Quadrilaterals

1
Quadrilaterals

• Chapter 5
• Pre-AP Geometry

2
Objectives

• Apply the definition of a parallelogram and the
  theorems about properties of a parallelogram.
• Prove that certain quadrilaterals are
  parallelograms.
• Apply theorems about parallel lines.
• Apply the midpoint theorems for triangles.
• Apply the definitions and identify the special
  properties of a rectangle, a rhombus, and a
  square.
• Determine when a parallelogram is a rectangle,
  rhombus, or square.
• Apply the definition and identify the properties
  of a trapezoid and an isosceles trapezoid.

3
Parallelograms

• Lesson 5.1
• Pre-AP Geometry

4
Objective

• Apply the definition of a parallelogram and the
  theorems about properties of a parallelogram.

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Definition

• Parallelogram
• A quadrilateral with two sets of parallel
  sides.

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Properties of Parallelograms

• The diagonals of a parallelogram bisect each
  other.
• Opposite sides of a parallelogram are congruent.
• Opposite angles of a parallelogram are congruent.
• Each diagonal bisects the parallelogram into two
  congruent triangles.

Parallelogram
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Theorems

• 5-1 Opposite sides of a parallelogram are
  congruent.
• 5-2 Opposite angles of a parallelogram are
  congruent.
• 5-3 Diagonals of a parallelogram bisect each
  other.

8
Practice

• Name all pairs of parallel lines.
• Name all pairs of congruent angles.
• Name all pairs of congruent segments.
• What is the sum of the measures of the interior
  angles of a parallelogram?
• What is the sum of the measures of the exterior
  angles of a parallelogram?

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Review True or False

• Every parallelogram is a quadrilateral.
• Every quadrilateral is a parallelogram.
• All angles of a parallelogram are congruent.
• All sides of a parallelogram are congruent.

10
Written Exercises

• Problem Set 5.1, p. 169 2 32 (even) skip 16

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Proving Quadrilaterals are Parallelograms

• Lesson 5.2
• Pre-AP Geometry

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Objective

• Prove that certain quadrilaterals are
  parallelograms.

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Quadrilaterals and Parallelograms

• A quadrilateral is a polygon with 4 sides.
• A parallelogram is a quadrilateral whose opposite
  sides are parallel (the top and bottom are
  parallel and the left and right are parallel).

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Theorem 5-4

• If both pairs of opposite sides of a
  quadrilateral are congruent, then the
  quadrilateral is a parallelogram.

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Theorem 5-5

• If one pair of opposite sides of a quadrilateral
  are both congruent and parallel, then the
  quadrilateral is a parallelogram.

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Theorem 5-6

• If both pairs of opposite angles of a
  quadrilateral are congruent, then the
  quadrilateral is a parallelogram.

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Theorem 5-7

• If the diagonals of a quadrilateral bisect each
  other, then the quadrilateral is a parallelogram.

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Five ways to Prove that a Quadrilateral is a
Parallelogram

• Show that both pairs of opposite sides are
  parallel.
• Show that both pairs of opposite sides are
  congruent.
• Show that one pair of opposite sides are both
  congruent and parallel.
• Show that both pairs of opposite angles are
  congruent.
• Show that the diagonals bisect each other.

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Review

• Answer with always, sometimes, or never.
• The diagonals of a quadrilateral bisect each
  other.
• If the measures of two angles of a quadrilateral
  are equal, then he quadrilateral is a
  parallelogram.
• If one pair of opposite sides of a quadrilateral
  is congruent and parallel, then the quadrilateral
  is a parallelogram.
• If both pairs of opposite sides of a
  quadrilateral are congruent, then the
  quadrilateral is a parallelogram.
• To prove a quadrilateral is a parallelogram, it
  is enough to show that one pair of opposite sides
  is parallel.

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Practice

• State the definition of theorem that enables you
  to deduce, from the information provided, that
  quadrilateral ABCD is a parallelogram.
• BE EX CE EA
• ?BAD ? ?DCB ?ADC ? ?CBA
• BC AD AB DC
• BC ? AD AB ? DC

21
Written Exercises

• Problem Set 5.2 p. 174 2, 4, 8, 10, 14, 20, 22

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Theorems Involving Parallel Lines

• Lesson 5.3
• Pre-AP Geometry

23
Objective

• Apply theorems about parallel lines and the
  segment that joins the midpoints of two sides of
  a triangle.

