Download Quadrilaterals in Euclidean Geometry

Quadrilaterals in Euclidean Geometry
The parallelogram
A convex quadrilateral ABCD is called a parallelogram if the pairs of opposite sides are parallel.
In the figure below, AB||CD and AD||BC
Special cases are the
• rectangle: a parallelogram with four right angles
• rhombus: a parallelogram having two adjacent sides congruent
• square: rhombus having two adjacent sides perpendicular
Key Theorem:
A diagonal of a parallelogram divides it into two congruent triangles.
Given ABCD a parallelogram with AB||CD and AD||BC, by the Z Property (alternate interior
angles) we have
∠1 ∼
and
∠3 ∼
= ∠2
= ∠4
Side AC ∼
= CA (reflexive), and by the ASA Theorem,
ACD ∼
= CAB
Properties of parallelograms
The properties are listed here:
(1) Opposite sides of a parallelogram are congruent, as are opposite angles.
(2) Adjacent angles are supplementary.
(3) If a convex quadrilateral has opposite sides congruent, then it is a parallelogram.
(4) If a convex quadrilateral has a pair of sides that are both congruent and parallel, then it is a
parallelogram.
(5) The diagonals of a parallelogram bisect each other.
(6) If the diagonals of a convex quadrilateral bisect each other, then the quadrilateral is a parallelogram.
(7) A parallelogram is a rhombus if and only if its diagonals are perpendicular.
(8) A parallelogram is a rectangle if and only if its diagonals are congruent.
(9) A parallelogram is a square if and only if its diagonals are both perpendicular and congruent.
The proofs of these are sketched out in the lecture narration - please listen, and write up proofs of
each of the properties. In some cases, the proof may be one line, as you work through the properties
in order. Others may be more involved (and be sure to go in both directions on the if and only if’s).