Constructions and Tilings · Lesson 3 of 12
Angle Bisection for a Design
“Use congruent triangles to divide an angle equally and build designs with exact repeated directions.”
• Explain why eight equal sectors around a point are 45° each. • Construct the internal bisector of a given angle. • Justify angle bisection using SSS congruence. • Distinguish a bisector line from its internal and opposite rays. • Use bisection to construct a design with equal directions.
Eight Equal Angles Around a Point
An eight-petal design needs eight directions spaced equally around its centre. If one gap is larger than another, the petals no longer repeat evenly. A full turn around a point is 360°, so the equal gaps must each be one eighth of that turn.
We already know how to construct a right angle of 90°. Dividing it into two equal parts gives 45°. An angle bisector is the ray that starts at the vertex and divides the angle into equal smaller angles. The word bisection refers to equal parts, not merely to putting another ray somewhere inside the angle.
A ray from the vertex of an angle that divides it into two equal angles.
Steps for Angle Bisection
We will construct the equal halves without measuring the original angle. The compass first gives two equal distances from the vertex. A second pair of equal distances then determines the bisector direction.
- Let O be the vertex, with the angle formed by two rays.
- With centre O, draw an arc crossing the rays at A and B. The one compass opening gives OA = OB.
- Choose a suitable radius and draw equal-radius arcs from A and B. Select their intersection C inside the given angle.
- Draw the ray OC. This is the internal angle bisector.
Why the Bisector Works
The arcs provide equal lengths, and congruence turns those length relationships into an angle relationship. This is why the method works for a narrow or wide angle, not just for the particular angle in a picture.
Compare triangles OAC and OBC. OA = OB because A and B lie on the same arc centred at O. AC = BC because C is the intersection of equal-radius arcs centred at A and B. OC is a side of both triangles. These three equalities give SSS congruence. Consequently ∠AOC = ∠BOC, which is precisely the required bisection.
The opening used for the second pair of arcs need not equal OA. It only needs to be the same from A and B and large enough to give a suitable intersection. Keep the first two points equally far from O; unequal OA and OB would remove one of the three side equalities.
Problem
An angle has measure 74°. What are its equal halves, and how would you construct them?
- 1.The two equal parts must add to 74°, so each is 74° ÷ 2 = 37°.
- 2.For the construction, draw one arc centred at the vertex to mark equal distances on the two rays.
- 3.Draw equal-radius arcs from those marked points and join the vertex to the interior crossing.
- 4.SSS congruence guarantees the halves are equal. The calculation tells us their measure, but the compass construction does not require a protractor.
Constructing a 45° Angle
A 45° angle can be produced from a right angle that has already been constructed exactly. This combines two methods: construct the perpendicular, then bisect the resulting right angle. The order matters because bisection needs a starting angle.
Problem
Construct a 45° angle and then halve it again.
- 1.Use the perpendicular construction to obtain 90°.
- 2.Bisect the right angle to obtain two angles of 90° ÷ 2 = 45°.
- 3.Apply the same bisection method to one 45° angle. Each half is 45° ÷ 2 = 22.5°.
- 4.Each new direction is justified by the same equal-side argument; it is not drawn by eye.
Petal Designs and Further Exploration
The support directions organise the design; the arcs form its visible boundary. Keeping these roles separate helps you construct an equal pattern instead of tracing a flower freehand. First establish the directions, then add matching curved shapes between them.
Draw a pair of perpendicular lines through the centre to give four right-angle sectors. Bisect each sector to obtain eight rays. Choose equal petal lengths along the rays and use the same compass opening for corresponding arcs. Repeat the same unit in each position.
For petals inside a given square, locate its centre using the diagonals and identify the symmetric support points. The edges limit how far the petals can extend. The largest version of the source design is obtained by using the appropriate side-midpoint arc centres and extending each corresponding arc until it reaches the prescribed corner and centre. The diagram below shows the intended pattern; do not enlarge one petal independently of the others.
Problem
Equal-radius arcs from A and B meet on the opposite side too. What happens if we join that point to O?
- 1.Every intersection of the equal-radius circles is equidistant from A and B. Both intersections therefore lie on the perpendicular bisector of AB.
- 2.O also satisfies OA = OB, so it lies on that same line.
- 3.The full line through O and the intersections is the bisector line.
- 4.However, a ray drawn away from the inside of the original angle is not its internal bisector ray. Use the ray pointing into the angle for the equal halves.
Knowing how to bisect a supplied angle does not mean every stated numerical angle can be constructed from a line using these methods. For example, 65.5° would be half of 131°, but you would still need an exact construction of 131° first. A protractor drawing and a ruler-and-compass construction are different tasks.
A rope can also locate the bisector. Mark A and B equally far from the vertex, attach equal usable rope lengths at A and B, and find a taut meeting point inside the angle. That point is equally far from A and B, reproducing the same SSS reasoning. Try four different given angles and compare their bisectors.
Check Your Understanding
Use these questions to check the conditions behind each method, as well as the result. For a diagram-based question, follow the labels and given information rather than judging by appearance.
Quiz
How large is each of eight equal angles around a point?
Which equalities justify the angle-bisector construction?
What does the ray OC do when C is the interior crossing?
What is half of 45°?
If a line contains the internal angle bisector, is its opposite ray also the internal bisector?
Practice Problems
- Draw four different angles and construct their internal bisectors.
- Construct a right angle and use bisection to obtain 45°.
- An angle of 118° is bisected. Find each half and explain why equal halves are required.
- Construct the eight-petal design using equal support directions.
- Explain how you could adapt the equal-distance idea to bisect an angle using a rope.
- Recreate the square-petal design and explain which geometric boundaries limit the petal size.
- Why is saying “halve 131°” not by itself a complete exact construction of 65.5° from a line?
Use a single vertex-centred arc to establish equal distances on both rays, then equal arcs from those points. Choose the intersection inside each angle.
Key Takeaways
• An angle bisector divides an angle into equal parts from its vertex. • Equal distances locate congruent triangles that justify bisection. • SSS gives equality of the two corresponding angles at the vertex. • Bisecting a constructed 90° angle gives 45°. • The internal ray points into the original angle, even though its supporting line extends both ways. • Repeated equal directions support symmetric petal designs.