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Objectives

Objectives. State the properties of rectangles, rhombuses, and squares Solve problems involving rectangles, rhombuses, and squares. Diagonals of Rhombus Bisect Angles. A parallelogram is a rhombus if and only if each diagonal bisects two angles of the rhombus. ∠1 ≅ ∠2 and ∠3 ≅ ∠4

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Objectives

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  1. Objectives • State the properties of rectangles, rhombuses, and squares • Solve problems involving rectangles, rhombuses, and squares

  2. Diagonals of Rhombus Bisect Angles • A parallelogram is a rhombus if and only if each diagonal bisects two angles of the rhombus. • ∠1 ≅ ∠2 and ∠3 ≅ ∠4 • Since ∠BCD and ∠BAD are opposite angles of a parallelogram, ∠1 ≅ ∠2 ≅ ∠3 ≅ ∠4

  3. Diagonals of a Rhombus are Perpendicular • A parallelogram is a rhombus if and only if the diagonals are perpendicular.

  4. Diagonals of a Rectangle • A parallelogram is a rectangle if and only if the diagonals are congruent.

  5. Squares • A square is a parallelogram, rectangle, and rhombus. All properties of parallelograms, rectangles, and rhombi apply to squares

  6. Example: Rhombus • Find m∠XTZ We need to solve for a before we can find m∠XTZ. 14a + 20 = 90 (diagonals of a rhombus are perpendicular) 14a = 70 a = 5 5a – 5 = 20° (substituting a = 5 in order to find m∠XTZ)

  7. Example: Rectangle Find FD in rectangle FEDG if FD = 2y + 4 and GE = 6y – 5 6y – 5 = 2y + 4 4y – 5 = 4 4y = 9 y = 9/4 FD = 2y + 4 = 2(9/4) + 4 = 8.5

  8. Example: Square Since EG = FH, Since , • Show that the figure is a square. • Strategy: • Show that the diagonals are perpendicular (rhombus) • Show that the diagonals are congruent (rectangle)

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