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5.7 Proving that figures are special quadrilaterals

5.7 Proving that figures are special quadrilaterals. Proving a Rhombus. First prove that it is a parallelogram then one of the following: If a parallelogram contains a pair of consecutive sides that are congruent , then it is a rhombus (reverse of the definition ).

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5.7 Proving that figures are special quadrilaterals

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  1. 5.7 Proving that figures are special quadrilaterals

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

  3. Proving a Rhombus • You can also prove a quadrilateral is a rhombus without first showing that it is a parallelogram. • If the diagonals of a quadrilateral are perpendicular bisectors of each other, then the quadrilateral is a rhombus.

  4. Proving a Rectangle • First, you must prove that the quadrilateral is a parallelogram and then prove one of the following conditions: • If a parallelogram contains at least one right angle, then it is a rectangle (reverse of the definition) • If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle.

  5. Proving a Rectangle • 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.

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

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

  8. Prove an Isosceles Trapezoid • If the nonparallel sides of a trapezoid are congruent, then it is isosceles (reverse of the definition). • If the lower or upper base angles of a trapezoid are congruent, then it is isosceles.

  9. Prove an Isosceles Trapezoid 3. If the diagonals of a trapezoid are congruent then it is isosceles.

  10. Given Opposite sides of a are ║. Given Given An alt. of a Δ is  to the side to which it is drawn. If 2 coplanar lines are  to a third line, they are ║. If both pairs of opposite sides of a quad are ║, it is a .  segments form a rt. . If a contains at least one rt, it is a rect. GJMP is a . OM ║ GK OH  GK MK is altitude of Δ MKJ. MK  GK OH ║ MK OHKM is a . OHK is a rt . OHKM is a rect.

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