Ch 3.5Standard 12.0:

Students find and use measures of interior and exterior angles of triangles to classify figures and solve problems.

Standard 13.0Students prove relationships between angles in polygons by using properties of complementary, supplementary,

vertical, and exterior angles.

Objective:1) To find the sums of the measures of the

interior and exterior angles of polygons.

DefinitionsA polygon is a closed plane figure formed by three or more segments that intersect only at their endpoints.A polygon is concave if any part of a diagonal contains points in the exterior of the polygon.

If no diagonal contains points in the exterior, then the polygon is convex. A regular polygon is always convex.

Theorem

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Example 1Find the value of b in polygon FGHJKL.

15b° + 18b° + 33b° + 16b° + 10b° + 28b° = 360° Polygon Ext. Sum Thm.

120b = 360 Combine like terms.

b = 3 Divide both sides by 120.

Example 2Find the measure of each interior angle of pentagon ABCDE.

(5 – 2)180° = 540°

Polygon Sum Thm.mA + mB + mC + mD + mE = 540°35c + 18c + 32c + 32c + 18c = 540 Substitute.

135c = 540 Combine like terms.c = 4 Divide both sides by 135.

mA = 35(4°) = 140°

mB = mE = 18(4°) = 72°

mC = mD = 32(4°) = 128°

Theorem

In the polygons below, an exterior angle has been measured at each vertex. Notice that in each case, the sum of the exterior angle measures is 360°.

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Example 3Find the value of r in polygon JKLM.

4r° + 7r° + 5r° + 8r° = 360° Polygon Ext. Sum Thm.

24r = 360 Combine like terms.

r = 15 Divide both sides by 24.

Example 4

Ann is making paper stars for party decorations. What is the measure of 1?

1 is an exterior angle of a regular pentagon. By the Polygon Exterior Angle Sum Theorem, the sum of the exterior angles measures is 360°.

A regular pentagon has 5 ext. , so divide the sum by 5.