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Lesson 8-R

This lesson provides a comprehensive review of Chapter 8, focusing on the properties of parallelograms and polygon angles. It includes exercises to complete statements about the parallelogram LMNO, find measures of interior angles, and solve problems related to interior and exterior angles of convex polygons. The lesson also discusses important polygon characteristics, including hierarchical classifications and relationships between angles. Students will gain a solid understanding of crucial geometry concepts in preparation for standardized tests.

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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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