Question 1: In a parallelogram , determine the sum of angles and .
Since is a parallelogram, therefore
Since, and transversal intersects them at and respectively.
Since the sum of the interior angles on the same side of the transversal is
Question 2: In a parallelogram , if , determine the measures of its other angles.
Also is a parallelogram.
Therefore and and
Question 3: is a square. and intersect at . State the measure of .
(side of a square)
(diagonals of a square bisect each other)
Question 4: is a rectangle with . Determine .
Given is a rectangle with
Question 5: The sides and, of a parallelogram are bisected at and . Prove that is a parallelogram.
Given: is a parallelogram, bisects and bisects
Therefore is a parallelogram.
Question 6: and are the points of trisection of the diagonal of parallelogram . Prove is parallel to . Prove also that bisects .
To prove: i) ii) bisects
Since diagonals of a parallelogram bisects each other
and … … … … … i)
Given … … … … … ii)
From i) and ii) we get
Now consider parallelogram
We know that diagonals of a parallelogram bisects each other
Hence is a parallelogram
Question 7: is a square and are points on and respectively, such that . Prove that is a square.
Since is a square
Now consider and
Similarly, we can prove that
Therefore is a square
is a parallelogram
Question 8: is a rhombus, is a straight line such that . Prove that and when produced meet at right angles.
We know that the diagonals of a rhombus bisect each other and are perpendicular to each other.
(sides of a rhombus)
In , since is midpoint of and is midpoint of
The line joining the midpoints i.e
Therefore in parallelogram and
Question 9: is a parallelogram, is produced to so that and produced meets produced in . Prove that .
… … … … … i)
… … … … … i)
From i) and ii)
Therefore or . Hence proved.