10 1 Polygons Geometry Objectives Identify name and

10. 1 Polygons Geometry

Objectives: • Identify, name, and describe polygons • Use the sum of the measures of the interior and exterior angles.

Example 1: Identifying Polygons • State whether the figure is a polygon. If it is not, explain why. • Not D – has a side that isn’t a segment – it’s 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 sides Type of Polygon 3 Triangle 4 Quadrilateral 5 Pentagon 6 Hexagon 7 Heptagon

Polygons are named by the number of sides they have – MEMORIZE Number of sides Type of Polygon 8 Octagon 9 Nonagon 10 Decagon 12 Dodecagon n n-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 this crosses a point on the inside? Concave. See how it doesn’t go on the Inside-- convex

Convex or concave? • Identify the polygon and state whether it is convex or concave. A polygon is EQUILATERAL If all of its sides are congruent. A polygon is EQUIANGULAR if all of its interior angles are congruent. A polygon is REGULAR if it is equilateral and equiangular.

Identifying Regular Polygons • Remember: Equiangular & equilateral • Decide 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 quadrilaterals • The sum of the measures of the interior angles of a quadrilateral is 2(180°), or 360°.

Theorem 6. 1: Interior Angles of a Quadrilateral • The sum of the measures of the interior angles of a quadrilateral is 360°. m 1 + m 2 + m 3 + m 4 = 360°

Ex. 4: Interior Angles of a Quadrilateral • Find m Q and m R. • 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. 80° 70° x° 2 x° x°+ 2 x° + 70° + 80° = 360°

Ex. 4: Interior Angles of a Quadrilateral x° x°+ 2 x° + 70° + 80° = 360° 3 x + 150 = 360 3 x = 210 x = 70 80° 70° 2 x° Sum of the measures of int. s of a quadrilateral is 360° Combine like terms Subtract 150 from each side. Divide each side by 3. Find m Q and m R. m Q = x° = 70° m R = 2 x°= 140° ►So, m Q = 70° and m R = 140°

Practice Together

= 157. 5 180 s – 360 = 157. 5 s 22. 5 s = 360 S = 16