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Lesson 8-7

Lesson 8-7. Coordinate Proof with Quadrilaterals. Transparency 8-7. 5-Minute Check on Lesson 8-6. ABCD is an isosceles trapezoid with median EF . Find m  D if m A = 110° . Find x if AD = 3x² + 5 and BC = x² + 27. Find y if AC = 9(2y – 4) and BD = 10y + 12.

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Lesson 8-7

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  1. Lesson 8-7 Coordinate Proof with Quadrilaterals

  2. Transparency 8-7 5-Minute Check on Lesson 8-6 • ABCD is an isosceles trapezoid with median EF. • Find mD if mA= 110°. • Find x if AD = 3x² + 5 and BC = x² + 27. • Find y if AC = 9(2y – 4) and BD = 10y + 12. • Find EF if AB = 10 and CD = 32. • Find AB if AB = r + 18, CD = 6r + 9 and EF = 4r + 10. • Which statement is always true about trapezoid LMNO with bases of LM and NO? B A 70° E F ± 4 D C 6 21 25 Standardized Test Practice: LO // MN LO  MN A B C C D LM // NO LM  NO Click the mouse button or press the Space Bar to display the answers.

  3. Objectives • Position and label quadrilaterals for use in coordinate proofs • Prove theorems using coordinate proofs

  4. Vocabulary • Kite – quadrilateral with exactly two distinct pairs of adjacent congruent sides.

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

  6. Example 7-2a Name the missing coordinates for the isosceles trapezoid. y D(?, ?) C(a-b, c) x A(0, 0) B(a, 0) The legs of an isosceles trapezoid are congruent and have opposite slopes. Point C is c units up and b units to the left of B. So, point D is c units up and b units to the right of A. Therefore, the x-coordinate of D is and the y-coordinate of D is 0 + c, or c 0 + b, or b. D (b, c) Answer:

  7. Answer: Example 7-2b Name the missing coordinates for the rhombus.

  8. 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

  9. Do you know your characteristics? • Extra Credit Assignment • Review Problems

  10. W P In the rectangle to the left, WA = 6x, AH = 24, AHB = 33°, WAP = y, and BAP = z – 5, solve for x, y and z A B H R S In the square to the right, RV = 5x, SV = 3y, VST = 9y, and RS = z solve for x, y and z x = 3 y = 5 z = 15√2 V U T J K In the rhombus to the left, JK = 6x, KM = 2y, LNM = 10y, JLN = 4z + 10, and JKN = 7z – 5, solve for x, y and z N L M 6x - 6 A B 12z 2y - 4 In the isosceles trapezoid to the right EF is a median, solve for x, y and z 21 E F y + 4 6z C D 2x + 8 x = 4 y = 114° z = 71° x = 3 y = 9 z = 5 x = 5 y = 8 z = 10

  11. Summary & Homework • Summary: • Position a quadrilateral so that a vertex is at the origin and a least one side lies along an axis. • Homework: • pg 450-451; 9, 11-14, 28, 29, 31-33

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