Back to Ganita Prakash: Class 7 Mathematics, Par...BackConcept

Bisecting Angles: 45° And The Eight-Petalled Flower

Bisect any angle using SSS congruence, and use repeated bisection to build 45° designs like the eight-petalled flower.

7 micro-lessons

Preview · progress not saved — Chapter 1 is free

Two Congruent Triangles Split an Angle

Make OA = OB and CA = CB, and OC bisects the angle.

Why we need it (Textbook p. 143)

The 8-petalled flower (Fig. 6.5) needs 8 support lines with equal angles between them: $360° \div 8 = 45°$ . 45° is half of 90°, so we need to bisect an angle.

1 / 3

Constructing a 45° Angle

Build 90°, then bisect it.

Problem. Construct a 45° angle with a ruler and compass. (Textbook p. 144)

1 / 5

Angle Bisection Card

Three steps and the reason.

Bisecting an angle

O A = OB, C A = CB \Rightarrow ∠ A OC = ∠ BOC

What each part means

Step 1

— arc from O gives A and B with OA = OB

Step 2

— same radius from A and B gives C

Step 3

— join OC

When it works

Reason: ΔOBC ≅ ΔOAC by SSS. Starting from 90° and 60°, bisecting gives 45°, 22.5°, 30°, 15°, 7.5°, and sums such as 67.5° = 45° + 22.5°.

Example

Bisect 90° → 45°; bisect 45° → 22.5°.

Angles from Bisection

Which angles can we make, and why?

1 / 4

Which of these can be made from 90° and 60° by bisecting (and adding)? (Textbook p. 144, Q4)

The Two Petal Designs

The 8-petalled flower and the petals in a square, for the next task.

The 8-petalled flower (Fig. 6.5, Textbook p. 144)

1 / 2

Petals in a Square

Construct the 8-petalled flower and the petals in a square.

Construct the designs

Figure it Out, Textbook p. 144, Q1, Q2 and Q6.

Construct at least 4 different angles, and draw their bisectors.
Construct the 8-petalled flower (Fig. 6.5) shown above.
Construct the petals inside a square, as large as possible, as shown above.

Your four angles: did each bisector look right? How did you check?

The 8-petalled flower: your support lines and how you made 45°

Start from 90°.

Petals in a square: where are the arc centres?

Which points of the square do the arcs pass through?

Bisect with a Rope

Invent a rope method to bisect an angle.

Bisect an angle with a rope

Figure it Out, Textbook p. 144, Q5. Come up with a method to construct the angle bisector using a rope. Explain why it works.

Your method

How do you get equal distances on both arms?

Why it works

Which congruence condition?

← previousRight Angles Anywhere: At A Point, From ...next →Copying Angles And Drawing Parallel Line...

Two Congruent Triangles Split an Angle

Make OA = OB and CA = CB, and OC bisects the angle.

Why we need it (Textbook p. 143)

The 8-petalled flower (Fig. 6.5) needs 8 support lines with equal angles between them: $360° \div 8 = 45°$ . 45° is half of 90°, so we need to bisect an angle.

1 / 3