problem solving properties of kites and trapezoids

Get in touch with us

Are you sure you want to logout?

Please select your grade.

• Earth and space

Properties of Trapezoids and Kites: Problem-Solving

• 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 = ∠BCDand ∠BAD = ∠ABC

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 = ∠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

In ΔADC and ΔBCD,

AD = BC                 [Isosceles trapezoid]

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

CD = CD                [Common]

Therefore, ΔAED ≌ ΔBFC   [SAS congruence rule]

So, AC = BD

Hence, the diagonals of an isosceles trapezoid are congruent.

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 a trapezoid ABCD, AB∥CD, and X is the midpoint of AD, and Y is the midpoint of BC.

To prove: XY = 1/2 x (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 ]

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

So, MY is the midsegment of ΔBCD.

Since XM ∥ AB and MY ∥ CD, so, XY

Now, XY=XM+MY

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

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: XZꞱWY

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]

So, ∠XYW= ∠ZYW       …(1)

In ΔOXY and ΔOZY,

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.

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

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

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

4. What must be the value of m if JKLM is a kite?

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

Concept Map:

 What We Have Learned

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

Related topics

Addition and Multiplication Using Counters & Bar-Diagrams

Introduction: We can find the solution to the word problem by solving it. Here, in this topic, we can use 3 methods to find the solution. 1. Add using counters 2. Use factors to get the product 3. Write equations to find the unknown. Addition Equation: 8+8+8 =? Multiplication equation: 3×8=? Example 1: Andrew has […]

Dilation: Definitions, Characteristics, and Similarities

Understanding Dilation A dilation is a transformation that produces an image that is of the same shape and different sizes. Dilation that creates a larger image is called enlargement. Describing Dilation Dilation of Scale Factor 2 The following figure undergoes a dilation with a scale factor of 2 giving an image A’ (2, 4), B’ […]

How to Write and Interpret Numerical Expressions?

Write numerical expressions What is the Meaning of Numerical Expression? A numerical expression is a combination of numbers and integers using basic operations such as addition, subtraction, multiplication, or division. The word PEMDAS stands for: P → Parentheses E → Exponents M → Multiplication D → Division  A → Addition S → Subtraction         Some examples […]

System of Linear Inequalities and Equations

Introduction: Systems of Linear Inequalities: A system of linear inequalities is a set of two or more linear inequalities in the same variables. The following example illustrates this, y < x + 2…………..Inequality 1 y ≥ 2x − 1…………Inequality 2 Solution of a System of Linear Inequalities: A solution of a system of linear inequalities […]

Other topics

How to Find the Area of Rectangle?

How to Solve Right Triangles?

Ways to Simplify Algebraic Expressions

Properties of Trapezoids and Kites

Now that we’ve seen several types of quadrilaterals that are parallelograms , let’s learn about figures that do not have the properties of parallelograms. Recall that parallelograms were quadrilaterals whose opposite sides were parallel. In this section, we will look at quadrilaterals whose opposite sides may intersect at some point. The two types of quadrilaterals we will study are called trapezoids and kites . Let’s begin our study by learning some properties of trapezoids.

Definition: A trapezoid is a quadrilateral with exactly one pair of parallel sides.

Since a trapezoid must have exactly one pair of parallel sides, we will need to prove that one pair of opposite sides is parallel and that the other is not in our two-column geometric proofs . If we forget to prove that one pair of opposite sides is not parallel, we do not eliminate the possibility that the quadrilateral is a parallelogram. Therefore, that step will be absolutely necessary when we work on different exercises involving trapezoids.

Before we dive right into our study of trapezoids, it will be necessary to learn the names of different parts of these quadrilaterals in order to be specific about its sides and angles. All trapezoids have two main parts: bases and legs . The opposite sides of a trapezoid that are parallel to each other are called bases. The remaining sides of the trapezoid, which intersect at some point if extended, are called the legs of the trapezoid.

The top and bottom sides of the trapezoid run parallel to each other, so they are the trapezoid’s bases. The other sides of the trapezoid will intersect if extended, so they are the trapezoid’s legs.

The segment that connects the midpoints of the legs of a trapezoid is called the midsegment . This segment’s length is always equal to one-half the sum of the trapezoid’s bases, or

Consider trapezoid ABCD shown below.

The midsegment, EF , which is shown in red, has a length of

The measurement of the midsegment is only dependent on the length of the trapezoid’s bases. However, there is an important characteristic that some trapezoids have that is solely reliant on its legs. Let’s look at these trapezoids now.

Isosceles Trapezoids

Definition: An isosceles trapezoid is a trapezoid whose legs are congruent.

By definition, as long as a quadrilateral has exactly one pair of parallel lines, then the quadrilateral is a trapezoid. The definition of an isosceles trapezoid adds another specification: the legs of the trapezoid have to be congruent.

