Worksheet: Triangle Inequality: Angle–Side Relationship Theorem

In this worksheet, we will practice using the angle–side inequality in triangles and more complex figures.

Q1:

Which is the correct relationship between 𝐴 𝐢 and 𝐴 𝐡 ?

  • A 𝐴 𝐢 = 𝐴 𝐡
  • B 𝐴 𝐢 < 𝐴 𝐡
  • C 𝐴 𝐢 > 𝐴 𝐡

Q2:

From the figure below, determine the correct inequality from the following.

  • A
  • B
  • C
  • D

Q3:

Complete the following using < , = , or > : If, in the triangle 𝐷 𝐸 𝐹 , 𝐷 𝐸 > 𝐸 𝐹 , then π‘š ∠ 𝐹 π‘š ∠ 𝐷 .

  • A =
  • B <
  • C >

Q4:

Given 𝐴 𝐡 = 9 2 c m , 𝐴 𝐢 = 9 1 c m , and 𝐢 𝐸 = 𝐡 𝐷 , choose the correct relationship between π‘š ∠ 𝐴 𝐸 𝐷 and π‘š ∠ 𝐴 𝐷 𝐸 .

  • A π‘š ∠ 𝐴 𝐸 𝐷 = π‘š ∠ 𝐴 𝐷 𝐸
  • B π‘š ∠ 𝐴 𝐸 𝐷 < π‘š ∠ 𝐴 𝐷 𝐸
  • C π‘š ∠ 𝐴 𝐸 𝐷 > π‘š ∠ 𝐴 𝐷 𝐸

Q5:

Choose the correct relationship between 𝐴 𝐷 and 𝐴 𝐡 .

  • A 𝐴 𝐷 = 𝐴 𝐡
  • B 𝐴 𝐷 > 𝐴 𝐡
  • C 𝐴 𝐷 < 𝐴 𝐡

Q6:

Use < , = , or > to fill in the blank: π‘š ∠ 𝐸 𝐷 𝐴 π‘š ∠ 𝐴 𝐢 𝐡 .

  • A =
  • B <
  • C >

Q7:

Use < , = , or > to fill in the blank: π‘š ∠ 𝐴 𝐡 𝐸 π‘š ∠ 𝐴 𝐢 𝐷 .

  • A <
  • B =
  • C >

Q8:

Use < , = , or > to complete the sentence: If 𝐴 𝐡 𝐢 is a triangle and a point 𝐷 lies outside the triangle, then 𝐴 𝐡 + 𝐴 𝐢 + 𝐡 𝐢 2 ( 𝐷 𝐴 + 𝐷 𝐡 + 𝐷 𝐢 ) .

  • A >
  • B =
  • C <

Q9:

Which of the following is true?

  • A 𝐢 𝐹 = 𝐡 𝐹
  • B 𝐢 𝐹 < 𝐡 𝐹
  • C 𝐢 𝐹 > 𝐡 𝐹

Q10:

Use < , = , or > to fill in the blank:

  • A =
  • B >
  • C <

Q11:

Which inequality is satisfied by this figure?

  • A π‘š ∠ 𝐡 𝐴 𝐢 < π‘š ∠ 𝐢 < π‘š ∠ 𝐷 𝐴 𝐢
  • B π‘š ∠ 𝐷 𝐴 𝐢 < π‘š ∠ 𝐡 < π‘š ∠ 𝐡 𝐴 𝐢
  • C π‘š ∠ 𝐷 𝐴 𝐢 < π‘š ∠ 𝐡 < π‘š ∠ 𝐢
  • D π‘š ∠ 𝐢 < π‘š ∠ 𝐡 < π‘š ∠ 𝐡 𝐴 𝐢
  • E π‘š ∠ 𝐡 < π‘š ∠ 𝐡 𝐴 𝐢 < π‘š ∠ 𝐢

Q12:

Arrange the side lengths of β–³ 𝐴 𝐢 𝐷 in descending order.

  • A 𝐢 𝐷 , 𝐴 𝐷 , 𝐴 𝐢
  • B 𝐢 𝐷 , 𝐴 𝐢 , 𝐴 𝐷
  • C 𝐴 𝐢 , 𝐢 𝐷 , 𝐴 𝐷
  • D 𝐴 𝐢 , 𝐴 𝐷 , 𝐢 𝐷
  • E 𝐴 𝐷 , 𝐢 𝐷 , 𝐴 𝐢

Q13:

Use < , = , or > to fill in the blank: 𝐴 𝐢 𝐷 𝐡 .

  • A <
  • B =
  • C >

Q14:

List the sides of from shortest to longest.

  • A
  • B
  • C
  • D
  • E

Q15:

Use < , = , or > to fill in the blank: If π‘š ∠ 𝐢 𝐡 𝐷 = π‘š ∠ 𝐢 𝐷 𝐡 , then π‘š ∠ 𝐴 𝐡 𝐢 π‘š ∠ 𝐴 𝐷 𝐢 .

  • A =
  • B >
  • C <

Q16:

Use < , = , or > to complete the following: In a triangle 𝐴 𝐡 𝐢 , if 𝐷 is any point on 𝐡 𝐢 , then 𝐡 𝐷 + 𝐷 𝐢 + 2 𝐴 𝐷 𝐴 𝐡 + 𝐴 𝐢 .

  • A <
  • B =
  • C >

Q17:

Use < , = , or > to fill in the blank: If π‘š ∠ 𝐴 𝐢 𝐡 > π‘š ∠ 𝐴 𝐡 𝐢 , then π‘š ∠ 𝐴 𝐡 𝑋 π‘š ∠ 𝐴 𝐢 π‘Œ .

  • A <
  • B =
  • C >

Q18:

Use < , = , or > to fill in the blank: 𝐡 𝐢 𝐴 𝐢 .

  • A <
  • B =
  • C >

Q19:

Use < , = , or > to complete the statement: If 𝐴 𝐡 = 6 2 , 𝐴 𝐢 = 6 3 , and π‘š ∠ 𝐴 𝑋 π‘Œ = π‘š ∠ 𝐴 π‘Œ 𝑋 , then π‘Œ 𝐢 𝑋 𝐡 .

  • A =
  • B <
  • C >

Q20:

Use < , = , or > to complete the following: If 𝐴 𝐸 > 𝐸 𝑋 , then π‘š ∠ 𝐢 π‘š ∠ 𝐷 𝐡 𝐢 .

  • A <
  • B =
  • C >

Q21:

Use < , = , or > to complete the statement: If π‘š ∠ 𝐴 𝐢 𝐡 = 6 2 ∘ and π‘š ∠ 𝐴 = 5 7 ∘ , then π‘š ∠ 𝐴 𝐡 𝐷 π‘š ∠ 𝐴 𝐢 𝐷 .

  • A =
  • B >
  • C <

Q22:

Is ∠ 𝐴 𝐷 𝐢 < , = , > ∠ 𝐴 𝐢 𝐡 ?

  • A >
  • B =
  • C <

Q23:

Arrange , , , in ascending order according to their lengths.

  • A
  • B
  • C
  • D
  • E

Q24:

Choose the correct relationship between 𝐴 𝐡 and 𝐴 𝐷 .

  • A 𝐴 𝐡 = 𝐴 𝐷
  • B 𝐴 𝐡 < 𝐴 𝐷
  • C 𝐴 𝐡 > 𝐴 𝐷

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