Lesson 8-R Chapter 8 Review

5-Minute Check on Chapter 8 Transparency 9-1 5-Minute Check on Chapter 8 Complete each statement about parallelogram LMNO LM  _______ MN  _______ OLM  _______ MP  _______ Find the measure of each interior angle What is the measure of each interior angle of a regular pentagon? ON LO ONM PO A = C = 65° B B = D = 115° L M P O N A B (4y + 5)° (8y - 5)° (8y - 5)° (4y + 5)° D C Standardized Test Practice: A 90° B 108° C 120° D 135° Click the mouse button or press the Space Bar to display the answers.

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 Trapezoids Parallelograms Rectangles Isosceles Trapezoids Trapezoids Rhombi Squares Kites

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 Rectangles Rhombi Isosceles Trapezoids All sides congruent Diagonals perpendicular Diagonals bisect opposite angles Angles all 90° Diagonals congruent Legs are congruent Base angle pairs congruent Diagonals are congruent Squares 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

Do you know your characteristics? Homework assignment Chapter 8 Review Problems

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