Quadrilaterals

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Quadrilaterals

• By Nicole Kwiatkowski
• EDC 601- Instructional Technologies
• 3/5/2006

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Main Menu

• Quadrilaterals
• What is a quadrilateral?
• Types of Quadrilaterals
• Parallelogram
• Rectangle
• Rhombus
• Square
• Trapezoid
• Isosceles Trapezoid
• Right Trapezoid

• Quadrilateral Family Tree
• See how quadrilaterals are connected
• Proofs
• Proving quadrilaterals
• Test Your Knowledge
• Quiz Time
• Bibliography
• Resources used

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What is a Quadrilateral?

• There are many different types of
  quadrilaterals, however there are specific
  properties that every quadrilateral has in
  common.
• One thing that all quadrilaterals have in common
  is that they have four sides. If a polygon does
  not have exactly four sides, then the polygon is
  not a quadrilateral.
• Due to the fact that quadrilaterals have four
  sides, they also have four angles. If you connect
  the two vertices of a quadrilateral to form a
  diagonal, you will have created two triangles.
  The sum of the angles in a triangle equals 180.
  Therefore, since a quadrilateral is made of two
  triangles, the sum of the four angles totals 360
  (180 2).
• These properties are what make quadrilaterals
  alike, but what makes them different?

Quadrilateral- any four sided figure (Source
Roberts, 2007)
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Parallelogram

• A parallelogram is one type of quadrilateral,
  because it has all the properties of a
  quadrilateral. However, there are additional
  properties that are specific to the
  parallelogram.
• Parallelograms have two sets of parallel sides
  in addition to two sets of congruent sides.
• Since a parallelogram has two sets of parallel
  sides, the angles that are opposite of each other
  in a parallelogram are congruent (equal measure).
  The consecutive angles in a parallelogram are
  supplementary (they add up to 180).
• The diagonals of a parallelogram bisect each
  other and the diagonals form two congruent
  triangles. Bisect means to cut in half.

Parallelogram- The properties of a parallelogram
come in pairs of two. There are two properties
about the angles, two properties about the sides,
and two properties about the diagonals. This is
helpful when remembering the properties. (Source
Roberts, 2007)
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Rectangle

• A rectangle is a type of quadrilateral because
  it also has all the properties that a
  quadrilateral has. However, a rectangle is also a
  type of parallelogram. The reason is because it
  has all the properties of parallelogram.
• So what makes a rectangle a rectangle? In
  addition to having all the properties of a
  quadrilateral and a parallelogram, a rectangle
  also has four right angles.
• A rectangle can be referred to as an equiangular
  parallelogram because all four of its angle are
  right, meaning they are all 90 (four equal
  angles).
• In addition to the diagonal bisecting each other
  and forming two congruent triangles the diagonals
  in a rectangle are also congruent.

Rectangle- four right angles/diagonals
congruent (Source Roberts, 2007)
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Rhombus

• A rhombus is sometimes referred to as a slanted
  square, because that is what it looks like.
• A rhombus has all the properties of a
  quadrilateral and all the properties of a
  parallelogram, in addition to three other
  properties.
• A rhombus is often referred to as a equilateral
  parallelogram, because it has four sides that are
  congruent (each side length has equal measure).
• The diagonals of a rhombus bisect each angle.
  Meaning each angle is cut in half by the
  diagonals. The diagonals of a rhombus are also
  perpendicular. Perpendicular is defined as two
  lines or line segments that intersect to form a
  right angle.

Rhombus- four congruent sides, diagonals bisect
angles, diagonals perpendicular (Source Roberts,
2007)
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Square

• The square is the most specific member of the
  family of quadrilaterals. The square has the
  largest number of properties.
• Squares have all the properties of a
  quadrilateral, all the properties of a
  parallelogram, all the properties of a rectangle,
  and all the properties of a rhombus.
• A square can be called a rectangle, rhombus, or
  a parallelogram because it has all of the
  properties specific to those figures.

Square- a quadrilateral that has all the
properties of the rectangle, rhombus, and the
parallelogram. (Source Roberts, 2007)
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Trapezoid

• A trapezoid is one type of quadrilateral,
  because it has all the properties of a
  quadrilateral. However, there are additional
  properties that are specific to the trapezoid.
• Unlike a parallelogram, rectangle, rhombus, and
  square who all have two sets of parallel sides, a
  trapezoid only has one set of parallel sides.
  These parallel sides are opposite one another.
  The other set of sides are non parallel.
• In a trapezoid, each pair of consecutive angles
  formed by either non parallel side are
  supplementary.

Trapezoid- exactly one pair of parallel sides and
exactly one pair of non-parallel sides. (Source
Roberts, 2007)
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Isosceles Trapezoid

• An isosceles trapezoid is a type of
  quadrilateral because it also has all the
  properties that a quadrilateral has. More
  importantly, it is a type of trapezoid. One can
  never assume a trapezoid is isosceles unless they
  are given that the trapezoid has specific
  properties of an isosceles trapezoid.
• Isosceles is defined as having two equal sides.
  Therefore, an isosceles trapezoid has two equal
  sides. These equal sides are called the legs of
  the trapezoid, which are the non-parallel sides
  of the trapezoid.
• An isosceles trapezoid also has diagonals which
  are congruent.
• Both pair of base angles in an isosceles
  trapezoid are also congruent.

