Lesson Worksheet: Inequalities in Two Triangles Mathematics

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 𝑎 is congruent to 𝑑 and length 𝑏 is congruent to 𝑒. Given that 𝜙 is greater than 𝜃, what does the hinge theorem tell us about the lengths 𝑐 and 𝑓?

  • A𝑓 and 𝑐 are equal.
  • BThere is no relation between 𝑓 and 𝑐.
  • C𝑓 is less than 𝑐.
  • D𝑓 is greater than 𝑐.

Q2:

Given that 𝑚∠𝐴𝐶𝐷=(6𝑥−12)∘, use the hinge theorem to find the range of all possible values of 𝑥 in the figure.

  • A2≤𝑥≤11
  • B2<𝑥<7
  • C−2<𝑥<7
  • D−2≤𝑥≤7
  • E2<𝑥<11

Q3:

Consider triangles 𝐴𝐵𝐶 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 𝐴𝐶.

  • A𝐷𝐹 is greater than 𝐴𝐶.
  • B𝐷𝐹 is less than 𝐴𝐶.
  • C𝐷𝐹 and 𝐴𝐶 are equal.

Q4:

Given that 𝐶𝐸=5𝑥−11 and 𝐶𝐵=9, find the range of all possible values of 𝑥 using the Hinge theorem.

  • A−115<𝑥<4
  • B115<𝑥<4
  • C115≤𝑥≤4
  • D115≤𝑥<4
  • E−115≤𝑥≤4

Q5:

In the figure, 𝑚∠𝑋𝑍𝑊=(𝑎+20)∘. Use the hinge theorem to find the range of all possible values of 𝑎.

  • A22<𝑎<160
  • B22<𝑎<200
  • C62<𝑎<160
  • D22<𝑎<160
  • E62<𝑎<200

Q6:

In the triangles, using the converse of the hinge theorem, determine whether 𝜃 is greater than, less than, or equal to 𝜙.

  • A𝜃 is larger than 𝜙.
  • B𝜃 is equal to 𝜙.
  • C𝜃 is less than 𝜙.

Q7:

Consider the two triangles in the diagram.

Length 𝑎 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 𝜃 and 𝜙?

  • AThe measure of 𝜙 is smaller than the measure of 𝜃.
  • BThe measure of 𝜙 is equal to the measure of 𝜃.
  • CThe measure of 𝜙 is greater than the measure of 𝜃.

Q8:

Consider the triangles 𝐴𝐵𝐶 and 𝐷𝐸𝐹, where 𝐴𝐵=𝐷𝐸 and 𝐵𝐶=𝐸𝐹. Given that 𝐴𝐶>𝐷𝐹 and using the converse of the Hinge theorem, determine whether ∠𝐴𝐵𝐶 is greater than ∠𝐷𝐸𝐹.

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

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