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