A quadrilateral whose two pairs of sides are parallel to each and the four angles at the vertices are not equal to the right angle, then the quadrilateral is called a parallelogram. Also, the opposite sides are equal in length.
Here,
AD = BC (opposite sides)
AB = CD (opposite sides)
Sum of all the four angles = 360 degrees
The important properties of angles of a parallelogram are:
- If one angle of a parallelogram is a right angle, then all the angles are right angles
- Opposite angles of a parallelogram are equal (or congruent)
- Consecutive angles are supplementary angles to each other (that means they add up to 180 degrees)
Read more:
Parallel Lines Transversals Angle
Opposite Angles of a Parallelogram
In the above parallelogram, A, C and B, D are a pair of opposite angles.
Opposite Angles of a Parallelogram are equal
Theorem: Prove that the opposite angles of a parallelogram are equal.
Given: Parallelogram ABCD.
To prove: ∠B = ∠D and ∠A=∠C
Proof:
In the parallelogram ABCD,
AB \\ CD and AD \\ BC
Consider triangle ABC and triangle ADC,
AC = AC (common side)
We know that alternate interior angles are equal.
∠1 = ∠4
∠2 = ∠3
By ASA congruence criterion, two triangles are congruent to each other.
Therefore, ∠B = ∠D and ∠A=∠C
Hence, it is proved that the opposite angles of a parallelogram are equal.
Consecutive Angles of a Parallelogram
Theorem: Prove that any consecutive angles of a parallelogram are supplementary.
Given: Parallelogram ABCD.
To prove: ∠A + ∠B = 180 degrees, ∠C + ∠D = 180 degrees
Proof:
AB ∥ CD and AD is a transversal.
We know that interior angles on the same side of a transversal are supplementary.
Therefore, ∠A + ∠D = 180°
Similarly, ∠B + ∠C = 180°, ∠C + ∠D = 180° and ∠A + ∠B = 180°.
Therefore, the sum of any two adjacent angles of a parallelogram is equal to 180°.
Hence, it is proved that any two adjacent or consecutive angles of a parallelogram are supplementary.
If one angle is a right angle, then all four angles are right angles:
From the above theorem, it can be decided that if one angle of a parallelogram is a right angle (that is equal to 90 degrees), then all four angles are right angles. Hence, it will become a rectangle.
Since, the adjacent sides are supplementary.
For example, ∠A, ∠B are adjacent angles and ∠A = 90°, then:
∠A + ∠B = 180°
90° + ∠B = 180°
∠B = 180° – 90°
∠B = 90°
Similarly, ∠C = ∠D = 90°
Example 1:
In the adjoining figure, ∠D = 85° and ∠B = (x+25)°, find the value of x.
Solution:
Given,
∠D = 85° and ∠B = (x+25)°
We know that, opposite angles of a parallelogram are congruent or equal.
Therefore,
(x+25)° = 85°
x = 85° -25°
x = 60°
Hence, the value of x is 60.
