6.1 PolygonsGeometry

Objectives:Identify, name, and describe polygons such as the building shapes in Example 2.Use the sum of the measures of the interior angles of a quadrilateral.

Assignmentspp. 325-327 # 4-46 allDefinitionsPostulates/Theorems

Definitions:Polygona plane figure that meets the following conditions:It is formed by 3 or more segments called sides, such that no two sides with a common endpoint are collinear.Each side intersects exactly two other sides, one at each endpoint.Vertex each endpoint of a side. Plural is vertices. You can name a polygon by listing its vertices consecutively. For instance, PQRST and QPTSR are two correct names for the polygon above.SIDE

Example 1: Identifying PolygonsState whether the figure is a polygon. If it is not, explain why.Not D has a side that isnt a segment its an arc.Not E because two of the sides intersect only one other side.Not F because some of its sides intersect more than two sides/Figures A, B, and C are polygons.

Polygons are named by the number of sides they have MEMORIZE

Number of sidesType of Polygon3Triangle4Quadrilateral5Pentagon6Hexagon7Heptagon

Polygons are named by the number of sides they have MEMORIZE

Number of sidesType of Polygon8Octagon9Nonagon10Decagon12Dodecagonnn-gon

Convex or concave? Convex if no line that contains a side of the polygon contains a point in the interior of the polygon.Concave or non-convex if a line does contain a side of the polygon containing a point on the interior of the polygon.See how it doesnt go on theInside-- convexSee how this crossesa point on the inside?Concave.

Convex or concave?Identify the polygon and state whether it is convex or concave.A polygon is EQUILATERALIf all of its sides are congruent.A polygon is EQUIANGULARif all of its interior angles are congruent. A polygon is REGULAR if it isequilateral and equiangular.

Identifying Regular PolygonsRemember: Equiangular & equilateralDecide whether the following polygons are regular.Equilateral, but not equiangular, so it is NOT a regular polygon.Heptagon is equilateral, but not equiangular, so it is NOT a regular polygon.Pentagon is equilateral and equiangular, so it is a regular polygon.

Interior angles of quadrilateralsA diagonal of a polygon is a segment that joins two nonconsecutive vertices. Polygon PQRST has 2 diagonals from point Q, QT and QSdiagonals

Interior angles of quadrilateralsLike triangles, quadrilaterals have both interior and exterior angles. If you draw a diagonal in a quadrilateral, you divide it into two triangles, each of which has interior angles with measures that add up to 180. So you can conclude that the sum of the measures of the interior angles of a quadrilateral is 2(180), or 360.

Theorem 6.1: Interior Angles of a QuadrilateralThe sum of the measures of the interior angles of a quadrilateral is 360.m1 + m2 + m3 + m4 = 360

Ex. 4: Interior Angles of a QuadrilateralFind mQ and mR.Find the value of x. Use the sum of the measures of the interior angles to write an equation involving x. Then, solve the equation. Substitute to find the value of R.80702xxx+ 2x + 70 + 80 = 360

Ex. 4: Interior Angles of a Quadrilateralx+ 2x + 70 + 80 = 360

3x + 150 = 3603x = 210 x = 7080702xxSum of the measures of int. s of a quadrilateral is 360Combine like termsSubtract 150 from each side.Divide each side by 3.Find m Q and mR.mQ = x = 70mR = 2x= 140So, mQ = 70 and mR = 140

Reminder:Quiz after 6.3 and 6.5. Definitions 20 point assignment due by Friday this week. (5th period Thursday)Postulates the green boxes20 point assignment due by Friday this week. (5th period Thursday)