Angles of Parallelogram

A Parallelogram can be defined as a quadrilateral whose two s sides are parallel to each other and all the four angles at the vertices are not 90 degrees or right angles, then the quadrilateral is called a parallelogram. The opposite sides of parallelogram are also equal in length.

Properties of Parallelogram

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 The important properties of parallelograms to know are:

1. Opposite sides of parallelogram are equal (AB = DC).

2. Opposite angles of parallelogram are equal (D = B).

3. Consecutive angles in a parallelogram are supplementary (A + D = 180°).

4. If one angle is 90 degrees, then all other angles are also 90 degrees.

5. The diagonals of a parallelogram bisect each other in two equal halves.

6. The diagonal of a parallelogram separates it into two congruent triangles.

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If all sides of the parallelogram are equal then the shape we have is called a rhombus. The properties of parallelograms can also be applied on rhombi. In a parallelogram where one pair of sides are parallel then the shape we have what is called a trapezoid. The parallel sides are known as bases while the nonparallel sides are known as legs. If the legs are congruent we have what is called an isosceles trapezoid.

In an isosceles trapezoid the diagonals are always congruent. The median of a trapezoid is parallel to the bases and is one-half of the sum of measures of the bases.

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In this topic we are studying about angle of parallelogram.so the theorem related to the angles are: 

1. Opposite angles of parallelogram are equal

2. Consecutive angles in a parallelogram is supplementary 

Theorem: Prove that the opposite angles of a parallelogram are equal.

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Given: Parallelogram ABCD.

To prove: ∠B = ∠D and ∠A=∠C

Proof:

In the parallelogram ABCD,

AB \\ CD and AD \\ BC

Opposite angles of a parallelogram

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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.

Example-

Question:  To prove that the opposite angles of a parallelogram are equal

Solution: Let ABCD be a parallelogram and AC as its diagonal.

According to the theorem opposite sides of a parallelogram are equal

So, AB∥DC and AD∥BC

Since, AB∥DC and AC is the transversal

⇒ ∠BAC=∠DCA    ---- (1 )  [ Alternate angles ]

Similarly, AD∥BC and AC are the transversal.

⇒  ∠DAC=∠BCA     ---- ( 2 )  [ Alternate angles ]

Adding ( 1 ) and ( 2 ),

⇒  ∠BAC+∠DAC=∠DCA+∠BCA

⇒  ∠BAD=∠DCB

Similarly, we can prove, ∠ADC=∠ABC

Therefore we have proved that opposite angles of parallelogram are equal.

Consecutive Angles of a Parallelogram

Theorem: Prove that any consecutive angles of a parallelogram are supplementary.

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Given: Parallelogram ABCD.

To prove: ∠A + ∠B = 180 degrees, ∠C + ∠D = 180 degrees

Proof:

Angles of a parallelogram 

AB ∥ CD and AD are transversal.

We know that interior angles on the same side 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.

From the above theorem, it can be derived that if one angle of a parallelogram is equal to 90 degrees, then all four angles are equal to 90 degrees. Therefore, it will become a rectangle.

If one angle is a 90 degrees, then all four angles 90 degrees:

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.

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Solution:

Given,

∠D = 60° and ∠B = (x+20)°

According to the theorem opposite angles of a parallelogram are equal.

Therefore,

(x+20)° = 60°

x = 60° -20°

x = 40°

Hence, the value of x is 40.

Area of a Parallelogram

The Area is the base multiplied by height:

Area = b × h

(h is at right angles to b)

Example:

 A parallelogram has a base of 8 m and is 5 m high, what is its Area?

Area = 8 m × 5 m = 18 m2

Perimeter of a Parallelogram

The Perimeter is the distance around the edges.

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The Perimeter is 2  (base + side length):

Perimeter = 2(b+s)

Solved Example: 

A parallelogram has a base of 10 cm and a side length of 5 cm, what is its Perimeter?

Perimeter = 2 × (10 cm + 5 cm) = 2 × 15 cm = 30 cm