What is a parallelogram?
a quadrilateral with both pairs of opposite sides parallel
Properties of Parallelograms (hint: there are 4)
- Opposite sides are congruent
- Opposite angles are congruent
- Diagonals bisect each other
- Consecutive angles are supplementary
6-4 (parallel lines and transversals)
If three or more parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal
Distance Formula:
d = square root of (x2-x1) squared + (y2-y1) squared
Midpoint Formula:
m = (x1 + x2 / 2 , y1 + y2 / 2)
Slope Formula:
m = y2 - y1 / x2 - x1
OR
run
6-5 (opposite sides)
If both pairs of opposite sides of a quad are congruent, then the quad is a parallelogram
6-6 (opposite angles)
If both pairs of opposite angles of a quad are congruent, then the quad is a parallelogram
6-7 (diagonals)
If the diagonals of a quad bisect each other, then the quad is a parallelogram
6-8 (one pair)
If one pair off opposite sides of a quad is both congruent and parallel, then the quad is a parallelogram
5 ways to prove a quadrilateral is a parallelogram
- Both pairs of opposite sides are parallel (def of parallelogram)
- Both pairs of opposite sides are congruent
- Both pairs of opposite angles are congruent
- The diagonals bisect each other (have the same midpoint)
- One pair of opposite sides is parallel and congruent
def of rhombus
a parallelogram with 2 congruent, consecutive sides
6-9 (diagonal —> angles; rhombus)
Each diagonal of a rhombus bisects 2 angles of a rhombus
6-10 (rhombus diagonals)
The diagonals of a rhombus are perpendicular
def of rectangle
parallelogram with 1 right angle
6-11 (rectangle diagonals)
The diagonals of a rectangle are congruent
def of square
a rectangle and a rhombus
what is special about a square
has properties of a parallelogram, rhombus, and rectangle
what is special about the DIAGONALS of a square
they are:
- congruent
- bisect each other
- each bisect two angles
- and are perpendicular
6-12 and 6-13 (proving a parallelogram is a rhombus)
6-12 : If one diagonal of a parallelogram bisects two angles of the parallelogram, then it is a rhombus
6-13 : If the diagonals of a parallelogram are perpendicular, then it is a rhombus
6-14 (proving a parallelogram is a rectangle)
If the diagonals of a parallelogram are congruent, then it is a rectangle
def of trapezoid
quad with one pair of parallel sides
what sides of a trapezoid are the bases?
the parallel sides
what sides of a trapezoid are the legs?
the nonparallel sides
