17 Parallelograms II
By studying this lesson you will be able to
identify the conditions that need to be satisfied for a quadrilateral to be a
parallelogram.
Theorem: If the opposite sides of a quadrilateral are equal, then it is a
parallelogram.
A 8 cm B
For example, in the given figure, AB = DC and
AD = BC. 5 cm 5 cm
Therefore, ABCD is a parallelogram.
D 8 cm C
Let us engage in the following activity to establish the truth of the above theorem.
Activity 1
• Draw as shown in the figure, such that the sides of the angle are of length
5 cm and 3 cm.
D
• As shown in the second figure, obtain the point C
which is 3 cm from B and 5 cm from D. 3 cm
• Now complete the quadrilateral ABCD.
• Then it can be seen that AB = DC and AD = BC. A 5 cm B
D 5 cm
• By using a set square and a ruler or by measuring C
the angles and obtaining that the sum of a pair of 3 cm
3 cm
allied angles is 180o, observe that the opposite sides
of the quadrilateral are parallel. That is, obtain that A B
AB//DC and AD//BC. 5 cm
It can be observed that in a quadrilateral with opposite side equal, the opposite
sides are parallel too.
169
, Theorem: If the opposite angles of a quadrilateral are equal, then it is a
parallelogram.
S R
For example, in the given figure, since 70x 110x
and , by the
theorem, PQRS is a parallelogram.
110x 70x
P Q
Let us engage in the following activity to establish the truth of the above theorem.
Activity 2
• Measure all the angles in each of the following quadrilaterals.
R
N M
D C
S
K L A B
P Q
• Check whether the opposite pairs of angles of each quadrilateral are equal.
• Check whether the pairs of opposite sides are parallel in the quadrilateral with
opposite angles equal. (See whether the sum of the allied angles is 180o.)
Accordingly, observe that in the quadrilateral with opposite angles equal, the
opposite sides are parallel to each other.
Theorem: If the diagonals of a quadrilateral bisect each other, then it is a
parallelogram.
A D
For example, in the quadrilateral ABCD, since
AO = OC and BO = OD, according to the theorem,
ABCD is a parallelogram. O
B C
170
By studying this lesson you will be able to
identify the conditions that need to be satisfied for a quadrilateral to be a
parallelogram.
Theorem: If the opposite sides of a quadrilateral are equal, then it is a
parallelogram.
A 8 cm B
For example, in the given figure, AB = DC and
AD = BC. 5 cm 5 cm
Therefore, ABCD is a parallelogram.
D 8 cm C
Let us engage in the following activity to establish the truth of the above theorem.
Activity 1
• Draw as shown in the figure, such that the sides of the angle are of length
5 cm and 3 cm.
D
• As shown in the second figure, obtain the point C
which is 3 cm from B and 5 cm from D. 3 cm
• Now complete the quadrilateral ABCD.
• Then it can be seen that AB = DC and AD = BC. A 5 cm B
D 5 cm
• By using a set square and a ruler or by measuring C
the angles and obtaining that the sum of a pair of 3 cm
3 cm
allied angles is 180o, observe that the opposite sides
of the quadrilateral are parallel. That is, obtain that A B
AB//DC and AD//BC. 5 cm
It can be observed that in a quadrilateral with opposite side equal, the opposite
sides are parallel too.
169
, Theorem: If the opposite angles of a quadrilateral are equal, then it is a
parallelogram.
S R
For example, in the given figure, since 70x 110x
and , by the
theorem, PQRS is a parallelogram.
110x 70x
P Q
Let us engage in the following activity to establish the truth of the above theorem.
Activity 2
• Measure all the angles in each of the following quadrilaterals.
R
N M
D C
S
K L A B
P Q
• Check whether the opposite pairs of angles of each quadrilateral are equal.
• Check whether the pairs of opposite sides are parallel in the quadrilateral with
opposite angles equal. (See whether the sum of the allied angles is 180o.)
Accordingly, observe that in the quadrilateral with opposite angles equal, the
opposite sides are parallel to each other.
Theorem: If the diagonals of a quadrilateral bisect each other, then it is a
parallelogram.
A D
For example, in the quadrilateral ABCD, since
AO = OC and BO = OD, according to the theorem,
ABCD is a parallelogram. O
B C
170
