A point P is 8 cm and 8 cm from the two arms of ∠AOB, measured at right angles. P lies on the bisector of ∠AOB. True or false?

True. Only a point equally far from both arms lies on the bisector (converse of ∠ bisector theorem).

P is on the bisector of ∠AOB, PM ⊥ OA and PN ⊥ OB, with OP = 5 cm and OM = 4 cm. PN = □

3 cm. PM = √(5^2 − 4^2) = 3 cm (Pyth. thm.); PN = PM = 3 cm (∠ bisector theorem).

Centres of a triangle · Ages 14–15

The angle bisector's property

A point on the bisector is equally far (perpendicularly) from both arms.

How to do it

Method

A point on the bisector is equally far (perpendicularly) from both arms

PM = 3 cm: PN = 3 cm, ∠POM = 70° ÷ 2 = 35°

Another way

By congruent triangles

△OPM ≅ △OPN (AAS), PM = PN

In full

The angle bisector theorem: any point on an angle's bisector is the same distance from the angle's two arms (∠ bisector theorem). Distance means the perpendicular: drop perpendiculars from P to OA and OB, feet M and N, and PM = PN. The converse: a point equally far from the two arms lies on the bisector (converse of ∠ bisector theorem). One way: use the theorem straight. ∠AOB = 70°, P on the bisector and PM = 3 cm give PN = 3 cm and ∠POM = 70° ÷ 2 = 35°. Another: prove it by congruent triangles. In △OPM and △OPN, ∠PMO = ∠PNO = 90°, ∠POM = ∠PON and OP is common, so △OPM ≅ △OPN (AAS) and PM = PN. Careful: the distance is the perpendicular, not P to O or P to just any point of an arm.

Worked examples

Have a go first, then open each one for the answer and the working.

Example 1
In the picture, OP bisects ∠AOB. x =
Show the answer and working
Answer: 45°
A bisector cuts the angle into two equal halves: x = 45°.
Example 2
A point P is 8 cm and 8 cm from the two arms of ∠AOB, measured at right angles. P lies on the bisector of ∠AOB.
True or false?
Show the answer and working
Answer: True
Only a point equally far from both arms lies on the bisector (converse of ∠ bisector theorem).
Example 3
P is on the bisector of ∠AOB, PM ⊥ OA and PN ⊥ OB, with OP = 5 cm and OM = 4 cm. PN =
Show the answer and working
Answer: 3 cm
PM = 5² − 4² = 3 cm (Pyth. thm.); PN = PM = 3 cm (∠ bisector theorem).

Practice questions to try

Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.

1.In the picture, OP bisects ∠AOB. x =
Answer:
2.In the picture, OP bisects ∠AOB, PM ⊥ OA and PN ⊥ OB. So PM = PN ( ? ). Which reason goes in the gap?
- A. given
- B. converse of ∠ bisector theorem
- C. ∠ bisector theorem
- D. ⊥ bisector theorem
Choose the right answer
Answer:
3.In the picture, OP bisects ∠AOB, ∠AOB = 82° and PM ⊥ OA. x =
Answer:
4.P lies inside ∠AOB with PM ⊥ OA, PN ⊥ OB, PM = PN = 9 cm and ∠AOB = 92°. ∠POM = °, ∠OPM = °
Fill in each blank
Answer:
5.P is on the bisector of ∠AOB, PM ⊥ OA and PN ⊥ OB, with OP = 75 cm and OM = 72 cm. PN =
Answer:

All the answers

1. 50°
2. C. ∠ bisector theorem
3. 49°
4. 46, 44
5. 21 cm

Free worksheet: The angle bisector's propertyWorking out, multiple choice and word problems, with answers and working, to print or save as a PDF

VIP practice notes and papers

Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.

Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
Practice paper: Set out in parts like a school paper, marked, with answers and full working

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