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Understanding Rectangle Properties: Theorems, Diagonals, and Applications

Explore properties of rectangles in geometry, including proving theorems, recognizing parallelograms, and solving for unknown sides and angles in rectangles. Practice using formulas and theorems to determine if given quadrilaterals are rectangles.

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Understanding Rectangle Properties: Theorems, Diagonals, and Applications

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  1. Lesson 6-4 Rectangles

  2. Recognize and apply properties of rectangles. • Determine whether parallelograms are rectangles. • rectangle Standard 7.0 Students prove and use theorems involvingthe properties of parallel lines cut by a transversal, the properties of quadrilaterals,and the properties of circles. (Key)

  3. Rectangle • Def—A //ogram with 4 Right Angles

  4. Properties of a Rectangle • Rectangle  Diagonals are  • (Also has all the properties of a //ogram.) • Opposite sides  • Opposite angles  • Consecutive angles supplementary • Diagonals bisect each other

  5. A B Given ABCD is a Rectangle, list everything that must be true. E D C //ogram: Opp. Sides // Def: 4 rt. s #1: Diagonals are  #2: Opp. Sides are  #3: Opp. s are  #4: Consec. s are Supp. #5: Diagonals bisect each other.

  6. Diagonals of a Rectangle Quadrilateral RSTU is a rectangle. If RT = 6x + 4 and SU = 7x – 4, find x.

  7. The diagonals of a rectangle are congruent, Diagonals of a Rectangle Diagonals of a rectangle are . Definition of congruent segments Substitution Subtract 6x from each side. Add 4 to each side. Answer: 8

  8. Quadrilateral EFGH is a rectangle. If FH = 5x + 4 and GE = 7x – 6, find x. • A • B • C • D A.x = –1 B.x = 3 C.x = 5 D.x = 10

  9. Angles of a Rectangle Quadrilateral LMNP is a rectangle. Find x.

  10. Angles of a Rectangle Angle Addition Postulate Substitution Simplify. Subtract 10 from each side. Divide each side by 8. Answer: 10

  11. Quadrilateral EFGH is a rectangle. Find x. A. 6 B. 7 C. 9 D. 14 • A • B • C • D

  12. Reminder • Perpendicular lines have opposite reciprocal slopes. • Prove the sides of a quadrilateral are perpendicular and you have proven it is a rectangle.

  13. Rectangle on a Coordinate Plane Method 1: Use the Slope Formula, to see ifopposite sides are parallel and consecutive sides are perpendicular. Quadrilateral ABCD has vertices A(–2, 1), B(4, 3), C(5, 0), and D(–1, –2). Determine whether ABCD is a rectangle using the Slope Formula.

  14. Rectangle on a Coordinate Plane = Slopes  // lines Opp. Reciprocal Slopes   lines //ogram with 4 right angles  Rectangle

  15. Rectangle on a Coordinate Plane Method 2: Use the Distance Formula, to determine whether opposite sides are congruent.

  16. Rectangle on a Coordinate Plane Find the length of the diagonals. //ogram w/  Diagonals  Rectangle Opp. Sides   //ogram

  17. Quadrilateral WXYZ has vertices W(–2, 1), X(–1, 3), Y(3, 1), and Z(2, –1).Determine whether WXYZ is a rectangle using the Distance Formula. • A • B • C A. yes B. no C. cannot be determined

  18. A. B.4 C.5 D.25 Quadrilateral WXYZ has vertices W(–2, 1), X(–1, 3), Y(3, 1), and Z(2, –1).What are the lengths of diagonals WY and XZ? • A • B • C • D

  19. Homework • pg 344: 1, 2, 7, 8, 10, 13-21, 27-29

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