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Proving Special Quadrilaterals: Rectangle, Kite, Rhombus, Square, Isosceles Trapezoid

This text explains how to prove that a quadrilateral is a rectangle, kite, rhombus, square, or isosceles trapezoid using various conditions and properties.

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Proving Special Quadrilaterals: Rectangle, Kite, Rhombus, Square, Isosceles Trapezoid

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  1. Warm Up • Given BCDF is a kite • BC = 3x + 4y • CD = 20 • BF = 12 • FD = x + 2y • Find the PERIMETER OF BCDF P = 2(12) + 2(20) = 64

  2. 5.7 Proving that figures are special quadrilaterals

  3. B C A Given: AB || CD <ABC <ADC AB AD Prove: ABCD is a rhombus D  

  4. Proving a Rhombus • First prove that it is a parallelogram then one of the following • 1. If a parallelogram contains a pair of consecutive sides that are congruent, then it is a rhombus (reverse of the definition). • 2. If either diagonal of a parallelogram bisects two angles of the parallelogram, then it is a rhombus.

  5. Proving a rectangle • First you must prove that the quadrilateral is a parallelogram and then prove one of the following conditions.

  6. 1. If a parallelogram contains at least one right angle, then it is a rectangle (reverse of the definition) Or • 2. If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle

  7. You can also prove a quadrilateral is a rectangle without first showing that it is a parallelogram if you can prove that all four angles are right angles. • If all four angles are right angles, then it is a rectangle.

  8. Proving a kite • 1. If two disjoint pairs of consecutive sides of a quadrilateral are congruent, then it is a kite (reverse of the definition). • 2. If one of the diagonals of a quadrilateral is the perpendicular bisector of the other diagonal, then the quadrilateral is a kite.

  9. You can also prove a quadrilateral is a rhombus if the diagonals are perpendicular bisectors of each other, then the quadrilateral is a rhombus.

  10. Prove a square • If a quadrilateral is both a rectangle and a rhombus, then it is a square (reverse of the definition).

  11. Prove an isosceles trapezoid • 1. If the nonparallel sides of a trapezoid are congruent, then it is isosceles (reverse of the definition). • 2. If the lower or upper base angles of a trapezoid are congruent, then it is isosceles. • 3. If the diagonals of a trapezoid are congruent then it is isosceles.

  12. Rectangle

  13. Kite

  14. Rhombus

  15. Square

  16. Isosceles trapezoid

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