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Lesson 7Quadrilaterals 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.

AB

CD

and AC BD

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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.

A B

CD

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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?

• It’s 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:

5

12c

2 2 25 12c 2 25 144 169c

169 13c

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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.

s2.s

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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,

5 2AE ?EG

A

B

C

DE

F

G

P

APE5.EP AP

10.AD AB .PG AB 10PG

5 10 15.EG EP PG

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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

( 2) 180 .n

2180 .

n

n

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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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360 .

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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.

( 3)

2

n n