Chapter 6 Notes Packet Name: _______________________________
Chapter 6 - Quadrilaterals
Chapter 6: Quadrilaterals
6.1 – Parallelograms on the Coordinate Plane
Objectives:
Show that a quadrilateral is a parallelogram on the coordinate plane
Identify and verify parallelograms
Distance Formula:
Midpoint Formula:
Slope Formula:
Investigation:
Plot your assigned parallelogram on the
coordinate plane.
Find the following:
The slope of each side
The length of each side
The slope of both diagonals
The length of both diagonals
The midpoint of both diagonals
You may want to show your work on a
separate sheet of paper.
Record your findings in the table below.
Sides:
Diagonals:
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Chapter 6: Quadrilaterals
Observations:
What do you observe about the slopes of the sides of the parallelogram? What does this tell us?
What do you observe about the lengths of the sides of the parallelogram?
What do you observe about the slopes of the diagonals of the parallelogram?
What do you observe about the lengths of the diagonals of the parallelogram?
What do you observe about the midpoints of the diagonals of the parallelogram? What does this
tell us?
Parallelograms
A parallelogram is a quadrilateral with both pairs of opposite sides
parallel.
Proving Parallelograms on the Coordinate Plane
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 is both parallel AND congruent
Show that the diagonals bisect each other
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Chapter 6: Quadrilaterals
EXAMPLES:
1. Three vertices of parallelogram WXYZ are
. Graph these
vertices. Use slopes to find the coordinates of the
vertex Y.
2. Show that quadrilateral ABCD is a parallelogram using the definition of parallelogram: show that
both pairs of opposite sides are parallel.
3. Show that quadrilateral FGHJ is a parallelogram by showing that one pair of opposite sides are
both parallel and congruent.
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Chapter 6: Quadrilaterals
4. Show that quadrilateral KLMN is a parallelogram by showing that the diagonals bisect each other.
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Chapter 6: Quadrilaterals
6.2 – Properties of Parallelograms
Objectives:
Know and prove the properties of parallelograms
Apply the properties of parallelograms to find side lengths, segment lengths, and angle measures
Parallelograms
A parallelogram is a quadrilateral with both pairs of opposite
sides parallel.
Proving the Properties of Parallelograms
Given:
ABCD is a parallelogram
Prove:
A C & B D
Statements
Reasons
Properties of Parallelograms
If a quadrilateral is a parallelogram, then…
Both pairs of opposite sides are congruent
Both pairs of opposite angles are congruent
Q S & P R
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Chapter 6: Quadrilaterals
Consecutive angles are supplementary
The diagonals bisect each other
The sum of all four angles in any quadrilateral is 360.
Examples: Using the Properties of Parallelograms
1. Complete each statement about JKLM.
_____
_____
_____
_____
_____
_____
2. VRZA is a parallelogram.
Given:
Find: The perimeter of VRZA
3. WXYZ is a parallelogram. Find each measure.
a. WV
b. YW
c. XZ
d. ZV
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Chapter 6: Quadrilaterals
4. FEDY is a parallelogram. Find the value of each variable.
5. For the given parallelogram, set up and solve a system of
equations to find the value of the variables.
6. In STUV, mTSU = 32, mUSV =
, mTUV = , and
TUV is an acute angle. Find the value of x (that makes
sense) and mUSV.
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Chapter 6: Quadrilaterals
6.3 – Proving Quadrilaterals Are parallelograms
Objectives:
Prove that a quadrilateral is a parallelogram
Identify and verify parallelograms
Conditions for Parallelograms
If both pairs of opposite sides of a quadrilateral are parallel, then the quadrilateral is a
parallelogram. (Definition)
If one pair of opposite sides of a quadrilateral is parallel and congruent,
then the quadrilateral is a parallelogram.
If and , then ABCD is a parallelogram.
If both pairs of opposite sides of a quadrilateral are congruent,
then the quadrilateral is a parallelogram.
If and , then ABCD is a parallelogram.
If both pairs of opposite angles of a quadrilateral are congruent,
then the quadrilateral is a parallelogram.
If and , then ABCD is a parallelogram.
If an angle of a quadrilateral is supplementary to both of its
consecutive angles, then the quadrilateral is a parallelogram
If is supplementary to and is supplementary to ,
then ABCD is a parallelogram.
