Bisecting an Angle
To bisect an angle means to divide it into two equal halves. The ray that divides the angle is called the angle bisector.
If you bisect a 60° angle, you get two angles of 30° each. If you bisect 90°, you get 45° each.
In Class 6, you will learn to construct an angle bisector using a compass and ruler.
What is Bisecting an Angle - Grade 6 Maths (Practical Geometry)?
Definition: The angle bisector is a ray that divides an angle into two equal parts.
If ∠AOB = 80° and ray OC bisects it, then:
- ∠AOC = 40°
- ∠COB = 40°
- ∠AOC = ∠COB
Key property: Every point on the angle bisector is equidistant from both arms (sides) of the angle.
Bisecting an Angle Formula
Steps to bisect angle ∠AOB:
- With O (the vertex) as centre and any radius, draw an arc that cuts ray OA at point P and ray OB at point Q.
- With P as centre and a radius more than half of PQ, draw an arc in the interior of the angle.
- With Q as centre and the same radius, draw another arc. Let the two arcs meet at point R.
- Join OR and extend it. Ray OR is the angle bisector.
- ∠AOR = ∠ROB = half of ∠AOB.
Important:
- In step 1, the radius can be any convenient length.
- In steps 2 and 3, the radius from P and Q must be the same.
- The radius in steps 2-3 must be large enough for the arcs to intersect.
Types and Properties
Using bisection to construct new angles:
- Bisect 60° → 30°
- Bisect 90° → 45°
- Bisect 120° → 60°
- Bisect 30° → 15°
- Bisect 45° → 22.5°
Repeated bisection:
- Start with 60°. Bisect once → 30°. Bisect again → 15°. Bisect again → 7.5°.
- You can keep bisecting to get smaller and smaller angles.
Solved Examples
Example 1: Bisecting a 60° Angle
Problem: Construct a 60° angle and bisect it to get 30°.
Solution:
- Construct ∠AOB = 60° using the compass method.
- Arc from O cuts OA at P and OB at Q.
- Arcs from P and Q (same radius) meet at R.
- Join OR.
Answer: ∠AOR = ∠ROB = 30°.
Example 2: Bisecting a 90° Angle
Problem: Construct a 90° angle and bisect it.
Solution:
- Construct 90° using compass (perpendicular construction).
- Bisect the 90° angle using the steps above.
Answer: Each half = 45°.
Example 3: Bisecting a 120° Angle
Problem: Construct ∠AOB = 120° and bisect it.
Solution:
- Construct 120° using compass.
- Bisect: arc from O gives P and Q. Arcs from P and Q meet at R. Join OR.
Answer: Each half = 60°.
Example 4: Bisecting Any Given Angle
Problem: An angle ∠XYZ is given (measurement unknown). Bisect it.
Solution:
- Arc from Y (vertex) cuts YX at A and YZ at B.
- Arcs from A and B (equal radius) meet at C.
- Join YC.
Answer: YC bisects ∠XYZ. Each half is equal.
Example 5: Constructing 15° by Double Bisection
Problem: Construct a 15° angle.
Solution:
- Construct 60°.
- Bisect 60° to get 30°.
- Bisect 30° to get 15°.
Answer: The angle is 15°.
Example 6: Verifying the Bisection
Problem: After bisecting a 70° angle, verify with a protractor.
Solution:
Measure both halves with the protractor. Each should be 35°.
Real-World Applications
Where angle bisectors are used:
- Construction: Making precise angles like 15°, 22.5°, 45° for designs.
- Origami: Paper folding uses angle bisection to create symmetric patterns.
- Navigation: Finding a direction halfway between two directions.
- Geometry: The angle bisectors of a triangle meet at the incentre (centre of the inscribed circle).
- Mirror angles: Light reflects at equal angles — the normal is like an angle bisector.
Key Points to Remember
- An angle bisector divides an angle into two equal parts.
- Steps: arc from vertex → arcs from intersection points with equal radius → join vertex to meeting point.
- The arcs from P and Q must have the same radius.
- You can construct 30° (bisect 60°), 45° (bisect 90°), 15° (bisect 30°).
- Every point on the bisector is equidistant from both arms of the angle.
- Do not erase construction arcs.
Practice Problems
- Construct a 60° angle and bisect it to get 30°.
- Construct a right angle (90°) and bisect it to get 45°.
- Bisect an angle of 120° to get 60°.
- Construct a 15° angle by bisecting 30°.
- Draw any angle with a protractor and bisect it using compass. Verify with the protractor.
- Construct 90° and bisect it twice to get 22.5°.
Frequently Asked Questions
Q1. What does bisecting an angle mean?
It means dividing the angle into two equal halves using a ray from the vertex. The ray is called the angle bisector.
Q2. Can I bisect any angle?
Yes. The compass method works for any angle. You do not need to know the angle measurement to bisect it.
Q3. Why must the arcs from P and Q have equal radius?
Equal radii ensure the intersection point R is equidistant from both arms of the angle. This guarantees the bisector divides the angle exactly in half.
Q4. Can I trisect an angle with compass and ruler?
In general, no. Trisecting (dividing into three equal parts) an arbitrary angle using only compass and straightedge is impossible — this is a famous result in mathematics.
Q5. What is the incentre of a triangle?
The incentre is the point where all three angle bisectors of a triangle meet. It is the centre of the circle that fits inside the triangle (inscribed circle).
Related Topics
- Constructing Angles
- Bisecting a Line Segment
- Types of Angles
- Constructing Perpendicular Lines
- Constructing a Line Segment
- Constructing Parallel Lines
- Constructing Triangles (SSS)
- Constructing Triangles (SAS)
- Constructing Triangles (ASA)
- Constructing Triangles (RHS)
- Constructing Quadrilaterals
- Constructing Special Quadrilaterals
- Triangle Construction Problems
