Question 1: In a parallelogram , determine the sum of angles and .

Answer:

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

Therefore

Question 2: In a parallelogram , if , determine the measures of its other angles.

Answer:

Given

Also is a parallelogram.

Therefore and and

Question 3: is a square. and intersect at . State the measure of .

Answer:

Consider and

is common

(side of a square)

(diagonals of a square bisect each other)

Therefore

Hence

Since

Question 4: is a rectangle with . Determine .

Answer:

Given is a rectangle with

Question 5: The sides and, of a parallelogram are bisected at and . Prove that is a parallelogram.

Answer:

Given: is a parallelogram, bisects and bisects

Therefore and

also

and

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 .

Answer:

Given:

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

Therefore

Question 7: is a square and are points on and respectively, such that . Prove that is a square.

Answer:

Given:

Since is a square

Now consider and

Therefore

Therefore 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.

Answer:

To prove:

We know that the diagonals of a rhombus bisect each other and are perpendicular to each other.

Given: and

(sides of a rhombus)

In , since is midpoint of and is midpoint of

The line joining the midpoints i.e

Similarly

Therefore in parallelogram and

Therefore

Hence proved.

Question 9: is a parallelogram, is produced to so that and produced meets produced in . Prove that .

Answer:

To prove:

Given:

Consider and

In

Since

… … … … … i)

Also since

… … … … … i)

From i) and ii)

Therefore or . Hence proved.