## Key Concepts

- Define a trapezoid.
- Explain the properties of a trapezoid.
- Solve problems based on properties of a kite.

## Trapezoid

A quadrilateral having only one pair of parallel sides is called a **trapezoid**.

- A trapezium in which the non-parallel sides are equal is an
**isosceles trapezium**.

### Property of a trapezoid related to base angles

#### Theorem 1:

In an isosceles trapezoid, each pair of base angles is congruent.

**Given: **ABCD is a trapezoid where AB∥CD.

**To prove: **∠ADC = ∠BCD and ∠BAD = ∠ABC

**Proof:**

Draw perpendicular lines AE and BF between the parallel sides of the trapezoid.

In ΔAED and ΔBFC,

AD = BC [Isosceles trapezoid]

AE = BF [Distance between parallel lines will always be equal]

∠AEB = ∠BFC=90° [AEꞱCD and BFꞱCD]

If two right-angled triangles have their hypotenuses equal in length and a pair of shorter sides are equal in length, then the triangles are congruent.

∴ ΔAED ≌ ΔBFC [RHS congruence rule]

We know that the corresponding parts of congruent triangles are equal.

Hence, **∠ADC = ∠BCD**

And ∠EAD = ∠FBC

Now, ∠BAD = ∠BAE + ∠EAD

∠BAD = 90° + ∠EAD

∠BAD = 90° + ∠EAD

**∠BAD = ∠ABC**

Hence, each pair of base angles of an isosceles trapezoid is congruent.

### Property of trapezoid related to the length of diagonals

#### Theorem 2:

The diagonals of an isosceles trapezoid are congruent.

**Given:** In trapezoid ABCD, AB∥CD, and AD = BC

**To prove:** AC = BD

**Proof:**

In ΔADC and ΔBCD,

AD = BC [Isosceles trapezoid]

∠ADC = ∠BCD [Base angles of isosceles trapezoid]

CD = CD [Common]

Therefore, ΔAED ≌ ΔBFC [SAS congruence rule]

We know that the corresponding parts of congruent triangles are equal.

So, AC = BD

Hence, the diagonals of an isosceles trapezoid are congruent.

### Property of trapezoid related to the length of diagonals

#### Theorem 3:

In a trapezoid, the midsegment is parallel to the bases, and the length of the midsegment is half the sum of the lengths of the bases.

**Given:** In trapezoid ABCD, AB∥CD, and X is the midpoint of AD, Y is the midpoint of BC.

**To prove:** XY = 1/2 × (AB+CD)

**Proof: **Construct BD such that the midpoint of BD passes through XY.

In ΔADB, X is the midpoint of AD, and M is the midpoint of DB.

So, XM is the midsegment of ΔADB.

We know that a line segment joining the midpoints of two sides of the triangle is parallel to the third side and has a length equal to half the length of the third side. [Midsegment theorem]

∴ XM ∥ AB and XM = 1/2 × AB …(1)

In ΔBCD, Y is the midpoint of BC and M is the midpoint of BD.

So, MY is the midsegment of ΔBCD.

∴ MY ∥ CD and MY = 1/2 × CD …(2) [Midsegment theorem]

Since XM ∥ AB and MY ∥ CD, so, XY

Now, XY=XM+MY

= 1/2 × AB + 1/2 × CD

** XY = 1/2** **× (AB+CD)**

### Kite

A **kite **is a quadrilateral whose four sides can be grouped into two pairs of equal-length sides that are adjacent to each other.

(or)

A parallelogram also has two pairs of equal-length sides, but they are opposite each other in a **kite**.

- Only one diagonal of a kite bisects the other diagonal.

### Property of kite related to the angle between the diagonals

#### Theorem:* *

The diagonals of a kite are perpendicular.

**Given:** In kite WXYZ, XY=YZ, WX=ZW

**To prove:** XZ Ʇ WY

**Proof: **Draw diagonals XZ and WY. Let the diagonals intersect at O.

In ΔWXY and ΔWZY,

WX=WZ [Adjacent sides of kite]

XY=ZY [Adjacent sides of kite]

WY=YW [Reflexive property]

∴ ΔWXY ≌ ΔWZY [SSS congruence rule]

We know that the corresponding parts of congruent triangles are equal.

So, ∠XYW= ∠ZYW …(1)

In ΔOXY and ΔOZY,

XY=ZY [Adjacent sides of kite]

OY=YO [Reflexive property]

∠XYW= ∠ZYW [from (1)]

∴ ΔOXY ≌ ΔOZY [SAS congruence rule]

So, ∠YOX = ∠YOZ [CPCT]

But ∠YOX+∠YOZ = 180°

2∠YOX = 180°

∠YOX = 90°

Hence, the diagonals of a kite are perpendicular.

## Exercise

- Find the value of k if STUV is a trapezoid.

- If EFGH is an isosceles trapezoid, find the value of
*p*.

- Find the length of PQ if LMQP is a trapezoid and O is the midpoint of LP and N is the midpoint of MQ.

- What must be the value of
*m*if JKLM is a kite?

- If WXYZ is a kite, find the length of diagonal XZ.

### Concept Map

### What we have learned

- A quadrilateral having only one pair of parallel sides is called a
**trapezoid**. - A
**kite**is a quadrilateral whose four sides can be grouped into two pairs of equal-length sides that are adjacent to each other.

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