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Quadrilateral
Parallelogram
Rectangle
Square
Rhombus

Lesson 6-2 Parallelograms
A parallelogram is a quadrilateral with both pairs of opposite sides parallel.
THEOREMS:
Opposite sides of a parallelogram are congruent
Opposite angles in a parallelogram are congruent
Consecutive angles in a parallelogram are supplementary
The diagonals of a parallelogram bisect each other.
If a parallelogram has one right angle, it has four right angles.
Ex1) Complete each statement about DEFG. Justify your answer. a.
DG || ____ b.
DE ≅ ____ c.
GH ≅ ____ d.
∠ DEF ≅ ____ e.
∠ EFG is suppl. to ______. f.
Δ DGE ≈ _______.
Ex2) Use WXYZ to find each measure or value. a.
m ∠ XYZ = ______ b.
m ∠ WZY = ______ c.
m ∠ WXY = ______ d.
a = ______

6.3: Tests for Parallelograms
Tests for Parallelograms: A quadrilateral is a parallelogram if…..
1) Both pairs of opposite sides parallel
(Definition)
2) Both pairs of opposite sides of a quadrilateral are congruent. (Theorem)
3) Both pairs of opposite angles of a quadrilateral are congruent. (Theorem)
4) Diagonals of a quadrilateral bisect each other. (Theorem)
5) One pair of opposite sides of a quadrilateral is both congruent and parallel. (Theorem)
c.
Ex4) Determine whether each quadrilateral is a parallelogram. Justify your answer.
a. b.
d.

Ex5) COORDINATE GEOMETRY Determine whether a figure with the given vertices is a parallelogram. S (-2, 1), R (1, 3), T (2, 0), Z (-1, -2)
Both pairs of opposite sides of a quadrilateral are congruent
Both pairs of opposite angles of a quadrilateral are parallel.

Ex2:
6.4: Rectangles
A rectangle is a parallelogram with four right angles.
Properties of a Rectangle:
1) Opposite sides are parallel
2) Opposite angles are congruent
3) Opposite sides are congruent
4) Consecutive angles are supplementary
5) Diagonals bisect each other
6) Diagonals are congruent
7) All four angles are right angles
Ex1:

6.4 Rectangle Cont.
Proving Rectangles
Theorems:

If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle.
 If the angles of a parallelogram are all 90°, then the parallelogram is a rectangle.
Coordinate Geometry Determine whether BHGL is a rectangle given each set of vertices. Justify your answer.
Steps:
(1) Prove that the quadrilateral is a parallelogram. (Use any of the 4 methods from 6.3)
(2) Prove that the parallelogram is a rectangle. (Use either of the two theorems)
Easiest method: Find the slopes.
Ex3)
Ex4)
If the quadrilateral is not a parallelogram, it is not a rectangle. In those cases you can stop after step 1.

6.5: Rhombi and Squares
A rhombus is a quadrilateral with all four sides congruent.
Properties of a Rhombus:
1) Opposite sides are parallel
2) Opposite angles are congruent
3) Opposite sides are congruent
4) Consecutive angles are supplementary
5) Diagonals bisect each other
And….
6) The diagonals of a rhombus are perpendicular.
7) Each diagonal of a rhombus bisects a pair of opposite angles.

We are told that the figure is a parallelogram. So….
(1) Graph the parallelogram
(2) Find the distance and slope of each diagonal.
Diagonals are perpendicular:
The parallelogram is a _______________
Diagonals are congruent:
The parallelogram is a ___________________
Diagonals are BOTH perpendicular and congruent:
The parallelogram is a _____________________________.