimeter of triangle 2, what does the converse of the hinge theorem tell us about the measures of angles πœƒ and πœ™?

  • AThe measure of πœ™ is smaller than the measure of πœƒ.
  • BThe measure of πœ™ is equal to the measure of πœƒ.
  • CThe measure of πœ™ is greater than the measure of πœƒ.

Q8:

Consider the triangles 𝐴𝐡𝐢 and 𝐷𝐸𝐹, where 𝐴𝐡=𝐷𝐸 and 𝐡𝐢=𝐸𝐹. Given that 𝐴𝐢>𝐷𝐹 and using the converse of the Hinge theorem, determine whether ∠𝐴𝐡𝐢 is greater than ∠𝐷𝐸𝐹.

  • A ∠ 𝐴 𝐡 𝐢 is smaller than ∠𝐷𝐸𝐹.
  • B ∠ 𝐴 𝐡 𝐢 is greater than ∠𝐷𝐸𝐹.
  • C ∠ 𝐴 𝐡 𝐢 is equal to ∠𝐷𝐸𝐹.

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