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Lesson 8-R

Lesson 8-R. Chapter 8 Review. 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

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Lesson 8-R

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  1. Lesson 8-R Chapter 8 Review

  2. 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 L M LO P ONM O N PO A B (8y - 5)° (4y + 5)° A = C = 65° B = D = 115° (4y + 5)° (8y - 5)° D C Standardized Test Practice: A 90° B 108° C 120° D 135° B Click the mouse button or press the Space Bar to display the answers.

  3. 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 Interior angle Exterior angle

  4. Example Problems 1 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°.

  5. Polygon Hierarchy Polygons Quadrilaterals Parallelograms Kites Trapezoids IsoscelesTrapezoids Rectangles Rhombi Squares

  6. Polygon Venn Diagram Quadrilaterals Trapezoids IsoscelesTrapezoids Parallelograms Rhombi Kites Rectangles Squares

  7. 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 basesMedian = ½ (base + base) Opposite sides parallel and congruent Opposite angles congruent Consecutive angles supplementary Diagonals bisect each other Rectangles Rhombi IsoscelesTrapezoids 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

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

  9. Example Solutions 1 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

  10. 3x + 8 35° m° 2y -1 25 35 3x - 8 R S J K V 16 2k° N L M U T 4y + 4 W P A B H Example Solutions 2 2 pairs isosceles ∆ 35 + 35 + x = 180 x + m = 180 (L pr) m = 70 In the square, In the rectangle, 18 9z In the rhombus, Opposite sides = 35 = 3x + 8 27 = 3x 9 = x 24 all sides = 4y + 4 = 16 = 3x – 8 4y = 12 24 = 3x y = 3 8 = x t 3z 4x z w° 54° 2y diagonals = and bisected 25 = 2y – 1 26 = 2y 13 = 3 diagonals bisected z = t 8 = t all sides = 3z = 4x = 2y = 24 z = 8, x = 6, y = 12 diagonals  2k = 90 k = 45 diagonals bisect angles w = 54

  11. 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 isosceles base ’s = 6z = m 72 = m Consecutive ’s supplementary 8t + 5t + 2t + 3t = 180 18t = 180 t = 10 In the isoscelestrapezoid 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 A B 3x+5 6x 6z° m° 24 3y - 6 25 3y 2z + 6 35 E F 3t° 8t° y + 4 9z° 5t° 2t° S R C D 2x + 8

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

  13. Summary & Homework • Summary: • 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

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