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Properties and Angle Attributes of Polygons 6.1 Polygon Sum Theorems I can classify polygons based on their sides and angles. Do Now: 1. A ? is a three-sided polygon. 2. A ? is a four-sided polygon. Success Criteria: I can identify sum of interior angles I can identify polygons by sides and angles triangle quadrilateral Today’s Agenda New seats Do now Lesson HW # 6.1 Polygon Sum Theorems Properties andAngle Attributes of Polygons I can classify polygons based on their sides and angles. Do Now: 1. Name the polygon by the number of its sides. Then tell whether the polygon is regular or irregular, concave or convex. nonagon; irregular; concave 2. Find the sum of the interior angle measures of a convex 11-gon. 1620° Success Criteria: I can identify sum of interior angles I can identify polygons by sides and angles Today’s Agenda Do Now Check HW #42 Test corrections No homework Polygon – A closed plane formed by or more Properties and figure Attributes of 3Polygons segments Each segment that forms a polygon is a side of the polygon. The common endpoint of two sides is a vertex of the polygon. A segment that connects any two nonconsecutive vertices is a diagonal. Properties and You can name a polygon by the number of its sides. The table shows the names of some common polygons. Attributes of Polygons Remember! A polygon is a closed plane figure formed by three or more segments that intersect only at their endpoints. Example 1A: Identifying Polygons Properties and Attributes of Polygons Discuss with A/B partner whether the figure is a polygon. If it is a polygon, name it by the number of sides. polygon, hexagon not a polygon polygon, heptagon not a polygon not a polygon Properties and Attributes of Polygons All the sides are congruent in an equilateral polygon. All the angles are congruent in an equiangular polygon. A regular polygon is one that is both equilateral and equiangular. If a polygon is not regular, it is called irregular. Properties and Attributes of Polygons 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. Checkand It Out! Example 2aof Polygons Properties Attributes Discuss with A/B partner whether the polygon is regular or irregular and whether it is concave or convex. regular, convex irregular, concave Properties and Attributes of Polygons To find the sum of the interior angle measures of a convex polygon, draw all possible diagonals from one vertex of the polygon. This creates a set of triangles. The sum of the angle measures of all the triangles equals the sum of the angle measures of the polygon. Properties and Attributes of Polygons 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°. Properties and Attributes of Polygons Example 3A: Finding Interior Angle Measures and Sums in Polygons Find the sum of the interior angle measures of a convex heptagon (7-sides). (n – 2)180° Polygon Sum Thm. (7 – 2)180° A heptagon has 7 sides, so substitute 7 for n. 900° Simplify. Properties and Attributes of Polygons Example 3B: Finding Interior Angle Measures and Sums in Polygons Find the measure of each interior angle of a regular 16-gon. Step 1 Find the sum of the interior angle measures. (n – 2)180° Polygon Sum Thm. (16 – 2)180° = 2520° Substitute 16 for n and simplify. Step 2 Find the measure of one interior angle. The int. s are , so divide by 16. Properties and Attributes of Polygons Example 3C: Finding Interior Angle Measures and Sums in Polygons Find the measure of each interior angle of pentagon ABCDE. (5 – 2)180° = 540° Polygon Sum Thm. Polygon mA + mB + mC + mD + mE = 540° Sum Thm. 35c + 18c + 32c + 32c + 18c = 540 135c = 540 c=4 Substitute. Combine like terms. Divide both sides by 135. Properties and Attributes of Polygons Example 3C Continued mA = 35(4°) = 140° mB = mE = 18(4°) = 72° mC = mD = 32(4°) = 128° Properties and Attributes of Polygons 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°. Properties and Attributes of Polygons Remember! An exterior angle is formed by one side of a polygon and the extension of a consecutive side. Properties and Attributes of Polygons Example 4A: Finding Interior Angle Measures and Sums in Polygons Find the measure of each exterior angle of a regular 20-gon. A 20-gon has 20 sides and 20 vertices. sum of ext. s = 360°. measure of one ext. = Polygon Sum Thm. A regular 20-gon has 20 ext. s, so divide the sum by 20. The measure of each exterior angle of a regular 20-gon is 18°. Properties and Attributes of Polygons Example 5: Art Application 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. Properties and Attributes of Polygons Assignment #2 pg 356 #7 - 25 Properties and Attributes of Polygons Example 4B: Finding Interior Angle Measures and Sums in Polygons Find the value of b in polygon FGHJKL. Polygon Ext. Sum Thm. 15b° + 18b° + 33b° + 16b° + 10b° + 28b° = 360° 120b = 360 b=3 Combine like terms. Divide both sides by 120. Properties Attributes of Polygons Exteriorand Angles Activity Get graph paper or Cut out each exterior scratch paper. angle Using a straight edge Put the colored angles draw a polygon, each together person in your group must have different amount of sides. What happened? Discuss who will draw each shape. It does not What about the other polygons? have to be regular What can you conclude After the shape is about the exterior angle drawn extend each consecutive side. of polygons? 3 sides? 4 sides? 5 sides? Highlight the exterior angle Properties and Attributes of Polygons Exterior Angles Activity cont. Cut out each exterior angle Put the colored angles together What happened? What about the other polygons? What can you conclude about the exterior angle of polygons? 3 sides? 4 sides? 5 sides? Properties and Attributes of Polygons Take out your Polygon Exploration worksheet. With your group yesterday what observations can you make about the degrees of a polygon? The person with their birthday closest to today will write your groups conclusion or findings from your Polygon Exploration. Properties and Attributes of Polygons Do Now 1. Name the polygon by the number of its sides. Then tell whether the polygon is regular or irregular, concave or convex. nonagon; irregular; concave 2. Find the sum of the interior angle measures of a convex 11-gon. 1620° 3. Find the measure of each interior angle of a regular 18-gon. 160° 4. Find the measure of each exterior angle of a regular 15-gon. 24° Properties and Attributes of Polygons