Quadrilaterals
Unit 6
6-1
Objectives
• To identify any quadrilateral, by name, as
specifically as you can, based on its
characteristics
Quadrilateral
• a quadrilateral is a polygon with 4 sides.
Specific Quadrilaterals
• There are several specific types of
quadrilaterals. They are classified based
on their sides or angles.
2 pairs adjacent
congruent sides
1 pair parallel sides2 pairs parallel
sides
4 right angles4 congruent sidesnon-parallel sides
are congruent
Quadrilaterals
Parallelogram
Rectangle
Rhombus
Square
Isosceles
Trapezoid
Trapezoid
Kite
2 pairs adjacent
congruent sides
1 pair parallel sides2 pairs parallel
sides
4 right angles4 congruent sidesnon-parallel sides
are congruent
Quadrilaterals
Parallelogram
Rectangle
Rhombus
Square
Isosceles
Trapezoid
Trapezoid
Kite
A quadrilateral simply has 4 sides –
no other special
requirements.
Examples of Quadrilaterals
2 pairs adjacent
congruent sides
1 pair parallel sides2 pairs parallel
sides
4 right angles4 congruent sidesnon-parallel sides
are congruent
Quadrilaterals
Parallelogram
Rectangle
Rhombus
Square
Isosceles
Trapezoid
Trapezoid
Kite
A parallelogram
has two pairs of
parallel sides.
Parallelogram
• Two pairs of parallel sides
• opposite sides are actually congruent.
2 pairs adjacent
congruent sides
1 pair parallel sides2 pairs parallel
sides
4 right angles4 congruent sidesnon-parallel sides
are congruent
Quadrilaterals
Parallelogram
Rectangle
Rhombus
Square
Isosceles
Trapezoid
Trapezoid
Kite
A rhombus is a parallelogram
that has four congruent sides.
Rhombus
• Still has two pairs of parallel sides; with
opposite sides congruent.
4 in.
4 in.
4 in.
4 in.
2 pairs adjacent
congruent sides
1 pair parallel sides2 pairs parallel
sides
4 right angles4 congruent sidesnon-parallel sides
are congruent
Quadrilaterals
Parallelogram
Rectangle
Rhombus
Square
Isosceles
Trapezoid
Trapezoid
Kite
A rectangle has
four right angles.
Rectangle
• Still has two pairs of parallel sides; with
opposite sides congruent.
• Has four right angles
2 pairs adjacent
congruent sides
1 pair parallel sides2 pairs parallel
sides
4 right angles4 congruent sidesnon-parallel sides
are congruent
Quadrilaterals
Parallelogram
Rectangle
Rhombus
Square
Isosceles
Trapezoid
Trapezoid
Kite
A square is a specific case of both a
rhombus AND a rectangle, having four
right angles and 4 congruent sides.
Square
• Still has two pairs of parallel sides.
• Has four congruent sides
• Has four right angles
2 pairs adjacent
congruent sides
1 pair parallel sides2 pairs parallel
sides
4 right angles4 congruent sidesnon-parallel sides
are congruent
Quadrilaterals
Parallelogram
Rectangle
Rhombus
Square
Isosceles
Trapezoid
Trapezoid
Kite
A trapezoid has
only one pair of
parallel sides.
2 pairs adjacent
congruent sides
1 pair parallel sides2 pairs parallel
sides
4 right angles4 congruent sidesnon-parallel sides
are congruent
Quadrilaterals
Parallelogram
Rectangle
Rhombus
Square
Isosceles
Trapezoid
Trapezoid
Kite
An isosceles
trapezoid is a
trapeziod with the
non-parallel sides
congruent.
Trapezoid
• has one pair of parallel sides.
Isosceles trapezoid trapezoids
(Each of these examples shown has top and bottom sides parallel.)
2 pairs adjacent
congruent sides
1 pair parallel sides2 pairs parallel
sides
4 right angles4 congruent sidesnon-parallel sides
are congruent
Quadrilaterals
Parallelogram
Rectangle
Rhombus
Square
Isosceles
Trapezoid
Trapezoid
Kite
An kite is a quadrilateral
with NO parallel sides but 2
pairs of adjacent congruent
sides.
Example of a Kite
2 in.
4 in.
4 in.
2 in.
Practice Pg. 290-292
# 1-12 all, 13, 17, 19, 23
#29-34
#36 can be turned in for 5 extra credit points!
Properties of Parallelograms
6-2
In this lesson . . .
And the rest of the unit, you will study special
quadrilaterals. A parallelogram is a quadrilateral
with both pairs of opposite sides parallel.
When you mark diagrams of quadrilaterals, use
matching arrowheads to indicate which sides are
parallel. For example, in the diagram to the
right, PQ║RS and QR║SP. The symbol
PQRS is read “parallelogram PQRS.”
Theorems about parallelograms
• 6.1—If a
quadrilateral is a
parallelogram,
then its opposite
sides are
congruent.
