Presentation on theme: "Geometry 6.1 Prop. & Attributes of Polygons"— Presentation transcript:
1 Geometry 6.1 Prop. & Attributes of Polygons 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.
2 3 Triangle 8 Octagon 4 9 Nonagon 5 Pentagon 10 Decagon 6 12 Hexagon You can name a polygon by the number of its sides. The table shows the names of some common polygons.# of sidesPolygon3Triangle8Octagon49NonagonQuadrilateral5Pentagon10Decagon612HexagonDodecagon7nHeptagonN-gon
3 A polygon is a closed plane figure formed by three or more segments that intersect only at their endpoints.Remember!Example 1Tell whether the figure is a polygon. If it is a polygon, name it by the number of sides.polygon, hexagonpolygon, heptagonnot a polygonpolygon, nonagonnot a polygon
4 All the sides are congruent in an equilateral polygon 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.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.
5 Example 2:Tell whether the polygon is regular or irregular. Tell whether it is concave or convex.irregular, convexregular, convexregular, convex
6 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 trianglesis (n — 2)180°.Example 3A:Find the sum of the interior angle measures of aconvex heptagon.(n – 2)180°Polygon Sum Thm.(7 – 2)180°A heptagon has 7 sides, so substitute 7 for n.900°Simplify.
7 Example 3B: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.Substitute 16 for n and simplify.(16 – 2)180° = 2520°Step 2 Find the measure of one interior angle.The int. s are , so divide by 16.
8 Find the measure of each interior angle of pentagon ABCDE. Example 3C:Find the measure of each interior angle of pentagon ABCDE.(5 – 2)180° = 540°Polygon Sum Thm.mA + mB + mC + mD + mE = 540°Polygon Sum Thm.35c + 18c + 32c + 32c + 18c = 540Substitute.135c = 540Combine like terms.c = 4Divide both sides by 135.mA = 35(4°) = 140°mB = mE = 18(4°) = 72°mC = mD = 32(4°) = 128°
9 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°.An exterior angle is formed by one side of a polygon and the extension of a consecutive side.Remember!
10 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°.Polygon Sum Thm.A regular 20-gon has 20 ext. s, so divide the sum by 20.measure of one ext. =The measure of each exterior angle of a regular 20-gon is 18°.
11 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 = 360Combine like terms.b = 3Divide both sides by 120.
12 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.