PARALLELOGRAMS
Definition & Properties
1) Both pair of opposite sides are parallel.
2) Both pair of opposite sides are congruent.
3) Both pair of opposite angles are congruent.
4) Each pair of consecutive angles are supplementary.
5) The diagonals bisect each other.
6) A diagonal cuts the parallelogram into two congruent triangles.
Examples
1) If DE = 10, KC = _______
2) If DC = 18, DT = _______
3) If mEDK = 100º, mECK = _______
4) If mDEC = 75º, mKDE = _______
5) If m1 = 30º and m2 = 40º, mKCE = _______, m3 = ________
6) If m3 = 36º and m2 = 44º, mKDE = ________
7) If DT = 7 and KT = 9, CD = ________
Determine the value of x and/or y.
8) DE = 5x and KC = 3x + 12
9) ET = x + 3 and EK = 22
10) DT = ½ x and TC = 10
11) mKCE = 6y – 20 and mEDK = 2y + 8
12) m1 = y + 10, m2 = 3y, and m3 = ½ y + 15
13) mDEC = y and mECK = 2y + 60
Proving that a quadrilateral is a parallelogram:
•
If both pairs of opposite sides are parallel
, then the quadrilateral is a parallelogram.
•
If both pairs of opposite sides are congruent
, then the quadrilateral is a parallelogram.
•
If one pair of opposite sides are both parallel and congruent
•
If both pairs of opposite angles are congruent
•
If one angle is supplementary to both of its consecutive angles, then the quadrilateral is a parallelogram.
•
If the diagonals bisect each other
, then the quadrilateral is a parallelogram.
, then the quadrilateral is a parallelogram.
, then the quadrilateral is a parallelogram.
SPECIAL PARALLELOGRAMS
RECTANGLE
Definition: a parallelogram with 4 right angles
Examples:
Properties:
-all parallelogram properties
- diagonals are congruent
a) If TA = 13, then TE =____ and RA = _____.
b) Solve for x if AC = 4x + 4 and EA = 7x - 23.
c) Solve for x if m ATC = 6x - 4 and mRTA = 10x - 2
RHOMBUS
Definition: a parallelogram with 4 congruent sides
Properties:
-all parallelogram properties
- diagonals are perpendicular
-each diagonal bisects the opposite angles
a) Solve for x if mUPT = 4x + 18.
b) Solve for x and y if UT = 5x + 4, TA = 2x + y, HA = 2y – 8
and UH = 24.
c) Solve for x if mPAH = 8x + 2 and mPAT = 10x - 10.
SQUARE
Definition: a parallelogram with 4 congruent sides and 4 right angles
Properties:
Examples:
- all parallelogram properties
mARU = ___, mRUQ= ___
- all rectangle and rhombus properties
Property
Opposite sides are ||
Opposite sides are 
Opposite angles are 
All sides are 
All angles are right angles
A diagonal forms 2  s
Diagonals are perpendicular
Diagonals are 
Diagonals bisect each other
A diagonal bisects opposite s
Parallelogram
Rectangle
Rhombus
Square
COORDINATE GEOMETRY
You will need to use the slope formula to determine if opposite sides are parallel and the distance formula to determine if
any sides are congruent.
Slope formula:
𝒎=
𝒚𝟐 −𝒚𝟏
𝒙𝟐 −𝒙𝟏
Distance formula: 𝒅 = √(𝒙𝟐 − 𝒙𝟏 ) + (𝒚𝟐 − 𝒚𝟏 )
Graph quad USMC with coordinates
U(1, 1), S(4, 5), M(9, 5) andC(6, 1).
•
Find the slope of each side of the quadrilateral.
•
Is the quadrilateral a parallelogram?
•
Find the length of each side of the parallelogram.
•
Is the parallelogram a rhombus?
•
How else could we determine if the parallelogram is a
rhombus?
Given Quad WHAT with vertices W(2, 4), H(5, 8), A(9, 5) and T(6, 1).
What is the best name for this quadrilateral?
a. parallelogram
b. rhombus
c. rectangle
COORDINATE GEOMETRY
You will need to use the slope formula to determine if opposite sides are parallel and the distance formula to determine if
any sides are congruent.
Slope formula:
𝒎=
𝒚𝟐 −𝒚𝟏
𝒙𝟐 −𝒙𝟏
Distance formula: 𝒅 = √(𝒙𝟐 − 𝒙𝟏 )𝟐 + (𝒚𝟐 − 𝒚𝟏 )𝟐
Graph quad USMC with coordinates
U(1, 1), S(4, 5), M(9, 5) andC(6, 1).
•
Find the slope of each side of the quadrilateral.
•
Is the quadrilateral a parallelogram?
•
Find the length of each side of the parallelogram.
•
Is the parallelogram a rhombus?
•
How else could we determine if the parallelogram is a
rhombus?
Given Quad WHAT with vertices W(2, 4), H(5, 8), A(9, 5) and T(6, 1).
What is the best name for this quadrilateral?
a. parallelogram
b. rhombus
c. rectangle