Quadrilaterals and Other Polygons

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Lesson 7

• Quadrilaterals and Other Polygons

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Quadrilaterals

• A quadrilateral is a four-sided figure like the
  one shown. The sides are line segments connected
  together at their endpoints, called the vertices
  (plural of vertex) of the quadrilateral.
• We call this quadrilateral ABCD or BADC or CDAB
  etc. (We start with a vertex and then proceed
  around the quadrilateral, either clockwise or
  counterclockwise.)
• The line segments shown
  are called diagonals.

D
C
A
B
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Rectangles

• A rectangle is a quadrilateral with four right
  angles.
• The opposite sides of a rectangle are congruent
  and parallel.
• Each diagonal of a rectangle divides the
  rectangle into two congruent triangles.
• The diagonals bisect each other and are congruen
  to each other.
• The width of a rectangle is the common length of
  one pair of opposite sides.
• The length of a rectangle is the common length of
  another pair of opposite sides.
• Instead of width and length, we sometimes use
  base and height.

C
D
A
B
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Example

• The length of a rectangle is 12 and the width is
  5. What is the length of one of its diagonals?
• Its a good idea to draw a picture.
• The diagonal is the hypotenuse of a right
  triangle whose legs measure 5 and 12. So, we use
  the Pythagorean Theorem

c
12
5
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Squares

• A square is a rectangle in which all four sides
  are congruent.
• Each diagonal divides the square into two
  45-45-90 triangles. Both diagonals together
  divide the square into four 45-45-90 triangles.
• If a side of the square measures then each
  diagonal measures
• The diagonals bisect each other and are congruent
  to each other.

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Example

• In the figure, ABCD and AFDE are squares.
• If then what is
• First note that is a 45-45-90
  triangle. So,
• So,
• But So, too.
• So,

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Parallelograms

• A parallelogram is a quadrilateral in which both
  pairs of opposite sides are parallel.
• Each pair of opposite sides in a parallelogram
  are not only parallel, but congruent as well.
• Each diagonal divides the parallelogram into two
  congruent triangles.
• The diagonals of a parallelogram bisect each
  other.
• Opposite angles of a parallelogram are congruent.
• Consecutive angles of a parallelogram are
  supplementary.

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Rhombi

• A rhombus (plural rhombi) is a parallelogram with
  four congruent sides.
• Since a rhombus is a parallelogram, it has all
  the properties a parallelogram has.
• In addition, the diagonals of a rhombus are
  perpendicular to each other and they divide the
  rhombus into four congruent right triangles.

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Trapezoids

• A trapezoid is a quadrilateral such that one pair
  of opposite sides is parallel.
• The parallel sides are called the bases of the
  trapezoid.
• The nonparallel sides are called the legs of the
  trapezoid.
• If the legs of a trapezoid are congruent, the the
  trapezoid is called isosceles.
• In an isosceles trapezoid, the diagonals bisect
  each other.

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Kites

• A kite is a quadrilateral that has two pairs of
  congruent sides. The congruent sides are
  consecutive, not opposite.
• A kite has one pair of opposite angles that are
  congruent.
• One of the diagonals bisects the other.
• The diagonals are perpendicular to each other.

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Polygons

• A polygon is a closed figure, like the one shown,
  made up of joining line segments together at
  their endpoints.
• Triangles and quadrilaterals are examples of
  polygons.
• A polygon with 5 sides is called a pentagon.
• A polygon with 6 sides is called a hexagon.
• A polygon with 7 sides is called a heptagon.
• A polygon with 8 sides is called an octagon.
• A polygon with 9 sides is called a nonagon.
• A polygon with 10 sides is called a decagon.
• A regular polygon is a polygon in which all of
  its sides are congruent and all of its interior
  angles are congruent.

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Interior Angles of a Polygon

• If a polygon has n sides, then the sum of the
  measures of its n angles is
• Each interior angle of a regular polygon with n
  sides measures

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Exterior Angles of a Polygon

• An exterior angle of a polygon is formed by a
  side of the polygon and an extension of an
  adjacent side.
• in the figure is an exterior angle of a
  regular hexagon.
• An exterior angle is the supplement of an
  interior angle.
• In any polygon if we draw one exterior angle at
  each vertex then the sum of the measures of these
  exterior angles is

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Diagonals of a Polygon

• A diagonal of a polygon is a line segment
  connecting two non-consecutive vertices.
• For example, a triangle has no diagonals and a
  quadrilateral has two diagonals.
• In general, if a polygon has n sides, then it has
• diagonals.