# Kinds of Quadrilaterals

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Now, let's look into the classifications of quadrilaterals based on the type of sides and angles it consists of. We can look at the

## (1) Trapezium

The **consists of a pair of parallel sides**.

Usually, **they consist of only one pair of opposite parallel sides**. If the quadrilateral has two pairs of opposite parallel sides, the quadrilateral is further classified, as we will see further on.

Trapeziums **can further be divided into** the following types:

**Scalene Trapezium** : The non-parallel lines are unequal in length.

**Isosceles Trapezium** : Non-parallel lines are equal in length.

**Right Trapezium** : Has atleast two adjacent right angles.

Let's try out an **activity**.

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Using paper, make identical cut-outs of a triangle with sides 3 cm, 4 cm, 5 cm. If we take three cut-outs and try to form a quardrilateral with it, we will get a

**Which are the parallel sides here?****Are the non-parallel sides be equal?**(Yes/No) - You have answered all the questions. Moving on.

**Activity 2:** Take four set-squares from the geometry box. As in the earlier example try to form different types of trapeziums.

Can we obtain an isoceles trapezium while carrying out this activity?

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Using all the four set squares, we can get an iscosceles trapezium by arranging as given below:

## Kite

Kite is the type of **a quadrilateral which has exactly two distinct consecutive pairs of sides of equal length**.

**Note**: The sides with the same markings in the above figure are equal. For example AB = AD and BC = CD.

(i) A kite has 4 sides (It is a quadrilateral).

(ii) There are exactly two distinct consecutive pairs of sides of equal length.

**Think:** Is a square also a kite?

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## Parallelogram

A parallelogram is a **quadrilateral having opposite parallel sides**.

**Now**, is a rectangle also a parallelogram ?

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## Elements of a parallelogram

Now, let's discuss the properties of a parallelogram.

In the parallelogram ABCD figure given above:

**AB and DC**and**AD and BC**are the.

2. **∠A and ∠C** and **∠B and ∠D** form the pair of

3. AB and BC are

4. **∠A and ∠B** and **∠B and ∠C** are ** angles**.

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Upon studying parallelograms, we find that:

**Property**: **The opposite sides of a parallelogram are of equal length**.

- Considering the triangles: ∆ABC and ∆ADC
- We see: ∠DAC = ∠
and ∠DCA = ∠ as they are angles - AC = AC as it is a
side - Therefore, ∆ ABC ≅ ∆ CDA by
congruency criterion. - Hence, AB =
and BC = - Therefore, we see that the opposite sides of a parallelogram are equal.

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### Try These

**Take two identical set squares with angles 30° – 60° – 90° and place them adjacently to form a parallelogram as shown in Fig. Does this help you to verify the above property?**

**Property:** The opposite sides of a parallelogram are of

As we can see in the figure above, the opposite sides of figure are equal.

The figure above is a

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**Example 3: Find the perimeter of the parallelogram PQRS.**

**Solution:** In a parallelogram, the opposite sides have same length.

Therefore, PQ = SR =

So, Perimeter = PQ + QR + RS + SP = 12 cm + 7 cm + 12 cm + 7 cm =

**Thus, the perimeter of the parallelogram PQRS is 38 cm.**

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## Angles of a parallelogram

Moving on to the property concerning the (opposite) angles of a parallelogram.

**Property:** **The opposite angles of a parallelogram are of equal measure.**

- Considering the triangles- ∆ABC and ∆ADC , we know that ∆ ABC ≅ ∆ CDA by
congruency criterion. - Therefore, ∠ADC = ∠
. Similarly, if we join BD and consider the triangles- ∆ADB and ∆CBD, we get: ∠BAD = ∠ - Hence, opposite angles are equal for a parallelogram.

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### Try These

**Take two identical 30° – 60° – 90° set squares and place them adjacently to form a parallelogram. Does the figure obtained help you to verify the above property?**

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**Example 4: In the figure, BEST is a parallelogram. Find the values x, y and z.**

**Solution:** S is

So, x =

y =

z =

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For the **adjacent angles of a parallelogram**:

**Property:** **The adjacent angles in a parallelogram are supplementary.**

- We know that, the co-interior angles in a pair of parallel lines are
to each other. - As AB||DC with AD being the transverse cutting it, we can say: ∠A + ∠D =
° - Similarly, the side BC also acts like a
for the parallel sides AB and DC, giving ∠B + ∠C = ° - Now, as AD||BC are a pair of parallel lines with AB and DC as the transverse, we get: ∠A + ∠B = ∠D + ∠C =
° - Therefore, the sum of any two adjacent angles of a parallelogram is equal to 180°

Let's try out a related problem.

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**Example 5: In a parallelogram RING, if m∠R = 70°, find all the other angles.**

- From the figure, we know that: ∠N =
° **As the opposite angles of a parallelogram are equal**. We also know that,*Sum of any two adjacent angles of a parallelogram is equal to 180°*.- Therefore, ∠N + ∠I =
° - ∠I =
° - °. Thus, ∠I = ° - Thus, m∠I =
° m∠N = ° m∠G = °

**After showing m∠R = m∠N = 70°, can you find m∠I and m∠G by any other method?**

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**Alternate Solution:** Using the formula for sum of interior angles of a polygon:

*Sum of interior angles of a polygon = (n - 2) x 180°*

where n is the number of sides

Putting n =

Sum of interior angles =

We know that, ∠I = ∠G from the parallelogram properties.

Therefore we have,

∠R + ∠I + ∠N + ∠G = 360°

70° + 2∠I + 70° = 360°

∠I =

Therefore,

∠I = ∠G = 110°

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## Diagonals of a parallelogram

Generally speaking, the **diagonals of a parallelogram are not of equal length**. However, the diagonals of a parallelogram share an **interesting property**.

**Property:** **The diagonals of a parallelogram bisect each other with respect to the point of intersection.**

- Considering the triangles- ∆PQT and ∆SRT, we see that: PQ =
as they are sides of a parallelogram - We also have: ∠TPQ = ∠
and ∠TQP = ∠ as they are angles - Thus, by
congruency criterion- ∆ PQT ≅ ∆ - This means PT =
and RT = - Therefore, the diagonals of a parallelogram bisect each other.

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**Example 6: In the figure, HELP is a parallelogram. (Lengths are in cms). Given that OE = 4 and HL is 5 more than PE? Find OH.**

**Solution:** If OE = 4 then OP also is

So, PE =

Therefore, HL = 8 +

Hence OH =

**Thus, OH = 6.5 (cms)**