Worksheet: Isosceles Triangles

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In this worksheet, we will practice using the properties of isosceles triangles to find a missing angle or side length and solving algebraic expressions or equations.

Q1:

Find 𝑚 ∠ 𝐴 𝐵 𝐶 .

Q2:

Find 𝑚 ∠ 𝐵 𝐴 𝐶 .

Q3:

Find 𝑚 ∠ 𝐶 𝐵 𝐴 and 𝑚 ∠ 𝐷 𝐴 𝐶 .

  • A 𝑚 ∠ 𝐶 𝐵 𝐴 = 7 1 ∘ , 𝑚 ∠ 𝐷 𝐴 𝐶 = 1 1 ∘
  • B 𝑚 ∠ 𝐶 𝐵 𝐴 = 1 0 9 ∘ , 𝑚 ∠ 𝐷 𝐴 𝐶 = 7 1 ∘
  • C 𝑚 ∠ 𝐶 𝐵 𝐴 = 4 9 ∘ , 𝑚 ∠ 𝐷 𝐴 𝐶 = 7 1 ∘
  • D 𝑚 ∠ 𝐶 𝐵 𝐴 = 1 0 9 ∘ , 𝑚 ∠ 𝐷 𝐴 𝐶 = 3 8 ∘

Q4:

Triangle 𝐴 𝐵 𝐶 is isosceles, with 𝑚 ∠ 𝐶 = 8 0 ∘ . Determine the measures of 𝑥 and 𝑦 .

  • A 𝑥 = 1 0 ∘ , 𝑦 = 1 7 0 ∘
  • B 𝑥 = 5 0 ∘ , 𝑦 = 1 3 0 ∘
  • C 𝑥 = 5 0 ∘ , 𝑦 = 1 0 0 ∘
  • D 𝑥 = 8 0 ∘ , 𝑦 = 1 0 0 ∘

Q5:

Which pair of segments have the same length?

  • A 𝐴 𝐵 , 𝐴 𝐶
  • B 𝐴 𝐶 , 𝐵 𝐶
  • C 𝐴 𝐵 , 𝐵 𝐶

Q6:

Find the value of 𝑥 .

Q7:

In the figure below, given that 𝐴 𝐵 = 𝐵 𝐶 , 𝐴 𝐷 = 𝐶 𝐷 , and 𝑚 ∠ 𝐵 𝐴 𝐷 = 1 0 8 ∘ , find 𝑚 ∠ 𝐴 𝐷 𝐶 .

Q8:

Find the values of 𝑥 and 𝑦 .

  • A 𝑥 = 8 3 , 𝑦 = 8 7
  • B 𝑥 = 9 5 , 𝑦 = 5
  • C 𝑥 = 4 1 , 𝑦 = 1 1
  • D 𝑥 = 4 1 , 𝑦 = 5
  • E 𝑥 = 9 5 , 𝑦 = 1 1

Q9:

In the figure, given that 𝐴 𝐷 = 𝐴 𝐶 = 𝐶 𝐵 and 𝑚 ∠ 𝐷 = 5 2 ∘ , find 𝑚 ∠ 𝐷 𝐴 𝐵 .

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

Q10:

In the displayed model of a house, what angle does the roof make with the horizontal, given that △ 𝐴 𝐸 𝐵 is isosceles?

Q11:

Find the value of 𝑥 in the figure below.

Q12:

Given that 𝑚 ∠ 𝐵 = ( 3 𝑥 + 5 2 . 5 ) ∘ and 𝑚 ∠ 𝐶 = ( 2 𝑥 + 5 3 . 5 ) ∘ , find the measures of the angles of △ 𝐴 𝐵 𝐶 .

  • A 𝑚 ∠ 𝐴 = 1 2 4 . 5 ∘ , 𝑚 ∠ 𝐵 = 5 5 . 5 ∘ , 𝑚 ∠ 𝐶 = 5 5 . 5 ∘
  • B 𝑚 ∠ 𝐴 = 5 5 . 5 ∘ , 𝑚 ∠ 𝐵 = 6 9 ∘ , 𝑚 ∠ 𝐶 = 5 5 . 5 ∘
  • C 𝑚 ∠ 𝐴 = 6 9 ∘ , 𝑚 ∠ 𝐵 = 5 5 . 5 ∘ , 𝑚 ∠ 𝐶 = 5 5 . 5 ∘

Q13:

In the following figure, find the value of 𝑥 .

