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Quadrilaterals
Chapter 5
Pre-AP Geometry
Objectives
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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.
Parallelograms
Lesson 5.1
Pre-AP Geometry
Objective
Apply the definition of a parallelogram and
the theorems about properties of a
parallelogram.
Definition
Parallelogram
A quadrilateral with two sets of parallel sides.
Properties of Parallelograms
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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
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.
Practice
1. Name all pairs of parallel lines.
2. Name all pairs of congruent angles.
3. Name all pairs of congruent segments.
4. What is the sum of the measures of the interior
angles of a parallelogram?
5. What is the sum of the measures of the exterior
angles of a parallelogram?
Review – True or False
1.
2.
3.
4.
Every parallelogram is a quadrilateral.
Every quadrilateral is a parallelogram.
All angles of a parallelogram are
congruent.
All sides of a parallelogram are
congruent.
Written Exercises
Problem Set 5.1, p. 169: # 2 – 32 (even)
skip # 16
Proving Quadrilaterals
are Parallelograms
Lesson 5.2
Pre-AP Geometry
Objective
Prove that certain quadrilaterals are
parallelograms.
Quadrilaterals and Parallelograms
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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).
Theorem 5-4
If both pairs of opposite sides of a
quadrilateral are congruent, then the
quadrilateral is a parallelogram.
Theorem 5-5
If one pair of opposite sides of a
quadrilateral are both congruent and
parallel, then the quadrilateral is a
parallelogram.
Theorem 5-6
If both pairs of opposite angles of a
quadrilateral are congruent, then the
quadrilateral is a parallelogram.
Theorem 5-7
If the diagonals of a quadrilateral bisect
each other, then the quadrilateral is a
parallelogram.
Five ways to Prove that a
Quadrilateral is a Parallelogram
1.
2.
3.
4.
5.
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.
Review
Answer with always, sometimes, or never.
1.
The diagonals of a quadrilateral bisect each other.
2.
If the measures of two angles of a quadrilateral are equal, then he
quadrilateral is a parallelogram.
3.
If one pair of opposite sides of a quadrilateral is congruent and
parallel, then the quadrilateral is a parallelogram.
4.
If both pairs of opposite sides of a quadrilateral are congruent,
then the quadrilateral is a parallelogram.
5.
To prove a quadrilateral is a parallelogram, it is enough to show
that one pair of opposite sides is parallel.
Practice
State the definition of theorem that enables you to
deduce, from the information provided, that
quadrilateral ABCD is a parallelogram.
1.
BE = EX; CE = EA
2.
BAD  DCB; ADC  CBA
3.
BC || AD; AB || DC
4.
BC  AD; AB  DC
B
C
E
A
D
Written Exercises
Problem Set 5.2 p. 174 # 2, 4, 8, 10, 14, 20,
22
Theorems Involving
Parallel Lines
Lesson 5.3
Pre-AP Geometry
Objective
Apply theorems about parallel lines and
the segment that joins the midpoints of two
sides of a triangle.
Theorem 5-8
If two lines are parallel, then all points on one
line are equidistant from the other line.
l
m
Theorem 5-9
If three parallel lines cut off congruent segments
on one transversal, then they cut off congruent
segments on every transversal.
l
m
n
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.
A
M
B
D
N
C
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.
A
M
B
BM = AM, CN = AN
BC = 2(MN)
MN = ½(BC)
N
C
Written Exercises
Problem Set 5.3 p. 180 # 2-20 even, p. 182
# 1-6
Special
Parallelograms
Lesson 5.4
Pre-AP Geometry
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.
Rectangle
A parallelogram with four right angles.
Both pairs of opposite angles are congruent.
Every rectangle is a parallelogram.
Rhombus
A quadrilateral with four congruent sides.
Both pairs of opposite sides are congruent.
Every rhombus is a parallelogram.
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.
Theorem 5-12
The diagonals of a rectangle are congruent.
Theorem 5-13
The diagonals of a rhombus are perpendicular.
Theorem 5-14
Each diagonal of a rhombus bisects two angles of the
rhombus.
Theorem 5-15
The midpoint of the hypotenuse of a right angle is
equidistant from the three vertices.
Theorem 5-16
If an angle of a parallelogram is a right angle, then the
parallelogram is a rectangle.
Theorem 5-17
If two consecutive sides of a parallelogram are
congruent, then the parallelogram is a rhombus.
Practice
Reply with always, sometimes, or never.
1.
2.
3.
4.
5.
6.
7.
8.
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.
Written Exercises
Problem Set 5.4 p. 187 # 1-10, 12-30 evens
Trapezoids
Lesson 5.5
Pre-AP Geometry
Objective
Apply the definitions and identify the
properties of a trapezoid and an isosceles
trapezoid.
Definition
Trapezoid
A quadrilateral with exactly one pair of parallel sides.
base
leg
leg
base
Definition
Isosceles Trapezoid
In an isosceles trapezoid, the base angles are
equal, and so are the other pair of opposite sides
AD and BC.
D
A
C
B
Theorem 5-18
Base angles of an isosceles trapezoid are
congruent.
Median of a Trapezoid
The segment that joins the midpoints of the legs
of a trapezoid.
Median of a Trapezoid
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.
Practice
In trapezoid ABCD, EF is a median.
1. If AB = 25 and DC = 13, then EF = _____.
2. If AE = 11 and FB = 8, then AD = _____ and BC =
_____.
3. If AB = 29 and EF = 24, then DC = _____.
4. If AB = 7y + 6 and EF = 5y – 3 and DC = y – 5,
then y = _____.
Written Exercises
Problem Set 5.5 p. 192 # 2-26 evens