1.  Sides of a parallelogram if you know diagonals and angle between them or other side

 

length sides parallelogram D

     a    - smaller  side

     b    - larger  side

    DD blue , d blue      -  diagonals

    alpha red , beta green     -  angles between the diagonals

 

Find length of sides of a parallelogram if given diagonals and angle between the diagonals ( a b ) :

length sides parallelogram D Formula1

length sides parallelogram D Formula2

 

length sides parallelogram D Formula4

length sides parallelogram D Formula3

 

Find length of sides of a parallelogram if given diagonals and other side ( a b ) :

length sides parallelogram D Formula5

length sides parallelogram D Formula6

 

 

2.  Sides of a parallelogram if you know angle and height

 

length sides parallelogram H

     alpha red , beta blue     -  angles

     a , b     -  sides

   Ha  - height  measured at right angles to the base  

   Hb green  - height  measured at right angles to the base  

 

Find sides of a parallelogram if given angle and height ( a b ) :

length sides parallelogram H Formula1

length sides parallelogram H Formula2



 

 


1.  Diagonals of a parallelogram if you know sides and angle or sides and other diagonal

 

diagonal parallelogram1

     d blue1    - smaller  diagonal

     DD blue    - larger  diagonal

    a black , b black      -  sides

     alpha red  -  acute angle

     beta green  -  obtuse angle

 

Find length of diagonals of a parallelogram if given sides and angle ( D d /) :

diagonal parallelogram1 F1 a

diagonal parallelogram1 F2 a

 

diagonal parallelogram1 F3 a

diagonal parallelogram1 F4 a

 

Find length of diagonal of a parallelogram if given sides and other diagonal ( D d ) :

diagonal parallelogram1 F d

diagonal parallelogram1 F1 d

 

 

2.  Diagonals of a parallelogram if you know area, angle between the diagonals and other diagonal

 

diagonal parallelogram2

     d blue1    - smaller  diagonal

     DD blue    - larger  diagonal

    alpha red , beta green     -  angles between the diagonals

   A18 black  - area  of a parallelogram

 

Find length of diagonal of a parallelogram if given area, angle between the diagonals and other diagonal ( D d ) :

diagonal parallelogram2 A F1

diagonal parallelogram2 A F2



 

 


angles parallelogram

     d blue1    - smaller  diagonal

     DD blue    - larger  diagonal

    a black , b black     -  sides

     alpha red  -  acute angle

     beta green  -  obtuse angle

   A18 black    -  area of a parallelogram

 

Find the angle of a parallelogram if given sides and diagonal  :

angles parallelogram F

angles parallelogram F1

 

Find the angle of a parallelogram if given sides and area  :

angles parallelogram F3

 

Equations for the relation of the acute and obtuse angles:  :

angles parallelogram F4



 

 


angles between diagonals parallelogram

     d blue1    - smaller  diagonal

     DD blue    - larger  diagonal

    a black , b black     -  sides

     alpha red  -  acute angle  between diagonals

     beta green  - obtuse angle  between diagonals

   A18 black    -  area of a parallelogram

 

Find the angle between diagonals if given sides and diagonals  :

angles between diagonals parallelogram F1

angles between diagonals parallelogram F2

 

angles between diagonals parallelogram F3

angles between diagonals parallelogram F4

 

Find the angle between diagonals if given area of a parallelogram and diagonals  :

angles between diagonals parallelogram F5

 

Identities for the relation of the acute and obtuse angles between diagonals:  :

angles between diagonals parallelogram F6



 

 


height parallelogram

    

  •  , b black      -  sides

         alpha red  -  acute angle

         beta2 blue  -  obtuse angle

       Ha    -  height to the base  

       Hb green    -  height to the base  b black

     

    Find the length of the height of a parallelogram without area if given side and angle  :

    height parallelogram F2

    height parallelogram F1

     

    The acute angle of intersection of the heights is equal to the acute angle of the parallelogram.  :
    The obtuse angle of intersection of the heights is equal to the obtuse angle of a parallelogram.  :


     

     


    sum squared diagonals parallelogram

         d blue1    - smaller  diagonal

         DD blue    - larger  diagonal

         a black  -  smaller side

         b black  -  larger side

         alpha red  -  acute angle between the diagonals

     

    Equation for the sum of the squared diagonals  :

    parallelogram F

     

    Equation for the difference of squares of the sides:  :

    parallelogram F1



     

     


    1.  Properties of a bisector of a parallelogram  
    By definition, the bisector divides the angle in halves  :
    The bisector cuts out an isosceles triangle (in this case, the triangles ABF and DKC)  :
    The bisectors of the supplementary angles intersect at a right angle  :
    The bisectors of the opposite angles are equal and parallel  :

     

    bisector parallelogram

        A16 brown F16 violet     -  bisector of an acute angle

       D16 brown K16 violet   -  bisector of an obtuse angle

       a black , b black    -  sides of a parallelogram

      alpha red  -  acute angle

      beta2 blue  -  obtuse angle

     

    Since ABF and DKC are isosceles triangles, the following identities hold  :

    bisector parallelogram F1

    bisector parallelogram F2

     

     

    2.  The length of a bisector of a parallelogram

    length bisector parallelogram

     

        a black , b black      -  sides of a parallelogram

         alpha red  -  acute angle

         beta2 blue  -  obtuse angle

       L  - bisector

     

    Find length of bisector of a parallelogram if given side and angle ( L  ) :

    length bisector parallelogram F1

    length bisector parallelogram F12

    length bisector parallelogram F2