A parallelogram is a type of quadrilateral with four sides, four vertices, and four angles. And it has two diagonals that bisect each other. We know that the opposite sides of a parallelogram are equal and parallel because they never intersect. In this article, we are going to prove that Pair of Opposite Sides of a Parallelogram are Equal and Parallel. Thoroughly read the entire page to find the proof of the given statement with suitable examples.
Pair of Opposite Sides of a Parallelogram are Equal and Parallel Theorem
Let us discuss about the statement “Pair of Opposite Sides of a Parallelogram are Equal and Parallel” with step by step explanation here.
Statement:
Prove that, A quadrilateral is a parallelogram if a pair of opposite sides are equal and parallel.
Proof:
Given: ABCD is quadrilateral and AB║CD, AB=CD.
To prove: ABCD is a parallelogram
Proof: AC is a transversal and also AB║CD, therefore
∠BAC=∠DCA(Alternate angles)
In ΔADC and ΔCBA, we have
AB=CD(Given)
∠BAC=∠DCA(Alternate angles)
AC=CA(Common)
ΔADC≅ΔCBA by the SAS rule.
Hence, by CPCT, DA=BC
Thus, Both the pair of opposite sides are equal in the quadrilateral ABCD, therefore ABCD is a parallelogram.
Hence proved.
Problems on Pair of Opposite Sides of a Parallelogram are Equal and Parallel
Example 1.
In the parallelogram PQRS ∠PQR = 60° find the measures of ∠QRS, ∠RSP and ∠SPQ
Solution:
Given that
∠PQR = 60°
In a parallelogram sum of any two opposite angles is equal to 180°
PQ//RS,∠PQR + ∠QRS = 180°
∠QRS = 180° – ∠PQR
= 180° – 60°
= 120°
In the parallelogram opposite angles are equal so
∠RSP = ∠PQR = 60° and
∠QRS = ∠SPQ = 120°
Example 2.
The two angles of a parallelogram have measures of (3x – 12)° and (2x + 16)° what is the measures of all the angles of this parallelogram
Solution:
Given that
Two angles are (3x – 12)° and (2x + 16)°
We know that
The two opposite angles of a parallelogram are equal to each other
Here
3x – 12 = 2x + 16
3x – 12 – 2x – 16
x = 28
(3x – 12)° = (3(28) – 12)
= 72°
(180 – 72) = 108°
Two opposite angles are 72° and the other two are 108°.
Example 3.
In the figure, PQRS is a parallelogram in which ∠P = 25° find the measures of each of the angles ∠Q, ∠R, and ∠S
Solution:
It is given that PQRS is a parallelogram in which ∠P = 85°
We know that
Sum of any two adjacent angles of a parallelogram is 180°
∠P + ∠Q = 180°
25° + ∠Q = 180°
∠Q = (180A° – 25°) = 165°
∠P + ∠Q = 165°
∠R + ∠S = 180°
25° + ∠S = 180°
∠S = (180° – 25°) =165°
Therefore
∠Q = 165°, ∠R = 25, and ∠S = 165°
Example 4.
In the parallelogram PQRS ∠PQR = 80° find the measures of ∠QRS, ∠RSP and ∠SPQ
Solution:
PQ//RS,∠PQR + ∠QRS = 180°
∠QRS = 180° – ∠PQR
= 180° – 80°
= 100°
In the parallelogram opposite angles are equal so
∠RSP = ∠PQR = 80° and
∠QRS = ∠SPQ = 100°
Example 5.
The two angles of a parallelogram have measures of (4x – 6)° and (3x + 13)° what is the measures of all the angles of this parallelogram
Solution:
Given that
Two angles are (4x – 6)° and (3x + 13)°
We know that
The two opposite angles of a parallelogram are equal to each other
Here
4x – 6 = 3x + 13
4x – 6 – 3x – 13
x = 19
(4x – 6)° = (4(19) – 12)
= 64°
(180 – 64) = 116°
The two opposite angles are 64° and the other two are 116°.
FAQs on Pair of Opposite Sides of a Parallelogram are Equal and Parallel
1. Are both pairs of opposite sides parallel in a parallelogram?
A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel.
2. Why are opposite sides of a parallelogram equal?
If one pair of opposite sides of a quadrilateral is equal and parallel, then the quadrilateral is a parallelogram.
3. How many pairs of parallel sides does a parallelogram have?
A parallelogram has two pairs of parallel sides.