5.7 Proving that figures are special quadrilaterals

5.7 Proving that figures are special quadrilaterals

Proving a RhombusProving 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.

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.

You can also prove a quadrilateral is a rhombus without first showing that it is a parallelogram.

1. If the diagonals of a quadrilateral are perpendicular bisectors of each other, then the quadrilateral is a rhombus.

Proving a Rhombus

Proving a RectangleProving a Rectangle

First, you must prove that the quadrilateral is a parallelogram and then prove one of the following conditions:

First, you must prove that the quadrilateral is a parallelogram and then prove one of the following conditions:

1. If a parallelogram contains at least one right angle, then it is a rectangle (reverse of the definition)

2. If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle.

1. If a parallelogram contains at least one right angle, then it is a rectangle (reverse of the definition)

2. If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle.

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.

Proving a KiteProving a Kite1. 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.

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.

Prove a SquareProve a Square

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

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

Prove an Isosceles TrapezoidProve 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.

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.

Prove an Isosceles Trapezoid

1. GJMP is a .2. OM ║ GK3. OH GK4. MK is altitude of

Δ MKJ.5. MK GK6. OH ║ MK7. OHKM is a .8. OHK is a rt .9. OHKM is a rect.

1. Given2. Opposite sides of a are ║.3. Given4. Given5. An alt. of a Δ is to the side to

which it is drawn.6. If 2 coplanar lines are to a third

line, they are ║.7. If both pairs of opposite sides of

a quad are ║, it is a . 8. segments form a rt. .9. If a contains at least one rt ,

it is a rect.