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 ║.
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.