►PQ≅RS and
SP≅QR P
Q R
S
Theorems about parallelograms
• 6.2—If a quadrilateral is a parallelogram, then its opposite angles are congruent.
P ≅ R and
Q ≅ S P
Q R
S
Theorems about parallelograms
• 6.3—If a
quadrilateral is a
parallelogram, then
its consecutive
angles are
supplementary (add
up to 180°).
mP +mQ = 180°,
mQ +mR = 180°,
mR + mS = 180°,
mS + mP = 180°
P
Q R
S
Theorems about parallelograms
• 6.4—If a
quadrilateral is a
parallelogram,
then its
diagonals bisect
each other.
QM ≅ SM and
PM ≅ RM P
Q R
S
Ex. 1: Using properties of
Parallelograms
• FGHJ is a
parallelogram. Find
the unknown length.
Explain your
reasoning.
a. JH
b. JK
F G
J H
K
5
3
b.
Ex. 1: Using properties of
Parallelograms • FGHJ is a
parallelogram. Find the unknown length. Explain your reasoning.
a. JH
b. JK
SOLUTION:
a. JH = FG Opposite sides of a are ≅.
JH = 5 Substitute 5 for FG.
F G
J H
K
5
3
b.
Ex. 1: Using properties of
Parallelograms • FGHJ is a
parallelogram. Find the unknown length. Explain your reasoning.
a. JH
b. JK
SOLUTION:
a. JH = FG Opposite sides of a are ≅.
JH = 5 Substitute 5 for FG.
F G
J H
K
5
3
b. b. JK = GK Diagonals of a
bisect each other.
JK = 3 Substitute 3 for GK
Ex. 2: Using properties of parallelograms
PQRS is a parallelogram.
Find the angle measure.
a. mR
b. mQ
P
R Q
70°
S
Ex. 2: Using properties of parallelograms
PQRS is a parallelogram.
Find the angle measure.
a. mR
b. mQ
a. mR = mP Opposite angles of a are ≅.
mR = 70° Substitute 70° for mP.
P
R Q
70°
S
Ex. 2: Using properties of parallelograms
PQRS is a parallelogram.
Find the angle measure.
a. mR
b. mQ
a. mR = mP Opposite angles of a are ≅.
mR = 70° Substitute 70° for mP.
b. mQ + mP = 180° Consecutive s of a are
supplementary.
mQ + 70° = 180° Substitute 70° for mP.
mQ = 110° Subtract 70° from each side.
P
R Q
70°
S
Ex. 3: Using Algebra with Parallelograms
PQRS is a
parallelogram. Find
the value of x.
mS + mR = 180°
3x + 120 = 180
3x = 60
x = 20
Consecutive s of a □ are
supplementary.
Substitute 3x for mS and 120 for
mR.
Subtract 120 from each side.
Divide each side by 3.
S
Q P
R 3x° 120°
Ex. 4: Proving Facts about Parallelograms
Given: ABCD and AEFG are
parallelograms.
Prove 1 ≅ 3.
1. ABCD is a □. AEFG is a ▭. 2. 1 ≅ 2, 2 ≅ 3
3. 1 ≅ 3
1. Given
3
2
1
C
D
A
G
BE
F
Ex. 4: Proving Facts about Parallelograms
Given: ABCD and AEFG are
parallelograms.
Prove 1 ≅ 3.
1. ABCD is a □. AEFG is a □.
2. 1 ≅ 2, 2 ≅ 3
3. 1 ≅ 3
1. Given
2. Opposite s of a ▭ are
≅
3
2
1
C
D
A
G
BE
F
Ex. 4: Proving Facts about Parallelograms
Given: ABCD and AEFG are
parallelograms.
Prove 1 ≅ 3.
1. ABCD is a □. AEFG is a □.
2. 1 ≅ 2, 2 ≅ 3
3. 1 ≅ 3
1. Given
2. Opposite s of a ▭ are
≅
3. Transitive prop. of
congruence.
3
2
1
C
D
A
G
BE
F
Ex. 5: Proving Theorem 6.2
Given: ABCD is a parallelogram.
Prove AB ≅ CD, AD ≅ CB.
1. ABCD is a .
2. Draw BD.
3. AB ║CD, AD ║ CB.
4. ABD ≅ CDB, ADB ≅ CBD
5. DB ≅ DB
6. ∆ADB ≅ ∆CBD
7. AB ≅ CD, AD ≅ CB
1. Given
A
D
B
C
Ex. 5: Proving Theorem 6.2
Given: ABCD is a parallelogram.
Prove AB ≅ CD, AD ≅ CB.
1. ABCD is a .
2. Draw BD.
3. AB ║CD, AD ║ CB.
4. ABD ≅ CDB, ADB ≅ CBD
5. DB ≅ DB
6. ∆ADB ≅ ∆CBD
7. AB ≅ CD, AD ≅ CB
1. Given
2. Through any two points, there exists exactly one line.
A
D
B
C
Ex. 5: Proving Theorem 6.2
Given: ABCD is a parallelogram.