Q14:

Find the values of 𝑥 and 𝑦 .

  • A 𝑥 = 3 0 , 𝑦 = 8
  • B 𝑥 = 1 5 , 𝑦 = 2
  • C 𝑥 = 1 5 , 𝑦 = 8
  • D 𝑥 = 7 . 5 , 𝑦 = 6
  • E 𝑥 = 3 0 , 𝑦 = 2

Q15:

Given that 𝐴 𝐷 = 𝐷 𝐶 = 𝐶 𝐵 = 𝐵 𝐷 , find the value of 𝑥 .

  • A 2 4
  • B − 6
  • C 1 8
  • D 8 5

Q16:

Find the values of 𝑥 and 𝑦 .

  • A 𝑥 = 7 8 , 𝑦 = 9
  • B 𝑥 = 1 2 , 𝑦 = 3
  • C 𝑥 = 1 2 , 𝑦 = 9
  • D 𝑥 = 3 0 , 𝑦 = 3
  • E 𝑥 = 3 0 , 𝑦 = 9

Q17:

Find all possible values of 𝑥 .

  • A 𝑥 = − 1 0 or 𝑥 = 4
  • B 𝑥 = 1 0 or 𝑥 = 4
  • C 𝑥 = 1 0 or 𝑥 = − 4
  • D 𝑥 = 3 4 or 𝑥 = 4 6
  • E 𝑥 = − 1 0 or 𝑥 = − 4

Q18:

In the figure, determine all possible values of 𝑦 .

  • A 𝑦 = − 2 or 𝑦 = − 3
  • B 𝑦 = 3 or 𝑦 = − 1
  • C 𝑦 = − 3 or 𝑦 = − 1
  • D 𝑦 = − 3 or 𝑦 = 1
  • E 𝑦 = 3 or 𝑦 = 1

Q19:

In the figure below, given that 𝐴 𝐷 = 𝐷 𝐶 = 𝐴 𝐶 , 𝐴 𝐵 = 𝐵 𝐶 , and 𝑚 ∠ 𝐴 𝐵 𝐶 = 5 4 ∘ , determine 𝑚 ∠ 𝐵 𝐴 𝐷 .

Q20:

Find 𝑚 ∠ 𝐵 𝐸 𝐴 and 𝑚 ∠ 𝐸 𝐴 𝐶 .

  • A 𝑚 ∠ 𝐵 𝐸 𝐴 = 1 3 2 ∘ , 𝑚 ∠ 𝐸 𝐴 𝐶 = 7 2 ∘
  • B 𝑚 ∠ 𝐵 𝐸 𝐴 = 7 2 ∘ , 𝑚 ∠ 𝐸 𝐴 𝐶 = 3 6 ∘
  • C 𝑚 ∠ 𝐵 𝐸 𝐴 = 7 2 ∘ , 𝑚 ∠ 𝐸 𝐴 𝐶 = 1 3 2 ∘
  • D 𝑚 ∠ 𝐵 𝐸 𝐴 = 3 6 ∘ , 𝑚 ∠ 𝐸 𝐴 𝐶 = 1 3 2 ∘

Q21:

If △ 𝐴 𝐵 𝐶 has one line of symmetry, and 𝑚 ∠ 𝐴 𝐵 𝐶 = 1 2 8 ∘ , find 𝑚 ∠ 𝐴 .

Q22:

In an isosceles triangle 𝐴 𝐵 𝐶 , 𝐴 𝐵 = 𝐴 𝐶 , 𝐵 𝐶 = 6 0 c m , 𝐴 𝐷 is perpendicular to 𝐵 𝐶 and intersecting the segment at 𝐷 , and 𝑚 ∠ 𝐵 𝐴 𝐷 = 2 6 ∘ . Find the length of 𝐵 𝐷 and 𝑚 ∠ 𝐵 .

  • A 30 cm, 6 4 ∘
  • B 60 cm, 6 4 ∘
  • C 30 cm, 2 6 ∘
  • D 60 cm, 2 6 ∘

Q23:

Which of the following is true?

  • A 𝐵 𝐶 = 𝐵 𝐴
  • B 𝐴 𝐵 = 𝐴 𝐶
  • C 𝐶 𝐴 = 𝐶 𝐵

Q24:

Is 𝑀 𝐵 𝐶 an isosceles triangle?

  • AYes
  • BNo

Q25:

Find 𝑚 ∠ 𝐵 𝐷 𝐸 .

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