ABCD is not an isosceles trapezoid because AD and BC are not congruent. Because EH and FG are congruent, trapezoid EFGH is an isosceles trapezoid.

There are several theorems we can use to help us prove that a trapezoid is isosceles. These properties are listed below.

(1) A trapezoid is isosceles if and only if the base angles are congruent.

(2) A trapezoid is isosceles if and only if the diagonals are congruent.

(3) If a trapezoid is isosceles, then its opposite angles are supplementary.

Definition: A kite is a quadrilateral with two distinct pairs of adjacent sides that are congruent.

Recall that parallelograms also had pairs of congruent sides. However, their congruent sides were always opposite sides. Kites have two pairs of congruent sides that meet at two different points. Let’s look at the illustration below to help us see what a kite looks like.

Segment AB is adjacent and congruent to segment BC. Segments AD and CD are also adjacent and congruent.

Kites have a couple of properties that will help us identify them from other quadrilaterals.

(1) The diagonals of a kite meet at a right angle.

(2) Kites have exactly one pair of opposite angles that are congruent.

These two properties are illustrated in the diagram below.

Notice that a right angle is formed at the intersection of the diagonals, which is at point N. Also, we see that ?K??M. This is our only pair of congruent angles because ?J and ?L have different measures.

Let’s practice doing some problems that require the use of the properties of trapezoids and kites we’ve just learned about.

Find the value of x in the trapezoid below.

Because we have been given the lengths of the bases of the trapezoid, we can figure out what the length of the midsegment should be. Let’s use the formula we have been given for the midsegment to figure it out. (Remember, it is one-half the sum of the bases.)

So, now that we know that the midsegment’s length is 24 , we can go ahead and set 24 equal to 5x-1 . The variable is solvable now:

Find the value of y in the isosceles trapezoid below.

In the figure, we have only been given the measure of one angle, so we must be able to deduce more information based on this one item. Because the quadrilateral is an isosceles trapezoid, we know that the base angles are congruent. This means that ?A also has a measure of 64° .

Now, let’s figure out what the sum of ?A and ?P is:

Together they have a total of 128° . Recall by the Polygon Interior Angle Sum Theorem that a quadrilateral’s interior angles must be 360° . So, let’s try to use this in a way that will help us determine the measure of ?R . First, let’s sum up all the angles and set it equal to 360° .

Now, we see that the sum of ?T and ?R is 232° . Because segment TR is the other base of trapezoid TRAP , we know that the angles at points T and R must be congruent to each other. Thus, if we define the measures of ?T and ?R by variable x , we have

This value means that the measure of ?T and ?R is 116° . Finally, we can set 116 equal to the expression shown in ?R to determine the value of y . We have

So, we get x=9 .

While the method above was an in-depth way to solve the exercise, we could have also just used the property that opposite angles of isosceles trapezoids are supplementary. Solving in this way is much quicker, as we only have to find what the supplement of a 64° angle is. We get

Once we get to this point in our problem, we just set 116 equal to 4(3y+2) and solve as we did before.

After reading the problem, we see that we have been given a limited amount of information and want to conclude that quadrilateral DEFG is a kite. Notice that EF and GF are congruent, so if we can find a way to prove that DE and DG are congruent, it would give us two distinct pairs of adjacent sides that are congruent, which is the definition of a kite.

We have also been given that ?EFD and ?GFD are congruent. We learned several triangle congruence theorems in the past that might be applicable in this situation if we can just find another side or angle that are congruent.

Since segment DF makes up a side of ?DEF and ?DGF , we can use the reflexive property to say that it is congruent to itself. Thus, we have two congruent triangles by the SAS Postulate .

Next, we can say that segments DE and DG are congruent because corresponding parts of congruent triangles are congruent. Our new illustration is shown below.

We conclude that DEFG is a kite because it has two distinct pairs of adjacent sides that are congruent. The two-column geometric proof for this exercise is shown below.

• How many hours until I reach 40?
• If you bought a stock last year for a price of $57, and it has gone down 5% since then, how much is the stock worth now, to the nearest cent?
• How do I determine whether two circles intersect?
• Find the intersection points of two circles.

Home / United States / Math Classes / 4th Grade Math / Parallelograms, Trapezoids and Kites

Parallelograms, Trapezoids and Kites

Parallelograms, trapezoids, and kites are special cases of quadrilaterals. A quadrilateral is a polygon containing four sides. It has four vertices and angles. We will learn about parallelograms, trapezoids, and kites along with their properties and then solve example problems for a better understanding of the concept. ...Read More Read Less

Table of Contents

What is a Parallelogram?

The properties of a parallelogram, what is a trapezoid, the properties of a trapezoid, what is a kite, the properties of a kite, solved examples.

