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Properties and Attributes of Polygons Chapter 6 - Lesson 1 Properties and Attributes of Polygons

Objective: The students will be able to find the measures of interior and exterior angles of polygons.

Names of Polygons Some polygons have special names A polygon with n sides is an n-gon

Definition Diagonal a segment that connects two nonconsecutive vertices of a polygon Consecutive means they are next to each other. So, nonconsecutive means they are not next to each other

Definition Interior Angle [of a polygon] angles inside the polygon that are formed by 2 sides of a polygon Ex. This Pentagon has 5 interior angles

To find the sum of the interior angle measures of a convex polygon, cut the polygon up into triangles. You must start at a vertex and not cross any diagonals. By the Triangle Sum Theorem, the sum of the interior angle measures of a triangle is 180°. Remember! 180 o 360 o 540 o o 720

In each convex polygon, the number of triangles formed is two less than the number of sides n. So the sum of the angle measures of all these triangles is (n — 2)180°.

Theorem 6.1 Polygon Angle Sum Theorem If a convex polygon has n sides and the sum of the measures of the interior angles = S, then S = (n - 2)180

Polygon Angle Sum Theorem ARCHITECTURE A mall is designed so that five walkways meet at a food court that is in the shape of a regular pentagon. Find the sum of measures of the interior angles of the pentagon. Polygon Angle Sum Theorem Answer: The sum of the measures of the angles is 540. Example 1-1a

Polygon Angle Sum Theorem A decorative window is designed to have the shape of a regular octagon. Find the sum of the measures of the interior angles of the octagon. Polygon Angle Sum Theorem 8 8 6 Answer: 1080 Example 1-1c

Polygon Angle Sum Theorem The measure of an interior angle of a regular polygon is 135. Find the number of sides in the polygon. Polygon Angle Sum Theorem Substitution Distributive Property - 45n = - 360 Subtraction n = 8 Division Answer: The polygon has 8 sides. Example 1-2a

Answer: The polygon has 10 sides. The measure of an interior angle of a regular polygon is 144. Find the number of sides in the polygon. Answer: The polygon has 10 sides. Example 1-2b

Find the measure of each interior angle. Use the value of x to find the measure of each angle. Answer: Example 1-3a

Find the measure of each interior angle. Answer: Example 1-3d

Definition Exterior Angle [of a polygon] the angles formed between the extended side of a polygon and an adjacent side

Theorem 6.2 Exterior Angle Sum If a polygon is convex then the sum of the exterior angles is 360

The sum of the measures of the exterior angles is 360. Find the measures of an exterior angle and an interior angle of convex regular nonagon ABCDEFGHJ. The sum of the measures of the exterior angles is 360. Answer: The measure of each exterior angle is 40. The measure of the interior angle is 180 – 40 or 140. Example 1-4a

Find the measures of an exterior angle and an interior angle of convex regular hexagon ABCDEF. Answer: 60; 120 Example 1-4c

Kahoot!

Lesson Summary: Objective: The students will be able to find the measures of interior and exterior angles of polygons.

Preview of the Next Lesson: Objective: The students will be able to prove and apply properties of parallelograms to solve problems.

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