Worksheet: Isosceles Triangles

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

Q1:

Find 𝑚∠𝐴𝐵𝐶.

Q2:

Find 𝑚∠𝐵𝐴𝐶.

Q3:

Find 𝑚∠𝐶𝐵𝐴 and 𝑚∠𝐷𝐴𝐶.

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

Q4:

Triangle 𝐴𝐵𝐶 is isosceles, with 𝑚∠𝐶=80∘. Determine the measures of 𝑥 and 𝑦.

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

Q5:

Which pair of segments have the same length?

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

Q6:

Find the value of 𝑥.

Q7:

In the figure below, given that 𝐴𝐵=𝐵𝐶, 𝐴𝐷=𝐶𝐷, and 𝑚∠𝐵𝐴𝐷=108∘, find 𝑚∠𝐴𝐷𝐶.

Q8:

Find the values of 𝑥 and 𝑦.

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

Q9:

In the figure, given that 𝐴𝐷=𝐴𝐶=𝐶𝐵 and 𝑚∠𝐷=52∘, find 𝑚∠𝐷𝐴𝐵.

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

Q10:

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

Q11:

Find the value of 𝑥 in the figure below.

Q12:

Given that 𝑚∠𝐵=(3𝑥+52.5)∘ and 𝑚∠𝐶=(2𝑥+53.5)∘, find the measures of the angles of △𝐴𝐵𝐶.

  • A 𝑚 ∠ 𝐴 = 5 5 . 5 ∘ , 𝑚 ∠ 𝐵 = 6 9 ∘ , 𝑚 ∠ 𝐶 = 5 5 . 5 ∘
  • B 𝑚 ∠ 𝐴 = 1 2 4 . 5 ∘ , 𝑚 ∠ 𝐵 = 5 5 . 5 ∘ , 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 = 1 5 , 𝑦 = 8
  • B 𝑥 = 1 5 , 𝑦 = 2
  • C 𝑥 = 3 0 , 𝑦 = 2
  • D 𝑥 = 7 . 5 , 𝑦 = 6
  • E 𝑥 = 3 0 , 𝑦 = 8

Q15:

Given that 𝐴𝐷=𝐷𝐶=𝐶𝐵=𝐵𝐷, find the value of 𝑥.

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

Q16:

Find the values of 𝑥 and 𝑦.

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

Q17:

Find all possible values of 𝑥.

  • A 𝑥 = − 1 0 or 𝑥=4
  • B 𝑥 = 3 4 or 𝑥=46
  • C 𝑥 = 1 0 or 𝑥=4
  • D 𝑥 = 1 0 or 𝑥=−4
  • E 𝑥 = − 1 0 or 𝑥=−4

Q18:

In the figure, determine all possible values of 𝑦.

  • A 𝑦 = − 3 or 𝑦=1
  • B 𝑦 = − 2 or 𝑦=−3
  • 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 𝐴𝐷=𝐷𝐶=𝐴𝐶, 𝐴𝐵=𝐵𝐶, and 𝑚∠𝐴𝐵𝐶=54∘, determine 𝑚∠𝐵𝐴𝐷.

Q20:

Find 𝑚∠𝐵𝐸𝐴 and 𝑚∠𝐸𝐴𝐶.

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

Q21:

If △𝐴𝐵𝐶 has one line of symmetry, and 𝑚∠𝐴𝐵𝐶=128∘, find 𝑚∠𝐴.

Q22:

In an isosceles triangle 𝐴𝐵𝐶, 𝐴𝐵=𝐴𝐶, 𝐵𝐶=60cm, 𝐴𝐷 is perpendicular to 𝐵𝐶 and intersecting the segment at 𝐷, and 𝑚∠𝐵𝐴𝐷=26∘. Find the length of 𝐵𝐷 and 𝑚∠𝐵.

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

Q23:

Which of the following is true?

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

Q24:

Is 𝑀𝐵𝐶 an isosceles triangle?

  • ANo
  • BYes

Q25:

Find 𝑚∠𝐵𝐷𝐸.

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