If the diagonals of a quadrilateral bisect each other, then the
quadrilateral is a parallelogram.
If and , then ABCD is a parallelogram.
Examples: Identifying Parallelograms
1. For each quadrilateral QUAD, state the property or definition that proves that QUAD is a
parallelogram.
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Chapter 6: Quadrilaterals
Examples: Proving Parallelograms
2.
Given:
ACDF is a parallelogram
Prove:
FBCE is a parallelogram
Statements
Reasons
Prove this property:
If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a
parallelogram.
3.
Given:
Prove:
ABCD is a parallelogram
Statements
Reasons
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Chapter 6: Quadrilaterals
4.
Given:
CAR is isosceles w/base
Prove:
BARK is a parallelogram
Statements
Reasons
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Chapter 6: Quadrilaterals
6.4 – Rectangles, Rhombi, & Squares
Objectives:
Apply the properties of rectangles, rhombi, and squares to find side lengths, segment lengths, and
angle measures
Find areas of rectangles, rhombi, and squares
Special Parallelograms
Rectangle
A quadrilateral with opposite sides congruent and with four right angles
Rhombus
A quadrilateral with all sides congruent
Square
A quadrilateral with four right angles and all sides congruent
Properties of Rectangles
All properties of a parallelogram apply:
Both pairs of opposite sides are parallel & congruent
Both pairs of opposite angles are congruent
Consecutive angles are supplementary
The diagonals bisect each other
All angles are right angles
The diagonals are congruent
Examples: Using the Properties of Rectangles
1.
Given:
RECT is a rectangle
Find:
The length of to the nearest tenth
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Chapter 6: Quadrilaterals
Examples: Using the Properties of Rectangles
2. Given: Rectangle QRST
a. Set up and solve a system of equations to find the value
of the variables.
b. Find the rectangle’s base and height.
c. What is the area of rectangle QRST?
3. Given: Rectangle ABCD
a. Find the values of x and y.
b. Find mBAC & mDBC.
Properties of Rhombuses
All properties of a parallelogram apply
Both pairs of opposite sides are parallel & congruent
Both pairs of opposite angles are congruent
Consecutive angles are supplementary
The diagonals bisect each other
All sides are congruent—that is, a rhombus is equilateral
The diagonals bisect the vertex angles
The diagonals are perpendicular bisectors of each other
The diagonals divide the rhombus into four congruent right triangles
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Chapter 6: Quadrilaterals
Examples: Using the Properties of Rhombuses
4. Given: Rhombus HIJK
a. Find the value of the variables.
b. Find mJ & mK.
c. What is the perimeter of rhombus HIJK?
5. Given: Rhombus WXYZ
a. Find the value of .
b. Find the area of rhombus WXYZ.
c. Find the WX. What is the perimeter of WXYZ?
12
5
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Chapter 6: Quadrilaterals
Properties of Squares
All the properties of a parallelogram apply:
Both pairs of opposite sides are parallel & congruent
Both pairs of opposite angles are congruent
Consecutive angles are supplementary
The diagonals bisect each other
All the properties of a rectangle apply:
All angles are right angles
The diagonals are congruent
All the properties of a rhombus apply:
All sides are congruent
The diagonals bisect the vertex angles
The diagonals are perpendicular bisectors of each other
The diagonals form four isosceles right triangles
Examples: Using the Properties of Squares
6.
Given:
EFGH is a square with a perimeter of 36
Find:
The area of square EFGH
A
B
C
D
E
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Chapter 6: Quadrilaterals
6.5 – Kites & Trapezoids
Objectives:
Apply the properties of kites and trapezoids to find side lengths, segment lengths, and angle
measures
Find areas of kites and trapezoids
Kites
A quadrilateral with two pairs of consecutive congruent sides
with opposite sides that are NOT congruent.
Properties of Kites
The diagonals are perpendicular to each other
One diagonal is the perpendicular bisector of the other
One of the diagonals bisects a pair of opposite angles
One pair of opposite angles are congruent
Examples: Using the Properties of Kites
1. Given: Kite ABCD
a. Find the value of x.
b. Find the perimeter of ABCD.
2. Given: Kite KITE
a. Find the values of x and y in the kite shown.
b.
