Worksheet: Angle Bisector Theorem and Its Converse

In this worksheet, we will practice using the theorem of angle bisector and its converse to find a missing side length in a triangle.


In the given figure, 𝐴𝐵=35, 𝐴𝐶=30, and 𝐶𝐷=12. If 𝐵𝐷=𝑥+10, what is the value of 𝑥?


In the figure, 𝐴𝐷 bisects 𝐵𝐴𝐶, 𝐵𝐷=8, 𝐷𝐶=11, and the perimeter of 𝐴𝐵𝐶 is 57. Determine the lengths of 𝐴𝐵 and 𝐴𝐶.

  • A𝐴𝐵=22, 𝐴𝐶=16
  • B𝐴𝐵=19, 𝐴𝐶=22
  • C𝐴𝐵=16, 𝐴𝐶=19
  • D𝐴𝐵=16, 𝐴𝐶=22


Given that angle 𝐴 is bisected by 𝐷𝐴, 𝐴𝐵=38, 𝐴𝐶=18, and 𝐵𝐶=28. Determine the lengths 𝐷𝐵 and 𝐷𝐶.

  • A𝐷𝐵=14, 𝐷𝐶=14
  • B𝐷𝐵=9, 𝐷𝐶=19
  • C𝐷𝐵=19, 𝐷𝐶=9
  • D𝐷𝐵=12, 𝐷𝐶=16


If 𝐴𝐵=30cm, 𝐵𝐶=40cm, and 𝐴𝐶=45cm, find the ratio between the areas of the 𝐴𝐸𝐷 and the 𝐴𝐸𝐶.

  • A45
  • B89
  • C815
  • D34


Find the lengths of 𝐴𝐶 and 𝐴𝐷 in the figure.

  • A𝐴𝐶=55cm, 𝐴𝐷=50cm
  • B𝐴𝐶=55cm, 𝐴𝐷=58cm
  • C𝐴𝐶=50cm, 𝐴𝐷=40cm
  • D𝐴𝐶=58cm, 𝐴𝐷=55cm


If 𝐵𝐴𝐷 is a right triangle at 𝐴, 𝐴𝐶=10cm, 𝐶𝐸=12cm, and 𝐸𝐴=15cm, calculate the value of 𝑥.


If 𝐴𝐵𝐶𝐷 is a quadrilateral in which 𝐴𝐵=10cm, 𝐵𝐶=5cm, 𝐶𝐷=6cm, 𝐴𝐷=11cm, where 𝐴𝐸 bisects 𝐴 and 𝐵𝐷 intersects at 𝐸, find the value of the ratio 𝐵𝐸𝐸𝐷.

  • A1011
  • B56
  • C1110
  • D65


Given that 𝐴𝐵𝐶 is a triangle in which 𝐴𝐶=10cm, find the value of each of 𝑥 and 𝑦.

  • A𝑥=66, 𝑦=12
  • B𝑥=66, 𝑦=8
  • C𝑥=12, 𝑦=66
  • D𝑥=8, 𝑦=66


In the triangle 𝐴𝐵𝐶, 𝐴𝐵=76cm, 𝐴𝐶=57cm, and 𝐵𝐷=52cm. Given that 𝐴𝐷 bisects 𝐴 and intersects 𝐵𝐶 at 𝐷, determine the length of 𝐴𝐷.


Given that in the figure, 𝐴𝐵=8, 𝐵𝐶=15, and 𝐴𝐶=20, what is 𝐸𝐵?


If 𝐴𝐵𝐴𝐶=35 and 𝐵𝐷=27cm, determine the perimeter of 𝐴𝐵𝐶.


Using the figure below, find the length of 𝐴𝐷 to the nearest hundredth.


In triangle 𝐴𝐵𝐶, 𝐷 lies on 𝐴𝐶 such that 𝐵𝐷 bisects 𝐴𝐵𝐶. Given 𝐴𝐵=10, 𝐵𝐶=20, and 𝐴𝐷=6, determine 𝐴𝐶 to the nearest hundredth.


Use the figure to determine length 𝐴𝐷 to two decimal places.


Given that 𝐴𝐵=60, 𝐴𝐶=40, and 𝐵𝐶=31, what is 𝐶𝐷?


𝐴𝐵𝐶 is a right-angled triangle at 𝐵, where 𝐴𝐷 bisects 𝐴 and intersects 𝐵𝐶 at 𝐷. Given that 𝐵𝐷=18cm, 𝐵𝐴𝐴𝐶=45, determine the perimeter of 𝐴𝐵𝐶.


In the given figure, if 𝐴𝐵𝐴𝐶𝐵𝐶=6911, find 𝐵𝐷𝐷𝐶.

  • A23
  • B32
  • C611
  • D911


If 𝐴𝐵=25cm and 𝐴𝐶=21cm, find 𝐵𝐸𝐵𝐶. Leave your answer as a fraction in its simplest form.

  • A2521
  • B2546
  • C4625
  • D2125


Suppose that in the figure, 𝑚𝐷𝐴𝐶=34. What is 𝑚𝐸𝐴𝐹?


𝐴𝐵 is a chord in a circle. 𝐷 major arc 𝐴𝐵 such that 𝐴𝐷𝐷𝐵=12. 𝐸 is the midpoint of the minor arc 𝐴𝐵. 𝐷𝐸 is drawn to intersect 𝐴𝐵 at 𝐶. Determine the ratio between the areas of 𝐴𝐷𝐸 and 𝐵𝐷𝐸.

  • A4
  • B12
  • C2
  • D1


𝐴𝐵𝐶 is a triangle in which 𝐴𝐵=32cm, 𝐵𝐶=33cm, and 𝐴𝐶=16cm. 𝐷𝐵𝐶, where 𝐵𝐷=22cm, 𝐴𝐸𝐴𝐷 and intersects 𝐵𝐶 at 𝐸. If 𝐴𝐷 bisects 𝐵𝐴𝐶, find the length of 𝐶𝐸.


Given that 𝐴𝐵=𝑥+5cm, 𝐴𝐶=29cm, 𝐶𝐷=38cm, and 𝐵𝐶=38cm, find the numerical value of 𝑥.


In the shown figure, determine 𝐴𝐷𝐵𝐷.

  • A916
  • B169
  • C97
  • D79


𝐴𝐵𝐶 is a triangle, where 𝑋 is the midpoint of 𝐵𝐶, 𝐵𝑋=23cm and 𝐴𝑋=23cm. If the bisector of 𝐴𝑋𝐵 intersects 𝐴𝐵 at 𝐷, find the value of 𝐴𝐷𝐷𝐵.

  • A2723
  • B2327
  • C4627
  • D2746


In the figure, 𝐴𝐵𝐴𝐶=47. What is 𝐵𝐷𝐵𝐶?

  • A72
  • B34
  • C27
  • D43

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