6.1.1 Properties and Attributes of Polygons (continued)
The sum of the interior angles of a hexagon is determined in this example. The hexagon has 6 sides. According to the Polygon Angle Sum Theorem, the sum of the interior angles is ( n − 2)180°. Substitute 6 for n and simplify. The sum of the interior angles of a hexagon is 720°.
The measure of the interior angles of a regular octagon is determined in this example. The octagon has 8 sides. According to the Polygon Angle Sum Theorem, the sum of the interior angles is ( n − 2)180°. Substitute 8 for n and simplify. The sum of the interior angles of the octagon is 1080°. Since the octagon is regular, all the interior angles are congruent. The size of each angle is 1080°/8 = 135°. The measure of the interior angles of a quadrilateral is determined in this example. The measures of the angles of the quadrilateral are given as multiples of an unknown, x . According to the Polygon Angle Sum Theorem, the sum of the interior angles is ( n − 2)180°. Equate this value with the sum of the expressions we are given for the angle measures. Substitute 4 for n and solve for x . The value of x is 36°. This is the measure of angles P and R . The other two angles measure 4 x , or 144°.
Example 4 Finding Exterior Angle Measures in Polygons
According to the Polygon Exterior Angle Sum Theorem, the sum of the exterior angle measures of a convex polygon is 360°. Only one exterior angle per vertex is counted. The angle is made by extending the line segment that forms a side of the polygon.
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