• Frequently Asked Questions

A parallelogram is a quadrilateral with both pairs of opposite sides that are parallel.

• The opposite sides are of equal length in a parallelogram.
• The opposite angles are of equal measure.
• The consecutive angles of a parallelogram are supplementary.
• The diagonals bisect each other in a parallelogram.
• The diagonal of a parallelogram divides it into two triangles that are congruent to each other.

A trapezoid is a quadrilateral with exactly one pair of opposite parallel sides.

• The pairs of parallel sides are called bases.
• The non-parallel sides are called legs.
• The two consecutive angles whose common side is the base are called base angles.
• The angles formed between the parallel sides are supplementary.
• If the legs of a trapezoid are congruent, it is called an isosceles trapezoid.

A kite is a quadrilateral whose two pairs of adjacent sides are congruent but whose opposite sides are not congruent.

• The diagonals are perpendicular to each other in the kite.
• The angles where the unequal sides meet are equal in measure.
• The longer diagonal bisects the other diagonal.

Example 1: A playground is in the shape of a parallelogram. One of its side 20 yards, as shown in the figure below. The side opposite to it is x + 5 yards. Find the value of x .

Solution:  

Let the playground which is a parallelogram be represented by ABCD , 

AB = CD           The opposite sides of a parallelogram are equal

x + 5 = 20         Substitute 

x – 5 = 20 – 5    Subtract 5 from each side

So, the value of x is 15 yards.

Example 2: Find the measure of \(\angle\) BCD if the quadrilateral ABCD is an isosceles trapezoid.

ABCD is an isosceles trapezoid.

\(\angle\)ADC = \(\angle\) BCD     The base angles of an isosceles trapezoid are equal

  70° = \(\angle\) BCD       Substitute

So, the measure of BCD is 70° .

Example 3: Find the value of y if the given quadrilateral is a kite.

  ABCD is a kite

AO = CO             The longer diagonal bisects the other diagonal

y – 3 = 7              Substitute

y – 3 + 3 = 7 + 3   Add 3 to each side

So, the value of y is 10 .

Example 4: Find the measure of \(\angle\) ABC if the given quadrilateral is a parallelogram.

ABCD is a parallelogram, use the parallelogram opposite angles theorem.

\(\angle\)ABC = \(\angle\) ADC   The opposite angles of a parallelogram are equal

\(\angle\)ABC = 75°         Substitute

So, the measure of \(\angle\) ABC is 75° .

How is a parallelogram different from a trapezoid?

A parallelogram has two pairs of parallel sides. In a trapezoid, however, only one pair of the opposite sides is parallel.

Are all parallelograms trapezoids?

All the properties of a trapezoid are present in a parallelogram, so all parallelograms are trapezoids.

Name two quadrilaterals with each angle measuring 90°.

A rectangle and a square are two quadrilaterals with each angle measuring 90°.

What is the name of a trapezoid whose legs are of equal measure?

A trapezoid whose legs are of equal measure is an isosceles trapezoid.

How many types of parallelograms are there?

There are three types of parallelograms:

Check out our other courses

Grades 1 - 12

Level 1 - 10

Child Login

• Kindergarten
• Number charts
• Skip Counting
• Place Value
• Number Lines
• Subtraction
• Multiplication
• Word Problems
• Comparing Numbers
• Ordering Numbers
• Odd and Even
• Prime and Composite
• Roman Numerals
• Ordinal Numbers
• In and Out Boxes
• Number System Conversions
• More Number Sense Worksheets
• Size Comparison
• Measuring Length
• Metric Unit Conversion
• Customary Unit Conversion
• Temperature
• More Measurement Worksheets
• Writing Checks
• Profit and Loss
• Simple Interest
• Compound Interest
• Tally Marks
• Mean, Median, Mode, Range
• Mean Absolute Deviation
• Stem-and-leaf Plot
• Box-and-whisker Plot
• Permutation and Combination
• Probability
• Venn Diagram
• More Statistics Worksheets
• Shapes - 2D
• Shapes - 3D
• Lines, Rays and Line Segments
• Points, Lines and Planes
• Transformation
• Quadrilateral
• Ordered Pairs
• Midpoint Formula
• Distance Formula
• Parallel, Perpendicular and Intersecting Lines
• Scale Factor
• Surface Area
• Pythagorean Theorem
• More Geometry Worksheets
• Converting between Fractions and Decimals
• Significant Figures
• Convert between Fractions, Decimals, and Percents
• Proportions
• Direct and Inverse Variation
• Order of Operations
• Squaring Numbers
• Square Roots
• Scientific Notations
• Speed, Distance, and Time
• Absolute Value
• More Pre-Algebra Worksheets
• Translating Algebraic Phrases
• Evaluating Algebraic Expressions
• Simplifying Algebraic Expressions
• Algebraic Identities
• Quadratic Equations
• Systems of Equations
• Polynomials
• Inequalities
• Sequence and Series
• Complex Numbers
• More Algebra Worksheets
• Trigonometry
• Math Workbooks
• English Language Arts
• Summer Review Packets
• Social Studies
• Holidays and Events
• Worksheets >
• Geometry >
• Quadrilaterals >

