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Properties and Geometry of Rectangles

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Learn and apply properties of rectangles, including parallel sides, congruent angles, and congruent diagonals. Determine if given quadrilaterals are rectangles. Practice exercises included.

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Properties and Geometry of Rectangles

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  1. Geometry Lesson 6 – 4 Rectangles Objective: Recognize and apply properties of rectangles. Determine whether parallelograms are rectangles.

  2. Rectangle • A parallelogram with 4 right angles. • Opposite sides are parallel and congruent • Opposite angles are congruent (right) • Consecutive angles are supplementary • Diagonals bisect each other • NEW: Diagonals are congruent.

  3. Theorem • Diagonals of a Rectangle • If a parallelogram is a rectangle, then its diagonals are congruent.

  4. PR = QS • A rectangular park has two walking paths as shown. If PS = 180 meters and PR = 200 meters, find QT and RS. QS = 200 QT = (1/2)(QS) 180 200 QT = (1/2)(200) QT = 100 (PS)2 + (SR)2 = (PR)2 (SR)2 = 7600 (180)2 + (SR)2 = (200)2 32400 + (SR)2 = 40000

  5. 64 64 • If 26 26 26 64 64

  6. Quadrilateral JKLM is a rectangle. If measure of angle KJL is 2x + 4 and measure of angle JLK is 7x + 5, find x. (2x + 4)o (7x + 5)o 2x + 4 + 7x + 5 = 90 9x + 9 = 90 9x = 81 x = 9

  7. Quadrilateral JKLM is a rectangle. If JP = 3y – 5 and MK = 5y + 1, find y. 3y - 5 5y + 1 2(JP) = MK 2(3y – 5) = 5y + 1 6y – 10 = 5y + 1 y = 11

  8. Theorem • Theorem 6.14 • If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle.

  9. Quadrilateral PQRS has vertices P(-5, 3), R(-1, -4) Q(1, -1) and S (-7, 0). Determine whether PQRS is a rectangle by using the distance formula. First figure out if the quad is a parallelogram By doing one of the tests. Either prove opp sides are congruent or one set Of opp sides is parallel and congruent. Cont…

  10. Figure out if the parallelogram is a rectangle: Diagonals of a rectangle are congruent. Quadrilateral PQRS is a rectangle.

  11. Quadrilateral JKLM has vertices J(-10, 2) K(-8,-6) L(5, -3) M(2, 5) • Determine whether JKLM is a rectangle using the Slope formula. To be a rectangle, consecutive sides must be perpendicular (opp. reciprocals) Since the slopes of consecutive sides are not opposite reciprocals the figure is not a rectangle.

  12. Homework • Pg. 422 1 – 9 all, 10 – 18 E, 22 – 30 E, 50 – 60 E

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