Check whether angles fit around a point using the 360° total, and see why six equilateral triangles make a regular hexagon.
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Angles fit around a point exactly when they add up to 360°.
A regular polygon has equal sides and equal angles. With 3 sides it is an equilateral triangle; with 4, a square. A regular pentagon needs ideas from later classes, but a regular hexagon is within our reach.
Finding the gap angle ∠AOI in the book's figure.
Problem. Around point O there are angles of 40°, 60°, 50°, 30°, 40° and 90°, and a gap ∠AOI. Will a 70° angle fit into the gap? (Textbook p. 152)
Missing angles around a point.
In the book's figure, what is the gap angle?
Why six equilateral triangles fit perfectly around a point.
Instead of cutting a hexagon into triangles, ask: can six congruent equilateral triangles be placed together around a point O (Fig. 6.12)? If so, do they make a regular hexagon?
Using the fact that the side equals the distance from the centre.
Problem. Construct a regular hexagon of side 4 cm using a ruler and a compass. (Textbook p. 152)
Reasoning about the regular hexagon.
Each angle of a regular hexagon is…