Question

In the figure below , ABCD is a parallelogram . Complete each statement along with the definition or property used.

1) AD =
2) DC =
3) $\angle DCB$ =
4) $\angle ADC$=
5) $\angle DAB$=
6) OB =
7) OC =

Answer 1

Hint:
By the using the following properties we can find the required measurements
1) opposite sides of the the parallelogram are equal
2) property adjacent angles in a parallelogram sum upto to 180 degree
3) diagonals of a parallelogram bisect each other

Complete step by step solution:
We are given a parallelogram ABCD
AD = 6 cm
This is given by the property opposite sides of the the parallelogram are equal
And we are given BC = 6 cm
Hence AD = 6 cm
DC = 9 cm
This is given by the property opposite sides of the the parallelogram are equal
And we are given AB = 6 cm
Hence DC = 6 cm
$\angle DCB = {60^ \circ }$
This is given by the property adjacent angles in a parallelogram sum upto to 180 degree
Here $\angle DCB $ and $\angle ABC$ are adjacent angles
Hence
$
   \Rightarrow \angle DCB + \angle ABC = 180 \\
   \Rightarrow \angle DCB + 120 = 180 \\
   \Rightarrow \angle DCB = 180 - 120 = 60 \\
 $

$\angle ADC$=${120^ \circ }$
This is given by the property adjacent angles in a parallelogram sum upto to 180 degree
Here $\angle DCB$ and $\angle ADC$ are adjacent angles
Hence
$
   \Rightarrow \angle DCB + \angle ADC = 180 \\
   \Rightarrow \angle DCB + 60 = 180 \\
   \Rightarrow \angle DCB = 180 - 60 = 120 \\
 $

$\angle DAB$=${60^ \circ }$
This is given by the property adjacent angles in a parallelogram sum up to to 180 degree
Here $\angle DAB$ and $\angle ABC$ are adjacent angles
Hence
$
   \Rightarrow \angle DAB + \angle ABC = 180 \\
   \Rightarrow \angle DAB + 120 = 180 \\
   \Rightarrow \angle DAB = 180 - 120 = 60 \\
 $

OC = 7 cm
This is given by the property that diagonals of a parallelogram bisect each other
Here we have OA = 7 cm
Hence OC = 7 cm

OB = 5 cm
This is given by the property that diagonals of a parallelogram bisect each other
Here we have OD = 5 cm
Hence OB = 5 cm

Note:
1) Opposite sides are congruent (AB = DC).
2) Opposite angles are congruent (D = B).
3) Consecutive angles are supplementary (A + D = 180°).
4) If one angle is right, then all angles are right.
5) The diagonals of a parallelogram bisect each other.