Question 1: Three angles of a quadrilateral are respectively equal to and . Find its fourth angle.

Answer:

Sum of internal angles

Therefore

Question 2: In a quadrilateral , the angles and are in the ratio . Find the measure of each angles of the quadrilateral.

Answer:

The ratio of angles is

Sum of internal angles

Therefore

Therefore angles are and

Question 3: In a quadrilateral and are the bisectors of and respectively. Prove that .

Answer:

To prove:

In

Now,

Therefore

Question 4: The angles of a quadrilateral are in the ratio . Find all the angles of the quadrilateral.

Answer:

Angles are in ratio of

Sum of internal angles

Therefore

Therefore angles are and

Question 5: Two opposite angles of a parallelogram are and . Find the measure of each angle of the parallelogram.

Answer:

In a parallelogram, the sum of diagonally opposite angles are equal.

Therefore

Therefore angles and are

We know that

Therefore angles are

Question 6: If an angle of a parallelogram is two-third of its adjacent angle, find the angles of the parallelogram.

Answer:

Given

Therefore

Therefore

Hence the angles are

Question 7: Find the measure of all the angles of a parallelogram, if one angle is less than twice the smallest angle.

Answer:

Let the smallest angle

Therefore the other angle would be

We know that the

Hence

Hence the angles are

Question 8: The perimeter of a parallelogram is . If the longer side measures what is the measure of the shorter side?

Answer:

Perimeter

Let the shorter side

Therefore

Question 9: In a parallelogram , determine the measures of and .

Answer:

Given

Therefore

Question 10: is a parallelogram in which . Compute and .

Answer:

Given

Therefore

Question 11: In the adjoining figure, is a parallelogram in which . If the bisectors of and meet at , prove that and .

Answer:

To prove i) ii) iii)

Given

AP bisects Therefore

Also

Bisects Therefore

Also

i) Consider ,

Since

(sides opposite to equal angles in a triangle are equal)

… … … … … i)

ii) Similarly in (all angles are equal to as it is an equilateral triangle)

Hence … … … … … ii)

iii) Since … … … … … iii) (opposite sides of a parallelogram)

Adding i) and ii)

From iii)

Question 12: In the adjoining figure, is a parallelogram in which and . Compute and .

Answer:

Given

In

Question 13: In the adjoining figure, is a parallelogram and is the mid-point of side . If and when produced meet at , prove that .

Answer:

To prove:

Given: is a parallelogram, is the mid point of

Consider and

is mid point of

(vertically opposite angles)

(alternate angles)

Therefore (By A.A.S criterion)

Hence

Since

Adding on both sides

. Hence Proved.

Question 14: Which of the following statements are true (T) and which are false (F) ?

- In a parallelogram, the diagonals are equal – FALSE
- In a parallelogram, the diagonals bisect each other – TRUE
- In a parallelogram, the diagonals intersect each other at right angles– FALSE
- In any quadrilateral, if a pair of opposite sides is equal, it is parallelogram– TRUE
- If all the angles of a quadrilateral are equal, it is a parallelogram– TRUE
- If three sides of a quadrilateral are equal, it is a parallelogram– FALSE
- If three angles of a quadrilateral are equal, it is a parallelogram– FALSE
- If all the sides of a quadrilateral are equal it is a parallelogram– TRUE