Presentation on theme: "3.6 Angles in Polygons Objectives: Warm-Up:"— Presentation transcript:
1 3.6 Angles in Polygons Objectives: Warm-Up: Develop and use formulas for the sums of themeasures of interior and exterior angles of polygonsWarm-Up:Here’s a two part puzzle designed to prove that half of eleven is six. First rearrange two sticks to reveal the number eleven. Then remove half of the sticks to reveal the number six.
2 Convex Polygon:A polygon in which any line segment connecting two points of the polygon has no part outside the polygon.
4 Consider the following Pentagon: 𝟐𝟑𝟒𝟏Divide the polygon into three triangular regions by drawing all the possible diagonals from one vertex.𝟓𝟗𝟔𝟕𝟖Add the three expressions:Find each of the following:𝒎<𝟏+𝒎<𝟐+𝒎<𝟑= ______𝒎<𝟒+𝒎<𝟓+𝒎<𝟔= ______𝒎<𝟕+𝒎<𝟖+𝒎<𝟗= ______
5 Note: Polygon Triangle 3 1 180 Quadrilateral 4 2 360 Pentagon 5 3 540 You can form triangular regions by drawing all possible diagonals from a given vertex of any polygon# oftriangularregionsSum ofInterioranglesPolygon# ofsidesTriangle31180Quadrilateral42360Pentagon53540Hexagon64720n-gonnn−2180(n-2)
6 The sum of the measures of the interior angles of a polygon with n sides is:
7 Note: Polygon Triangle 3 180 60 Quadrilateral 4 360 90 Pentagon 5 540 Recall that a regular polygon is on in which all the angles are congruent.Sum ofInterioranglesMeasureof InterioranglesPolygon# ofsidesTriangle318060Quadrilateral436090Pentagon5540108Hexagon6720120n-gonn180(n-2)n180(n-2)
8 The measure of an Interior Angle of a Regular Polygon with n sides is:
11 TheoremSum of the measures of the Exterior Angles of a Polygon is:𝟑𝟔𝟎 𝟎
12 For a Convex PolygonFor a Regular PolygonPolygonNumberofSidesNumber of ΔRegionsSum of Interior AnglesSum of Exterior AnglesSum of Int & Ext AnglesMeasure ofInterior AnglesMeasureExterior AnglesTriangle3118036054060120Quadrilateral4272090Pentagon590010872Hexagon61080Heptagon71260128.651.4Octagon8144013545Nonagon9162014040Decagon1018001443611-gon111980147.332.7Dodecagon1221601503013-gon132340152.327.2n-gonnn-2180(n-2)180n𝟏𝟖𝟎(𝐧−𝟐) 𝐧𝟏𝟖𝟎− 𝟏𝟖𝟎(𝐧−𝟐) 𝐧
17 A rectangle An equilateral triangle A regular dodecagon An equiangular For each polygon determine the measure of an interior angle and the measure of an exterior angle.A rectangleAn equilateraltriangleA regulardodecagonAn equiangularpentagon
18 An interior angle measure of a regular polygon is given An interior angle measure of a regular polygon is given. Find the number of sides of the polygon𝟏𝟑𝟓 𝟎𝟏𝟓𝟎 𝟎𝟏𝟔𝟓 𝟎
19 An exterior angle measure of a regular polygon is given An exterior angle measure of a regular polygon is given. Find the number of sides of the polygon𝟔𝟎 𝟎𝟑𝟔 𝟎𝟐𝟒 𝟎