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.

  • A𝑚∠𝐶𝐵𝐴=109∘, 𝑚∠𝐷𝐴𝐶=38∘
  • B𝑚∠𝐶𝐵𝐴=109∘, 𝑚∠𝐷𝐴𝐶=71∘
  • C𝑚∠𝐶𝐵𝐴=71∘, 𝑚∠𝐷𝐴𝐶=11∘
  • D𝑚∠𝐶𝐵𝐴=49∘, 𝑚∠𝐷𝐴𝐶=71∘

Q4:

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

  • A𝑥=80∘, 𝑦=100∘
  • B𝑥=10∘, 𝑦=170∘
  • C𝑥=50∘, 𝑦=100∘
  • D𝑥=50∘, 𝑦=130∘

Q5:

Which pair of segments have the same length?

  • A𝐴𝐵, 𝐵𝐶
  • B𝐴𝐶, 𝐵𝐶
  • C𝐴𝐵, 𝐴𝐶

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.

  • A𝑥=83, 𝑦=87
  • B𝑥=41, 𝑦=5
  • C𝑥=95, 𝑦=11
  • D𝑥=41, 𝑦=11
  • E𝑥=95, 𝑦=5

Q9:

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

  • A128∘
  • B130∘
  • C102∘
  • D104∘
  • E52∘

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.

  • A𝑚∠𝐴=55.5∘, 𝑚∠𝐵=69∘, 𝑚∠𝐶=55.5∘
  • B𝑚∠𝐴=124.5∘, 𝑚∠𝐵=55.5∘, 𝑚∠𝐶=55.5∘
  • C𝑚∠𝐴=69∘, 𝑚∠𝐵=55.5∘, 𝑚∠𝐶=55.5∘

Q13:

In the following figure, find the value of 𝑥.

Q14:

Find the values of 𝑥 and 𝑦.

  • A𝑥=15, 𝑦=8
  • B𝑥=15, 𝑦=2
  • C𝑥=30, 𝑦=2
  • D𝑥=7.5, 𝑦=6
  • E𝑥=30, 𝑦=8

Q15:

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

  • A85
  • B24
  • C−6
  • D18

Q16:

Find the values of 𝑥 and 𝑦.

  • A𝑥=78, 𝑦=9
  • B𝑥=30, 𝑦=9
  • C𝑥=30, 𝑦=3
  • D𝑥=12, 𝑦=3
  • E𝑥=12, 𝑦=9

Q17:

Find all possible values of 𝑥.

  • A𝑥=−10 or 𝑥=4
  • B𝑥=34 or 𝑥=46
  • C𝑥=10 or 𝑥=4
  • D𝑥=10 or 𝑥=−4
  • E𝑥=−10 or 𝑥=−4

Q18:

In the figure, determine all possible values of 𝑦.

  • A𝑦=−3 or 𝑦=1
  • B𝑦=−2 or 𝑦=−3
  • C𝑦=3 or 𝑦=−1
  • D𝑦=−3 or 𝑦=−1
  • E𝑦=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.

  • A𝑚∠𝐵𝐸𝐴=72∘, 𝑚∠𝐸𝐴𝐶=36∘
  • B𝑚∠𝐵𝐸𝐴=36∘, 𝑚∠𝐸𝐴𝐶=132∘
  • C𝑚∠𝐵𝐸𝐴=132∘, 𝑚∠𝐸𝐴𝐶=72∘
  • D𝑚∠𝐵𝐸𝐴=72∘, 𝑚∠𝐸𝐴𝐶=132∘

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.

  • A60 cm, 64∘
  • B30 cm, 26∘
  • C30 cm, 64∘
  • D60 cm, 26∘

Q23:

Which of the following is true?

  • A𝐴𝐵=𝐴𝐶
  • B𝐵𝐶=𝐵𝐴
  • C𝐶𝐴=𝐶𝐵

Q24:

Is 𝑀𝐵𝐶 an isosceles triangle?

  • ANo
  • BYes

Q25:

Find 𝑚∠𝐵𝐷𝐸.

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