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Polygon Exterior Angles

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Polygon Exterior Angles

A coastal walkway turns at each vertex of a polygonal route. At every turn, the direction changes by an exterior angle. Use the facts that a complete turn is 360° and that an interior angle and its adjacent exterior angle add to 180°. Show calculations where asked.

Understand the turns

Explore how exterior angles add up and use the pattern to solve route-design problems.

1.A polygon is traced once, always turning in the same direction at each vertex. What is the sum of its exterior angles?

180°
360°
540°
It depends on the number of sides

2.A regular pentagonal lookout has equal exterior angles. Calculate the size of one exterior angle.

3.At one corner of a walkway, the interior angle is 147°. Find the adjacent exterior angle and explain the relationship you used.

4.A four-sided route has exterior angles of 82°, 103° and 91° at three turns. Find the exterior angle at the fourth turn.

5.Which combination correctly identifies all the true statements about convex polygons and their exterior angles, taking one exterior angle at each vertex in the same direction? The statements are: (1) Each exterior angle and its adjacent interior angle add to 180°. (2) The exterior angles always have equal sizes. (3) Their sum is 360°. (4) A regular polygon has equal exterior angles.

Statements 1, 2 and 3 are true.
Statements 1, 3 and 4 are true.
Statements 2, 3 and 4 are true.
All four statements are true.

Design and reason

Use the turning-angle rule to analyse larger polygonal routes.

6.A regular octagonal route has the same turn at every vertex. Calculate each exterior angle, then calculate each interior angle.

7.A six-sided route has five exterior angles measuring 48°, 67°, 72°, 81° and 39°. Calculate the sixth exterior angle.

8.A regular polygon has an exterior angle of 24°. How many sides does it have? Name the polygon.

9.Write a formula for the size E of each exterior angle of a regular polygon with n sides. Then use it to find E when n = 12.

10.A student says, “If a convex polygon has one exterior angle of 120°, it must be a regular triangle.” Is this conclusion valid? Explain what can and cannot be inferred.

11.Draw an irregular convex quadrilateral. Extend one side at each vertex, mark the four exterior angles in the same turning direction, and label them so their total is 360°.