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Theorem 5-8

• If two lines are parallel, then all points on
  one line are equidistant from the other line.

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Theorem 5-9

• If three parallel lines cut off congruent
  segments on one transversal, then they cut off
  congruent segments on every transversal.

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Theorem 5-10

• A line that contains the midpoint of one side of
  a triangle and is parallel to another side passes
  through the midpoint of the third side.

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Theorem 5-11

• The segment that joins the midpoints of two
  sides of a triangle
• (1) is parallel to the third side
• (2) is half as long as the third side.

BM AM, CN AN BC 2(MN) MN ½(BC)
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Written Exercises

• Problem Set 5.3 p. 180 2-20 even, p. 182 1-6

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Special Parallelograms

• Lesson 5.4
• Pre-AP Geometry

30
Objectives

• Apply the definitions and identify the special
  properties of a rectangle, a rhombus, and a
  square.
• Determine when a parallelogram is a rectangle,
  rhombus, or square.

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Rectangle

• A parallelogram with four right angles.
• Both pairs of opposite angles are congruent.
• Every rectangle is a parallelogram.

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Rhombus

• A quadrilateral with four congruent sides.
• Both pairs of opposite sides are congruent.
• Every rhombus is a parallelogram.

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Square

• A quadrilateral with four right angles and four
  congruent sides.
• Both pairs of opposite angles and opposite sides
  are congruent.
• A square is also a rectangle, a rhombus, and a
  parallelogram.

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Theorem 5-12

• The diagonals of a rectangle are congruent.

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Theorem 5-13

• The diagonals of a rhombus are perpendicular.

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Theorem 5-14

• Each diagonal of a rhombus bisects two angles of
  the rhombus.

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Theorem 5-15

• The midpoint of the hypotenuse of a right angle
  is equidistant from the three vertices.

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Theorem 5-16

• If an angle of a parallelogram is a right angle,
  then the parallelogram is a rectangle.

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Theorem 5-17

• If two consecutive sides of a parallelogram are
  congruent, then the parallelogram is a rhombus.

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Practice

• Reply with always, sometimes, or never.
• A square it a rhombus.
• The diagonals of a parallelogram bisect the
  angles of the parallelogram.
• A quadrilateral with one pair of sides congruent
  is a parallelogram.
• The diagonals of a rhombus are congruent.
• A rectangle has consecutive sides congruent.
• A rectangle has perpendicular diagonals.
• The diagonals of a rhombus bisect each other.
• The diagonals of a parallelogram are
  perpendicular bisectors of each other.

42
Written Exercises

• Problem Set 5.4 p. 187 1-10, 12-30 evens

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Trapezoids

• Lesson 5.5
• Pre-AP Geometry

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Objective

• Apply the definitions and identify the
  properties of a trapezoid and an isosceles
  trapezoid.

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Definition

• Trapezoid
• A quadrilateral with exactly one pair of
  parallel sides.

46
Definition
Isosceles Trapezoid In an isosceles trapezoid,
the base angles are equal, and so are the other
pair of opposite sides AD and BC.
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Theorem 5-18

• Base angles of an isosceles trapezoid are
  congruent.

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Median of a Trapezoid

• The segment that joins the midpoints of the legs
  of a trapezoid.

Median of a Trapezoid
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Theorem 5-19

• The median of a trapezoid
• (1) is parallel to the base
• (2) has a length equal to the average of the
  base lengths.

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Practice

• In trapezoid ABCD, EF is a median.
• If AB 25 and DC 13, then EF _____.
• If AE 11 and FB 8, then AD _____ and BC
  _____.
• If AB 29 and EF 24, then DC _____.
• If AB 7y 6 and EF 5y 3 and DC y 5,
• then y _____.

51
Written Exercises

• Problem Set 5.5 p. 192 2-26 evens