Lesson 7-R Chapter 7 Review

Angles in Convex Polygons Interior angle + exterior angle = 180° They are a Linear Pair Sum of Interior angles, S = (n-2)  180° One Interior angle = S / n = (n-2)  180°/n Sum of Exterior angles = 360° Number of sides, n = 360° / Exterior angle Exterior angle Interior angle

Example Problems 1 Find the sum of the interior angles in a 16-gon Sides Name Sum of Interior ’s One Interior  One Exterior  3 180 60 5 72 Heptagon 900 128.57 Find the sum of the interior angles in a 16-gon Find the sum of the exterior angles in a 16-gon Find the number of sides of a polygon if an interior angle is 140°.

Polygon Hierarchy Polygons Quadrilaterals Parallelograms Kites Sides Name 3 Triangle 4 Quadrilateral 5 Pentagon 6 Hexagon 7 Heptagon 8 Octagon 9 Nonagon 10 Decagon 12 Dodecagon n N-gon Quadrilaterals Parallelograms Kites Trapezoids Isosceles Trapezoids Rectangles Rhombi Squares

Polygon Venn Diagram Quadrilaterals Parallelograms Trapezoids Rectangles Kites Isosceles Trapezoids Trapezoids Rhombi Squares

Quadrilateral Characteristics Summary Convex Quadrilaterals 4 sided polygon 4 interior angles sum to 360 4 exterior angles sum to 360 Parallelograms Trapezoids Bases Parallel Legs are not Parallel Leg angles are supplementary Median is parallel to bases Median = ½ (base + base) Opposite sides parallel and congruent Opposite angles congruent Consecutive angles supplementary Diagonals bisect each other Kites 2 congruent sides (consecutive) Diagonals perpendicular Diagonals bisect opposite angles One diagonal bisected One pair of opposite angle congruent Rectangles Rhombi Isosceles Trapezoids Angles all 90° Diagonals congruent All sides congruent Diagonals perpendicular Diagonals bisect opposite angles Squares Legs are congruent Base angle pairs congruent Diagonals are congruent Diagonals divide into 4 congruent triangles

Example Problems 2 W P B H A R S T U V 9z 18 24 J K L M N t 3z z 4x w° In the rectangle, In the square, W P B H A 35° 35 3x + 8 m° 2y -1 25 R S T U V 2k° 3x - 8 16 4y + 4 9z 18 In the rhombus, 24 J K L M N t 3z z 4x w° 54° In the isosceles trapezoid EF is a median, 2y In the parallelogram, P Q 6x - 6 3x+5 6x A B 6z° m° 24 3y - 6 2z + 6 3y 35 25 E F 8t° 3t° y + 4 5t° 2t° 9z° R S C 2x + 8 D

Example Solutions 1 Find the sum of the interior angles in a 16-gon Sides Name Sum of Interior ’s One Interior  One Exterior  3 Triangle 180 60 120 5 Pentagon 540 108 72 7 Heptagon 900 128.57 51.43 Find the sum of the interior angles in a 16-gon Find the sum of the exterior angles in a 16-gon Find the number of sides of a polygon if an interior angle is 140°. S = (n – 2)  180 = (16 – 2)  180 = 14  180 = 2520 S = 360 Int  + Ext  = 180 so Ext  = 40 n = 360 / Ext  = 360 / 40 = 9

Example Solutions 2 2 pairs isosceles ∆ 35 + 35 + x = 180 x + m = 180 (L pr) m = 70 In the rectangle, In the square, W P B H A 35° 35 3x + 8 m° 2y -1 25 R S T U V 2k° 3x - 8 16 4y + 4 9z 18 In the rhombus, Opposite sides = 35 = 3x + 8 27 = 3x 9 = x 24 J K L M N all sides = 4y + 4 = 16 = 3x – 8 4y = 12 24 = 3x y = 3 8 = x t 3z z 4x w° 54° diagonals = and bisected 25 = 2y – 1 26 = 2y 13 = 3 2y all sides = 3z = 4x = 2y = 24 z = 8, x = 6, y = 12 diagonals bisected z = t 8 = t diagonals  2k = 90 k = 45 diagonals bisect angles w = 54

Example Solutions 2 Cont isosceles legs = y + 4 = 3y – 6 10 = 2y 5 = y diagonals bisected 35 = 3x + 5 30 = 3x 10 = x opposite sides = 24 = 2z + 6 18 = 2z 9 = z isosceles leg ’s supplementary 6z + 9z = 180 15z = 180 z = 12 diagonals bisected 3y = 6x 3y = 60 y = 20 Consecutive ’s supplementary 8t + 5t + 2t + 3t = 180 18t = 180 t = 10 isosceles base ’s = 6z = m 72 = m In the isosceles trapezoid EF is a median, In the parallelogram, median = ½(sum of bases) 25 = ½(6x – 6 + 2x + 8) 50 = 6x – 6 + 2x + 8 50 = 8x + 2 48 = 8x 6 = x P Q 6x - 6 3x+5 6x A B 6z° m° 24 3y - 6 2z + 6 3y 35 25 E F 8t° 3t° y + 4 5t° 2t° 9z° R S C 2x + 8 D

Summary & Homework Summary: Homework: Interior and Exterior angles make a linear pair (=180) Sum of interior angles = (n - 2)  180 Sum of exterior angles = 360 (no matter the size) Number of sides = 360 / exterior angle Quadrilateral characteristics are very important for solving problems and verifying figures Reminder: Sum of triangle angles = 180 Medians in trapezoids are similar to mid-segments Homework: study for the test