Kite Worksheets

Walk through this assortment of Kite worksheets that provide best-practice materials on topics like identifying kites, area and perimeter of a kite, printable property charts, angles, solving problems involving algebraic expressions and a lot more. The worksheets are diligently prepared and recommended for students of grade 3 through grade 8. Fly-start practice with our free worksheets!

State Whether the Given Shape is a Kite: With Measures

The 3rd grade and 4th grade worksheets consist of quadrilaterals depicted in three forms - with measures, indicated with congruent parts and in word form. Recognize and write, whether the given shape is a 'kite' or 'not a kite'.

• Download the set

State Whether the Given Shape is a Kite: No Measures

Identify the kite from the group of regular and irregular quadrilaterals presented with no measures in these printable worksheets for grade 4 and grade 5.

Find the Side Length from the Diagonal

The diagonals of a kite are perpendicular. Using this property and the given diagonal measures, find the indicated side length. Knowledge of the Pythagorean theorem is a prerequisite in solving these problems.

Solve for x - Side Length

One side of the congruent parts is shown in numeral and the other side in linear expression. Set up the equation and solve for x.

Solve for x - Diagonal

This set of pdf kite worksheets for 6th grade, 7th grade, and 8th grade students is based on the property - diagonals of a kite bisect each other. One part of the diagonal measure is given. The other part of the diagonal is shown as an algebraic expression. Set the equation and solve the problem.

Finding Indicated Measure by Solving x (Length): Sides

Heighten your skills in solving kite problems that involve algebraic expressions. One pair of congruent parts is provided in linear expressions. Equate the expression, solve for x and find the indicated lengths.

Properties of a Kite | Charts

This set of chart pdfs illustrates the fundamental properties of kites based on angles, diagonals and sides. Memorize the properties of a Kite to swiftly solve the problems.

Perimeter of a Kite

This compilation of printable worksheets features skills to find the perimeter of a kite involving integers, decimals and fractions, finding the missing length and a lot more!

(35 Worksheets)

Area of a Kite

Engage in this collection of worksheets that provides exercises like finding the area of a kite with attributes in integers, fractions and decimals, finding the area of a kite involving unit conversions and more.

(60 Worksheets)

Angles in a Kite

Get introduced to the angles of kites with this range of worksheets that contains skills like finding the indicated angles, angle properties, solving algebra problems and more.

(15 Worksheets)

Related Worksheets

» Rhombus

» Rectangles

» Squares

» Parallelograms

» Trapezoids

Become a Member

Membership Information

Privacy Policy

What's New?

Printing Help

Testimonial

Copyright © 2024 - Math Worksheets 4 Kids

This is a members-only feature!

PROBLEMS INVOLVING PARALLELOGRAMS, TRAPEZOIDS AND KITES

Related documents.

Add this document to collection(s)

You can add this document to your study collection(s)

Add this document to saved

You can add this document to your saved list

Suggest us how to improve StudyLib

(For complaints, use another form )

Input it if you want to receive answer

Grade 9 Mathematics Module: Trapezoids and Kite

This Self-Learning Module (SLM) is prepared so that you, our dear learners, can continue your studies and learn while at home. Activities, questions, directions, exercises, and discussions are carefully stated for you to understand each lesson.

Each SLM is composed of different parts. Each part shall guide you step-by-step as you discover and understand the lesson prepared for you.

Pre-tests are provided to measure your prior knowledge on lessons in each SLM. This will tell you if you need to proceed on completing this module or if you need to ask your facilitator or your teacher’s assistance for better understanding of the lesson. At the end of each module, you need to answer the post-test to self-check your learning. Answer keys are provided for each activity and test. We trust that you will be honest in using these.

Please use this module with care. Do not put unnecessary marks on any part of this SLM. Use a separate sheet of paper in answering the exercises and tests. And read the instructions carefully before performing each task.

LEARNING COMPETENCY

The learners will be able to:

• Prove theorems on Trapezoid and kite
• Solve problems involving Trapezoid and kite

Grade 9 Mathematics Quarter 3 Self-Learning Module: Trapezoids and Kite

Can't find what you're looking for.

We are here to help - please use the search box below.

1 thought on “Grade 9 Mathematics Module: Trapezoids and Kite”

Very entertainer website

Leave a Comment Cancel reply