Isosceles Trapezoid- non-parallel sides are
congruent/both pair of base angles
congrent (Source Roberts, 2007)
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Right Trapezoid

• A right trapezoid is a type of quadrilateral
  because it also has all the properties that a
  quadrilateral has. In addition to being a
  quadrilateral, a right trapezoid is a type of
  trapezoid.
• A right trapezoid also has one set of parallel
  sides, and one set of non-parallel sides.
• A right trapezoid has exactly two right angles.
  This means that two angles measure 90.
• There should be no problem identifying this
  quadrilateral correctly, because its just like
  its name. When you think of right trapezoid,
  think of right angles!

Right Trapezoid- a trapezoid with exactly two
right angles (Source adapted from Weisstein,
2007)
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Quadrilateral Family Tree

• Its important to have a good understanding of
  how each of the quadrilaterals relate to one
  another.
• Any quadrilateral that has two sets of parallel
  sides can be considered a parallelogram.
• A rectangle and rhombus are both types of
  parallelograms, and a square can be considered a
  rectangle, rhombus, and a parallelogram.
• Any quadrilateral that has one set of parallel
  sides is a trapezoid. Isosceles and Right are two
  types of trapezoids.

Quadrilateral
Parallelogram
Trapezoid
Rectangle
Rhombus
Isosceles Trapezoid
Right Trapezoid
Square
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Proofs

• In order to prove different quadrilaterals, you
  need to know the properties that the
  quadrilateral possesses.
• For example, we could prove a quadrilateral is a
  square if we can first prove that it is a rhombus
  AND a rectangle. Or we can prove a quadrilateral
  is a parallelogram if we can prove that it has
  two sets of parallel sides.
• Marilyn vos Savant talks about proofs in one of
  her articles in PARADE magazine. She says You
  will need them every day, I hope, without even
  knowing it. Geometry is beautifully logical, and
  it teaches you how to think and prove that things
  are so, step by step by step. Proofs are
  excellent lessons in reasoning. Without logic and
  reasoning, you are dependent on jumping to
  conclusions or- worse- having empty opinions.

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Quiz

• 1.) A square can also be called what?
• a.) a quadrilateral
• b.) a parallelogram
• c.) a rhombus
• d.) all of the above
• 2.) What must a rhombus have?
• a.) all angles congruent
• b.) two sets of parallel sides
• c.) all sides congruent
• d.) b and c only

• 3.) When is a square also a rectangle?
• a.) sometimes
• b.) always
• c.) never
• d.) all of the above
• 4.) Diagonals are perpendicular in which two
  quadrilaterals?
• a.) isosceles trapezoid and square
• b.) square and rectangle
• c.) rhombus and square
• d.) rhombus and isosceles trapezoid

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Bibliography

• Dressler, Isidore Keenan, Edward P. Amscos
  Mathematics A. (2002). Amsco School
  Publications, Inc.
• Math.com- A World of Math Online. (2000-2005).
  Retrieved March 1, 2007, from http//www.math.com/
  school/subject3/lesson/S3U2L3DP.html
• Roberts, Donna. Regents Exam Prep Center.
  (1995-2007). Retrieved March 1, 2007, from
  http//www.regentsprep.org.
• Weisstein, Eric W. Trapezoid From MathWorld- A
  Wolfram Web Resource. (1999-2007). Retrieved
  March 2, 2007, from http//mathworld.wolfram.com/T
  rapezoid.html.

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Try Again

• A square is a type of quadrilateral, however
  there is a better answer to this question.
• Think about the quadrilateral family tree.

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Try Again

• A square is a type of rhombus, however there is
  a better answer to this question.
• Think about the quadrilateral family tree.

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You got it!!!

• Congratulations!! Youre right!
• A square has the same properties as a rectangle,
  a rhombus, and a parallelogram!

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Try Again

• A square is a type of parallelogram, however
  there is a better answer to this question.
• Think about the quadrilateral family tree.

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Try Again

• Rhombi do NOT have angles that are congruent.
  Quadrilaterals that have congruent angles are the
  rectangle and the square.
• Think about the quadrilateral family.

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Try Again

• Although rhombi do have two sets of parallel
  sides, this is not the best answer to this
  question.
• Think about the quadrilateral family tree and
  check the choices again!

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Try Again

• Although rhombi do have congruent sides, this is
  not the best answer to the question.
• Think about the quadrilateral family tree and
  check the choices one more time.

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Try Again

• Sorry, but that answer is not correct.
• Think about the quadrilateral family tree and
  check the answer choices again!

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Try Again

• Sorry, but that answer is not correct.
• Think about the quadrilateral family tree and
  check the answer choices again!

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Try Again

• Sorry, but that answer is not correct.
• Think about the quadrilateral family tree and
  check the answer choices again!

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Try Again

• Sorry, thats not the correct choice.
• A square does have diagonals that are
  perpendicular, however the isosceles trapezoid
  does not.

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Try Again

• Sorry, thats not the correct choice.
• A square does have diagonals that are
  perpendicular, however the rectangle does not.

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Try Again

• Sorry, thats not the correct choice.
• A rhombus does have diagonals that are
  perpendicular, however the isosceles trapezoid
  does not.

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You got it!!!

• Congratulations! Youre right!
• A rhombus has four congruent sides AND two sets
  of parallel sides!!

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You got it!!

• Congratulations!! Youre right!
• A square is ALWAYS a rectangle because a square
  has all the properties of a rectangle.

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You got it!!!

• Congratulations!! You are correct!
• The rhombus has diagonals that are
  perpendicular. Because a square is a rhombus, the
  square has the same properties as the rhombus.

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