Need Help?

Get in touch with us

bannerAd

Properties of Trapezoids

Sep 12, 2022
link

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.

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

Property of a trapezoid related to base angles

Theorem 1:

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

Theorem 1

Given: ABCD is a trapezoid where AB∥CD.

To prove: ∠ADC = ∠BCD and ∠BAD = ∠ABC

Proof:

parallel

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]

parallel

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.

Theorem 2:

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.

Theorem 3

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

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.

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.

Theorem: kite

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.
Find the value of k if STUV is a trapezoid.
  • If EFGH is an isosceles trapezoid, find the value of p.
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.
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?
What must be the value of m if JKLM is a kite?
  • If WXYZ is a kite, find the length of diagonal XZ.
If WXYZ is a kite, find the length of diagonal XZ.

Concept Map

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.

Comments:

Related topics

Composite Figures – Area and Volume

A composite figure is made up of simple geometric shapes. It is a 2-dimensional figure of basic two-dimensional shapes such as squares, triangles, rectangles, circles, etc. There are various shapes whose areas are different from one another. Everything has an area they occupy, from the laptop to your book. To understand the dynamics of composite […]

Read More >>
special right triangles_01

Special Right Triangles: Types, Formulas, with Solved Examples.

Learn all about special right triangles- their types, formulas, and examples explained in detail for a better understanding. What are the shortcut ratios for the side lengths of special right triangles 30 60 90 and 45 45 90? How are these ratios related to the Pythagorean theorem?  Right Angle Triangles A triangle with a ninety-degree […]

Read More >>
simplify algebraic expressions

Ways to Simplify Algebraic Expressions

Simplify algebraic expressions in Mathematics is a collection of various numeric expressions that multiple philosophers and historians have brought down. Talking of algebra, this branch of mathematics deals with the oldest concepts of mathematical sciences, geometry, and number theory. It is one of the earliest branches in the history of mathematics. The study of mathematical […]

Read More >>
solve right triangles

How to Solve Right Triangles?

In this article, we’ll learn about how to Solve Right Triangles. But first, learn about the Triangles.  Triangles are made up of three line segments. These three segments meet to form three angles. The lengths of the sides and sizes of the angles are related to one another. If you know the size (length) of […]

Read More >>

Other topics