Exercise 13.2
Page-13.19Question 1:
Two opposite angles of a parallelogram are (3x − 2)° and (50 − x)°. Find the measure of each angle of the parallelogram.
Answer 1:
It is given that the two opposite angles of a parallelogram are 
and 
.
We know that the opposite angles of a parallelogram are equal.
Therefore,
…… (i)
Thus, the given angles become
Also, 
.
Therefore the sum of consecutive interior angles must be supplementary.
That is;
Since opposite angles of a parallelogram are equal.
Therefore,
And 
Hence the four angles of the parallelogram are 
, 
, and .
Question 2:
If an angle of a parallelogram is two-third of its adjacent angle, find the angles of the parallelogram.
Answer 2:
Let one of the angle of the parallelogram as 
Then the adjacent angle becomes 
We know that the sum of adjacent angles of the parallelogram is supplementary.
Therefore,
Thus, the angle adjacent to 
Since, opposite angles of a parallelogram are equal.
Therefore, the four angles in sequence are 
,
,and.
Question 3:
Find the measure of all the angles of a parallelogram, if one angle is 24° less than twice the smallest angle.
Answer 3:
Let the smallest angle of the parallelogram be
Therefore, according to the given statement other angle becomes 
.
Also, the opposite angles of a parallelogram are equal.
Therefore, the four angles become ,,and.
According to the angle sum property of a quadrilateral:
Thus, the other angle becomes
Hence, the four angles of the parallelogram are 
, 
, and .
Question 4:
The perimeter of a parallelogram is 22 cm. If the longer side measures 6.5 cm what is the measure of the shorter side?
Answer 4:
Let the shorter side of the parallelogram be 
cm.
The longer side is given as
cm.
Perimeter of the parallelogram is given as 22 cm
Therefore,
Hence, the measure of the shorter side is 
cm.
Question 5:
In a parallelogram ABCD, ∠D = 135°, determine the measures of ∠A and ∠B.
Answer 5:
It is given that ABCD is a parallelogram with 
We know that the opposite angles of the parallelogram are equal.
Therefore,
Also, 
and
are adjacent angles, which must be supplementary.
Therefore,
Hence , 
and 
.
Question 6:
ABCD is a parallelogram in which ∠A = 70°. Compute ∠B, ∠C and ∠D.
Answer 6:
It is given that ABCD is a parallelogram with 
We know that the opposite angles of the parallelogram are equal.
Therefore,
Also, and
are adjacent angles, which must be supplementary.
Therefore,
Also, and are opposite angles of a parallelogram.
Therefore,
Hence, the angles of a parallelogram are 
, 
, and .
Question 7:
In the given figure, ABCD is a parallelogram in which ∠DAB = 75° and ∠DBC = 60°. Compute ∠CDB and ∠ADB.
Answer 7:
The figure is given as follows:
It is given that ABCD is a parallelogram.
Thus 
And 
are
alternate interior opposite angles.
Therefore,
…… (i)
We know that the opposite angles of a parallelogram are equal. Therefore,
Also, we have 
Therefore,
…… (ii)
In 
By angle sum property of a triangle.
From (i) and (ii),we get:
Hence, the required value for is 
And 
is 
.
Question 8:
Which of the following statements are true (T) and which are false (F)?
(i) In a parallelogram, the diagonals are equal.
(ii) In a parallelogram, the diagonals bisect each other.
(iii) In a parallelogram, the diagonals intersect each other at right angles.
(iv) In any quadrilateral, if a pair of opposite sides is equal, it is a parallelogram.
(v) If all the angles of a quadrilateral are equal, it is a parallelogram.
(vi) If three sides of a quadrilateral are equal, it is a parallelogram.
(vii) If three angles of a quadrilateral are equal, it is a parallelogram.
(viii) If all the sides of a quadrilateral are equal it is a parallelogram.
Answer 8:
(i) Statement: In a parallelogram, the diagonals are equal.
False
(ii) Statement: In a parallelogram, the diagonals bisect each other.
True
(iii) Statement: In a parallelogram, the diagonals intersect each other at right angles.
False
(iv) Statement: In any quadrilateral, if a pair of opposite sides is equal, it is a parallelogram.
False
(v) Statement: If all the angles of a quadrilateral are equal, then it is a parallelogram.
True
(vi) Statement: If three sides of a quadrilateral are equal, then it is not necessarily a parallelogram.
False
(vii) Statement: If three angles of a quadrilateral are equal, then it is no necessarily a parallelogram.
False
(viii) Statement: If all sides of a quadrilateral are equal, then it is a parallelogram.
True
Question 9:
In the given figure, ABCD is a parallelogram in which ∠A = 60°. If the bisectors of ∠A and ∠B meet at P, prove that AD = DP, PC = BC and DC = 2AD.
Answer 9:
The figure is given as follows:
It is given that ABCD is a parallelogram.
Thus,
Opposite angles of a parallelogram are equal.
Therefore, 
Also, we have AP as the bisector of
Therefore,
…… (i)
Similarly,
…… (ii)
We have 
,
From (i)
Thus, sides opposite to equal angles are equal.
Similarly,
From (ii)
Thus, sides opposite to equal angles are equal.
Also,
Question 10:
In the given figure, ABCD is a parallelogram and E is the mid-point of side BC. If DE and AB when produced meet at F, prove that AF = 2AB.
Answer 10:
Figure is given as follows:
It is given that ABCD is a parallelogram.
DE and AB when produced meet at F.
We need to prove that 
It is given that
Thus, the alternate interior opposite angles must be equal.
In 
and 
, we have
(Proved above)
(Given)
(Vertically opposite angles)
Therefore,
(By ASA Congruency )
By corresponding parts of congruent triangles property, we get
DC = BF …… (i)
It is given that ABCD is a parallelogram. Thus, the opposite sides should be equal. Therefore,
…… (ii)
But,
From (i), we get:
From (ii), we get:
Hence proved.
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