. Set up and solve a
quadratic equation to find the value of n (that makes sense).
K
I
T
E
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Chapter 6: Quadrilaterals
Trapezoids
A quadrilateral with exactly one pair of parallel sides.
Properties of Trapezoids
Consecutive non-base angles are supplementary.
A is supplementary to B
C is supplementary to D
Midsegment of a Trapezoid
Parallel to the bases
Length is half the sum of the lengths of the bases:
Isosceles Trapezoids
A trapezoid with congruent non-parallel sides (legs)
Properties of Isosceles Trapezoids
All properties of a trapezoid apply
The base angles are congruent.
QPS RSP
PQR SRQ
The diagonals are congruent.
Examples: Using the Properties of Trapezoids
3. ABCD is a trapezoid. Find the value of .
Find mA & mD.
4. Find the value of t and the length of the midsegment of the
trapezoid shown:
5. Find the length of the midsegment AND the area of the trapezoid
shown:
X Y
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Chapter 6: Quadrilaterals
6. Given: Isosceles trapezoid EFGH
a. Find the value of .
b. If
, find the
value of x that makes sense. Then find EF.
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Chapter 6: Quadrilaterals
6.6 – Quadrilaterals in the Coordinate Plane
Objective:
Use the distance, slope, and midpoint formulas to prove that a figure graphed in the coordinate
plane is special quadrilateral: rectangle, rhombus, square, kite, or trapezoid
FORMULAS & THE COORDINATE PLANE
Formula
When to Use it
Distance Formula:
To determine whether…
Sides are congruent
Diagonals are congruent
Midpoint Formula:
To determine…
The coordinates of a midpoint of a side
Whether diagonals bisect each other
Slope Formula:
To determine whether…
Opposite sides are parallel
Diagonals are perpendicular
Sides are perpendicular
QUADRILATERAL
PROVE:
PARALLELOGRAM
Both pairs of opposite sides are parallel
Both pairs of opposite sides are congruent
One pair of opposite sides are parallel and congruent
Diagonals bisect each other
RECTANGLE
First prove it’s a parallelogram, and then prove…
The diagonals are congruent
Two consecutive sides of the parallelogram are
perpendicular
RHOMBUS
First prove it’s a parallelogram, and then prove…
Two consecutive sides are congruent
The diagonals are perpendicular
OR…
All four sides are congruent
SQUARE
It’s a rectangle and a rhombus (see above)
TRAPEZOID
Only one pair of sides are parallel
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Chapter 6: Quadrilaterals
Examples: Quadrilaterals in the Coordinate Plane
1. The vertices of KARI are
. Show that KARI is a
rectangle.
(Remember, you must first show that KARI is a
parallelogram.)
2. The vertices of DION are
. Prove that DION is a
parallelogram. Is DION a rhombus?
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Chapter 6: Quadrilaterals
3. Quadrilateral JACK has vertices
. Prove that JACK is a
trapezoid.
Day 2
4. What special quadrilateral is formed by the
intersection of these lines?
5. The coordinates of three vertices of parallelogram
RHOM are given. Find the coordinates of O so
that a rhombus is formed.
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Chapter 6: Quadrilaterals
Use the diagonals to determine whether a parallelogram with the given vertices is a rectangle, rhombus,
or square. Give all the names that apply.
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Chapter 6: Quadrilaterals
6.7 – Coordinate Proofs Bonus Topic
Objective:
Use the distance, slope, and midpoint formulas in a coordinate proof to show that a figure
graphed in the coordinate plane is special quadrilateral
You can use coordinate geometry and algebra to prove theorems in geometry. This kind of proof is
called a coordinate proof. Sometimes it’s easier to show that a theorem is true by using a
coordinate proof rather than a standard deductive proof.
1. Rectangle RECT is shown at the right. What are the coordinates of
point E?
2. Use coordinate geometry to prove that T is a right angle.
3. Given: Square ABCD
Prove that the diagonals are congruent and perpendicular using
coordinate geometry.
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Chapter 6: Quadrilaterals
4. Write a coordinate proof to prove that quadrilateral CDEF is an
isosceles trapezoid.
5. Write a coordinate proof to prove that quadrilateral ABCO is
a parallelogram.
