1. Lesson 8-7 Coordinate Proof with Quadrilaterals

2. 5-Minute Check on Lesson 8-6 Transparency 8-7 Click the mouse button or press the Space Bar to display the answers. ABCD is an isosceles trapezoid with median EF. 1.Find m  D if m  A= 110°. 2.Find x if AD = 3x² + 5 and BC = x² + 27. 3.Find y if AC = 9(2y – 4) and BD = 10y + 12. 4.Find EF if AB = 10 and CD = 32. 5.Find AB if AB = r + 18, CD = 6r + 9 and EF = 4r + 10. 6. Which statement is always true about trapezoid LMNO with bases of LM and NO? Standardized Test Practice: A C B D LO // MN LO  MN A B D C F E LM  NO LM // NO 70° ± 4 6 21 25 C

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 Squares RhombiRectangles ParallelogramsKitesTrapezoids Isosceles Trapezoids Quadrilaterals

6. Example 7-2a Name the missing coordinates for the isosceles trapezoid. 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. Answer: D (b, c) A(0, 0)B(a, 0) D(?, ?) C(a-b, c) y x

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

8. Quadrilateral Characteristics Summary Convex Quadrilaterals Squares RhombiRectangles ParallelogramsTrapezoids Isosceles Trapezoids Opposite sides parallel and congruent Opposite angles congruent Consecutive angles supplementary Diagonals bisect each other Bases Parallel Legs are not Parallel Leg angles are supplementary Median is parallel to bases Median = ½ (base + base) Angles all 90° Diagonals congruent Diagonals divide into 4 congruent triangles All sides congruent Diagonals perpendicular Diagonals bisect opposite angles Legs are congruent Base angle pairs congruent Diagonals are congruent 4 sided polygon 4 interior angles sum to 360 4 exterior angles sum to 360

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

10. In the isosceles trapezoid to the right EF is a median, solve for x, y and z 2x + 8 6x - 6 21 2y - 4 y + 4 6z 12z AB C D EF x = 5 y = 8 z = 10 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 JK LM N x = 3 y = 9 z = 5 In the rectangle to the left, WA = 6x, AH = 24,  AHB = 33°,  WAP = y, and  BAP = z – 5, solve for x, y and z W P B H A x = 4 y = 114° z = 71° In the square to the right, RV = 5x, SV = 3y,  VST = 9y, and RS = z solve for x, y and z RS TU V x = 3 y = 5 z = 15√2

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