Geometry/Trigonometry Unit 5: Polygon Notes Period:

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1 Geometry/Trigonometry Unit 5: Polygon Notes Name: Date: Period: # (1) Page #8 14 Even, #15 20, #27-32 (2) Page , #11 25 Odd (3) Page #12 30 Even (4) Page 283 #1-14 All (5) Page #15 24; FF #32, 34 (6) Page 290 #1-10 (7) Page 290 #11 26, Page 292 #39-42 (8) Page 296 #1-16 (9) Page #17 24, #34, 36 and 38 (10) Page 302 #1-14 (11) Page #15-26 (12) Page 309 #1-12 (13) Page #13 22, 27, 28, (14) Page 317 Chapter 6 Test

2 Geometry Notes 6.1 Exploring Polygons Polygon is formed by called such that the following are true: Each side intersects with, once at each. with a common endpoint are. if no line that contains a side of the polygon contains a point in of the polygon. one that is not convex. They are classified by the that they have. Every polygon has vertices, an interior, interior angles, an exterior, exterior angles, a perimeter and an area. This is polygon or polygon or polygon or Diagonal a segment that vertices. Equilateral of the are Equiangular of the are both equilateral and equiangular

3 Geometry Notes 6.2 Angles of Polygons Theorem 6.1 Polygon Interior Angles Theorem: The sum of the measures of the interior angles of a convex n-gon is Corollary to Theorem 6.1: The measure of each interior angle of a regular n-gon is Theorem 6.2 Polygon Exterior Angles Theorem: The sum of the measures of the exterior angles, one from each vertex, of a convex polygon is Corollary to Theorem 6.2: The measure of each exterior angle of a regular n-gon is To find the number of sides of a regular polygon, and are given (a) The measure of each interior angle (1) Find an (2) that number into (Ex1) If each interior angle is 135 degrees, how many sides does the regular polygon have? Solution: = /45 = 8 therefore it is an 8 sided figure (b) The measure of each exterior angle (1) that number into (Ex2) If each exterior angle is 30 degrees how many sides does the regular polygon have? Solution: 360/30 = 12 therefore it is a 12 sided figure To find the measure of each exterior angle in a regular polygon, and are given (a) The number of sides (1) that number into (Ex3) Find the measure of each exterior angle in a regular polygon if it has 20 sides. Solution: 360/20 = 18 therefore each exterior angle is 18 degrees

4 To find the measure of each interior angle in a regular polygon, and are given (a) The number of sides (1) that number into (2) that answer from (Ex4) Find the measure of each interior angle in a regular polygon if it has 15 sides. Solution: 360/15 = 24 so each exterior angle is 24 degrees = 156, so each interior angle is 156 degrees (P1) You are given the measure of each interior angle in a regular polygon. How many sides does the polygon have? (a) (b) (c) (P2) You are given the measure of each exterior angle in a regular polygon. How many sides does the polygon have? (a) (b) (c) (P3) You are given the number of sides in a regular polygon. Find the measure of each exterior angle of the polygon. (a) 10 (b) 20 (c) 36

5 (P4) You are given the number of sides in a regular polygon. Find the measure of each interior angle of the polygon. (a) 8 (b) 12 (c) 14 Geometry Notes 6.3 Properties of Parallelograms Special Quadrilaterals Parallelogram a whose are. Properties of Parallelograms Theorem 6.3: If a quadrilateral is a parallelogram, then its opposite sides are congruent. Theorem 6.4: If a quadrilateral is a parallelogram, then its opposite angles are congruent. Theorem 6.5: If a quadrilateral is a parallelogram, then its consecutive angles are supplementary. Theorem 6.6 Diagonals of a Parallelogram: If a quadrilateral is a parallelogram, then the diagonals bisect each other.

6 Geometry Notes 6.4 Proving Quadrilaterals are Parallelograms Ways to prove are Theorem 6.7: If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. Theorem 6.8: If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. Theorem 6.9: If an angle of a quadrilateral is supplementary to both consecutive angles then the quadrilateral is a parallelogram. Theorem 6.10: If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. Theorem 6.11: If one pair of opposite sides of a quadrilateral are congruent and parallel, then the quadrilateral is a parallelogram.

7 Geometry Notes 6.5 Special Parallelograms Rhombus a with that has all Rectangle a that has Square a that is Theorem 6.12: A parallelogram is a rhombus if and only if its diagonals are perpendicular. Theorem 6.13: A parallelogram is a rhombus if and only if each diagonal bisects a pair of opposite angles. Theorem 6.14: A parallelogram is a rectangle if and only if its diagonals are congruent. Theorem 6.15: A quadrilateral is a rhombus if and only if it has four congruent sides.

8 Theorem 6.16: A quadrilateral is a rectangle if and only if it has four right angles. Geometry Notes 6.6 Trapezoids Trapezoid (a) (b) (c) (d) (e) (f) of opposite sides The are called the The sides are called the if the are there are connects the midpoints of its legs Base Angles Theorem 6.17 Trapezoid Base Angles Theorem: If a trapezoid is isosceles, then each pair of base angles are congruent. Theorem 6.18 Trapezoid Diagonals Theorem: If a trapezoid is isosceles, then its diagonals are congruent. Theorem 6.19: If a trapezoid has one pair of congruent base angles, then it is an isosceles trapezoid. Theorem 6.20: If a trapezoid has congruent diagonals, then it is an isosceles trapezoid.

9 Theorem 6.21 Midsegment Theorem for Trapezoids: The midsegment of a trapezoid is parallel to each base, and its length is half the sum of the bases. Geometry Notes 6.7 Congruence and Kites (Optional) Two are if their and are. Theorem 6.22 Congruence Theorem: If three sides and the included angles of one quadrilateral are congruent to the corresponding three sides and included angles of another quadrilateral, then the quadrilaterals are congruent. Theorem 6.23 Congruence Theorem: If three angles and the included sides of one quadrilateral are congruent to the corresponding three angles and included sides of another quadrilateral, then the quadrilaterals are congruent. Kite (a) Two pairs of, but opposite sides are Theorem 6.24: If a quadrilateral is a kite, then its diagonals are perpendicular. Theorem 6.25: If a quadrilateral is a kite, then exactly one pair of opposite angles are congruent.

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