# 5.7 Proving that figures are special quadrilaterals

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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). If either diagonal of a parallelogram bisects two angles of the parallelogram, then it is a rhombus.

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.

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.

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 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.

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

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.

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

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.