Geometry and proof · Ages 14–15

The angle bisector's property worksheet

Part of: Centres of a triangle

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Geometry and proof · Ages 14–15

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Name:Date:Score:／48

Write each answer in the space. For word problems, show your working and write a sentence answer.

1.In the picture, OP bisects ∠AOB, PM ⊥ OA and PN ⊥ OB. x =
Answer:
2.In the picture, OP bisects ∠AOB. x =
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3.In the picture, OP bisects ∠AOB, ∠AOB = 98° and PM ⊥ OA. x =
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4.In the picture, OP bisects ∠AOB, ∠AOB = 54° and PM ⊥ OA. x =
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5.P lies inside ∠AOB with PM ⊥ OA, PN ⊥ OB, PM = PN = 8 cm and ∠AOB = 108°. ∠POM = °, ∠OPM = °
Fill in each blank
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6.P is on the bisector of ∠AOB, PM ⊥ OA and PN ⊥ OB, with OP = 30 cm and OM = 18 cm. PN =
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7.P lies inside ∠AOB with PM ⊥ OA, PN ⊥ OB, PM = PN = 9 cm and ∠AOB = 116°. ∠POM = °, ∠OPM = °
Fill in each blank
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8.P is on the bisector of ∠AOB, PM ⊥ OA and PN ⊥ OB, with OP = 13 cm and OM = 12 cm. PN =
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9.In the picture, OP bisects ∠AOB, PM ⊥ OA and PN ⊥ OB. So PM = PN ( ? ). Which reason goes in the gap?
- A. ⊥ bisector theorem
- B. given
- C. ∠ bisector theorem
- D. converse of ∠ bisector theorem
Choose the right answer
Answer:
10.A point P is 2 cm and 4 cm from the two arms of ∠AOB, measured at right angles. P lies on the bisector of ∠AOB.
True or false?
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11.A point P is 7 cm and 7 cm from the two arms of ∠AOB, measured at right angles. P lies on the bisector of ∠AOB.
True or false?
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12.A point P is 2 cm and 2 cm from the two arms of ∠AOB, measured at right angles. P lies on the bisector of ∠AOB.
True or false?
Answer:

13.A tap P is on the bisector of the angle between two straight fences, (2x + 6) m from one and (3x − 1) m from the other. What is x?
Working:
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14.Two straight walls meet at a corner O at 108°. A fountain P is equally far from both walls. What angle does OP make with a wall?
Working:
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15.Two straight roads meet at O at 60°. A lamp P stands on the bisector of that angle, 7 m from one road. P is m from the other road, and OP makes ° with a road.
Fill in each blank
Working:
Answer:
16.A park gate P is on the bisector of the angle between two straight roads, 39 m from the junction O. M is the foot of the perpendicular from P to one road, with OM = 36 m. P is m from that road and m from the other.
Fill in each blank
Working:
Answer:
17.A tap P is on the bisector of the angle between two straight fences, (2x + 1) m from one and (3x − 5) m from the other. What is x?
Working:
Answer:
18.Two straight roads meet at O at 78°. A lamp P stands on the bisector of that angle, 13 m from one road. P is m from the other road, and OP makes ° with a road.
Fill in each blank
Working:
Answer:

Show the answers and working

The angle bisector's property worksheet: Answers and working

1. 8 cm
P is on the bisector, so it is equally far from both arms: x = PM = 8 (∠ bisector theorem).
2. 58°
A bisector cuts the angle into two equal halves: x = 58°.
3. 41°
∠POM = 98° ÷ 2 = 49°; in △OPM, x = 180° − 90° − 49° = 41°.
4. 63°
∠POM = 54° ÷ 2 = 27°; in △OPM, x = 180° − 90° − 27° = 63°.
5. 54, 36
PM = PN puts P on the bisector (converse of ∠ bisector theorem): ∠POM = 108° ÷ 2 = 54°, ∠OPM = 90° − 54° = 36°.
6. 24 cm
PM = 30² − 18² = 24 cm (Pyth. thm.); PN = PM = 24 cm (∠ bisector theorem).
7. 58, 32
PM = PN puts P on the bisector (converse of ∠ bisector theorem): ∠POM = 116° ÷ 2 = 58°, ∠OPM = 90° − 58° = 32°.
8. 5 cm
PM = 13² − 12² = 5 cm (Pyth. thm.); PN = PM = 5 cm (∠ bisector theorem).

9. C. ∠ bisector theorem
P is on the bisector, so it is equally far from the arms: ∠ bisector theorem.
10. False
Only a point equally far from both arms lies on the bisector (converse of ∠ bisector theorem).
11. True
Only a point equally far from both arms lies on the bisector (converse of ∠ bisector theorem).
12. True
Only a point equally far from both arms lies on the bisector (converse of ∠ bisector theorem).

Marking: 2 for the working, 1 for the answer, 1 for the sentence.

13. 7
The two distances are equal (∠ bisector theorem): 2x + 6 = 3x − 1, so x = 7, each 20 m.
14. 54°
A point equally far from both arms is on the bisector (converse of ∠ bisector theorem): 108° ÷ 2 = 54°.
15. 7, 30
A point on the bisector is equally far from both arms (∠ bisector theorem): 7 m; the bisector halves 60° into two of 30°.
16. 15, 15
PM = 39² − 36² = 15 m (Pyth. thm.); P is on the bisector, so 15 m from the other road too (∠ bisector theorem).
17. 6
The two distances are equal (∠ bisector theorem): 2x + 1 = 3x − 5, so x = 6, each 13 m.
18. 13, 39
A point on the bisector is equally far from both arms (∠ bisector theorem): 13 m; the bisector halves 78° into two of 39°.

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