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Warm Up True or false? 1. Some trapezoids are parallelograms. 2. Some figures with 4 right angles are squares. 3. Some quadrilaterals have only one right angle. false true true

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Learn to identify regular and not regular polygons and to find the angle measures of regular polygons.

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Vocabulary polygon regular polygon diagonal

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A polygon is a closed plane figure formed by three or more line segments. A regular polygon is a polygon in which all sides are congruent and all angles are congruent. Polygons are named by the number of their sides and angles.

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**Additional Example 1A: Identifying Polygons**

Tell whether each shape is a polygon. If so, give its name and tell whether it appears to be regular or not regular. The shape is a closed plane figure formed by 3 or more line segments. polygon There are 5 sides and 5 angles. pentagon All 5 sides do not appear to be congruent. not regular

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**Additional Example 1B: Identifying Polygons**

Tell whether each shape is a polygon. If so, give its name and tell whether it appears to be regular or not regular. The shape is a closed plane figure formed by 3 or more line segments. polygon There are 8 sides and 8 angles. octagon The sides and angles appear to be congruent. regular

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Check It Out: Example 1A Tell whether each shape is a polygon. If so, give its name and tell whether it appears to be regular or not regular. There are 4 sides and 4 angles. quadrilateral The sides and angles appear to be congruent. regular

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Check It Out: Example 1B Tell whether each shape is a polygon. If so, give its name and tell whether it appears to be regular or not regular. There are 4 sides and 4 angles. quadrilateral All 4 sides do not appear to be congruent. not regular

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You can divide any quadrilateral into two triangles by drawing a diagonal, a line segment that connects two non-adjacent vertices. The sum of the interior angle measures in a triangle is 180°, so the sum of the interior angle measures in a quadrilateral is 360°.

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**Understand the Problem**

Additional Example 2: Problem Solving Application Malcolm designed a wall hanging that was a regular 9-sided polygon (called a nonagon). What is the measure of each angle of the nonagon? 1 Understand the Problem The answer will be the measure of each angle in a nonagon. List the important information: A regular nonagon has 9 congruent sides and t 9 congruent angles.

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**Additional Example 2 Continued**

Make a Plan Make a table to look for a pattern using regular polygons. Solve 3 Draw some regular polygons and divide each into triangles.

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**Additional Example 2 Continued**

720° The prefixes in the names of the polygons tell you how many sides and angles there are. tri- = three quad- = four penta- = five hexa- = six octa- = eight Reading Math

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**Additional Example 2 Continued**

Solve Cont. 3 The number of triangles is always 2 fewer than the number of sides. A nonagon can be divided into 9 – 2 = 7 triangles. The sum of the interior angle measures in a nonagon is 7 180° = 1,260°. So the measure of each angle is 1,260° ÷ 9 = 140°.

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**Additional Example 2 Continued**

4 Look Back Each angle in a nonagon is obtuse. 140° is a reasonable answer, because an obtuse angle is between 90° and 180°.

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**Understand the Problem**

Check It Out: Example 2 Sara designed a picture that was a regular 6-sided polygon (called a hexagon). What is the measure of each angle of the hexagon? 1 Understand the Problem The answer will be the measure of each angle in a hexagon. List the important information: A regular hexagon has 6 congruent sides and 6 congruent angles.

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**Check It Out: Example 2 Continued**

Make a Plan Make a table to look for a pattern using regular polygons. Solve 3 Draw some regular polygons and divide each into triangles.

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**Check It Out: Example 2 Continued**

Solve Cont. 3 The number of triangles is always 2 fewer than the number of sides A hexagon can be divided into 6 – 2 = 4 triangles. The sum of the interior angles in a octagon is 180° = 720°. So the measure of each angle is 720° ÷ 6 = 120°.

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**Check It Out: Example 2 Continued**

4 Look Back Each angle in a hexagon is obtuse. 120° is a reasonable answer, because an obtuse angle is between 90° and 180°.

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Lesson Quiz 1. Name each polygon and tell whether it appears to be regular or not regular. 2. What is the measure of each angle in a regular dodecagon (12-sided figure)? nonagon, regular; octagon, not regular 150°

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