Prove AB ≅ CD, AD ≅ CB.
1. ABCD is a .
2. Draw BD.
3. AB ║CD, AD ║ CB.
4. ABD ≅ CDB, ADB ≅ CBD
5. DB ≅ DB
6. ∆ADB ≅ ∆CBD
7. AB ≅ CD, AD ≅ CB
1. Given
2. Through any two points, there exists exactly one line.
3. Definition of a parallelogram
A
D
B
C
Ex. 5: Proving Theorem 6.2
Given: ABCD is a parallelogram.
Prove AB ≅ CD, AD ≅ CB.
1. ABCD is a .
2. Draw BD.
3. AB ║CD, AD ║ CB.
4. ABD ≅ CDB, ADB ≅ CBD
5. DB ≅ DB
6. ∆ADB ≅ ∆CBD
7. AB ≅ CD, AD ≅ CB
1. Given
2. Through any two points, there exists exactly one line.
3. Definition of a parallelogram
4. Alternate Interior s Thm.
A
D
B
C
Ex. 5: Proving Theorem 6.2
Given: ABCD is a parallelogram.
Prove AB ≅ CD, AD ≅ CB.
1. ABCD is a .
2. Draw BD.
3. AB ║CD, AD ║ CB.
4. ABD ≅ CDB, ADB ≅ CBD
5. DB ≅ DB
6. ∆ADB ≅ ∆CBD
7. AB ≅ CD, AD ≅ CB
1. Given
2. Through any two points, there exists exactly one line.
3. Definition of a parallelogram
4. Alternate Interior s Thm.
5. Reflexive property of congruence
6. ASA Congruence Postulate
A
D
B
C
Ex. 5: Proving Theorem 6.2
Given: ABCD is a parallelogram.
Prove AB ≅ CD, AD ≅ CB.
1. ABCD is a .
2. Draw BD.
3. AB ║CD, AD ║ CB.
4. ABD ≅ CDB, ADB ≅ CBD
5. DB ≅ DB
6. ∆ADB ≅ ∆CBD
7. AB ≅ CD, AD ≅ CB
1. Given
2. Through any two points, there exists exactly one line.
3. Definition of a parallelogram
4. Alternate Interior s Thm.
5. Reflexive property of congruence
6. ASA Congruence Postulate
7. CPCTC
A
D
B
C
Ex. 6: Using parallelograms in real life
FURNITURE DESIGN. A drafting
table is made so that the legs can
be joined in different ways to
change the slope of the drawing
surface. In the arrangement
below, the legs AC and BD do not
bisect each other. Is ABCD a
parallelogram?
B
C
DA
Ex. 5: Proving Theorem 6.2
Given: ABCD is a parallelogram.
Prove AB ≅ CD, AD ≅ CB.
1. ABCD is a .
2. Draw BD.
3. AB ║CD, AD ║ CB.
4. ABD ≅ CDB, ADB ≅ CBD
5. DB ≅ DB
6. ∆ADB ≅ ∆CBD
7. AB ≅ CD, AD ≅ CB
1. Given
2. Through any two points, there exists exactly one line.
3. Definition of a parallelogram
4. Alternate Interior s Thm.
5. Reflexive property of congruence
A
D
B
C
Ex. 6: Using parallelograms in real life
FURNITURE DESIGN. A drafting
table is made so that the legs can
be joined in different ways to
change the slope of the drawing
surface. In the arrangement
below, the legs AC and BD do not
bisect each other. Is ABCD a
parallelogram?
ANSWER: NO. If ABCD were a
parallelogram, then by Theorem
6.5, AC would bisect BD and BD
would bisect AC. They do not, so
it cannot be a parallelogram.
B
C
DA
Practice!
Pg.297-300
#2-16 even
#18,20,22
#24-33 all
#53-55
6.3 Proving Quadrilaterals
are Parallelograms
Theorems
Theorem 6.5: If the
diagonals of a
quadrilateral
bisect each other,
then the
quadrilateral is a
parallelogram. ABCD is a parallelogram.
A
D
B
C
Theorems
Theorem 6.6: If both
pairs of opposite
sides of a
quadrilateral are
congruent, then the
quadrilateral is a
parallelogram.
A
D
B
C
ABCD is a parallelogram.
Theorems
Theorem 6.7: If both
pairs of opposite
angles of a
quadrilateral are
congruent, then the
quadrilateral is a
parallelogram.
A
D
B
C
ABCD is a parallelogram.
Theorems
Theorem 6.8: If an
angle of a
quadrilateral is
supplementary
to both of its
consecutive
angles, then the
quadrilateral is a
parallelogram.
A
D
B
C
ABCD is a parallelogram.
x°
(180 – x)° x°
Ex. 1: Proof of Theorem 6.6
Statements:
1. AB ≅ CD, AD ≅ CB.
2. AC ≅ AC
3. ∆ABC ≅ ∆CDA
4. BAC ≅ DCA, DAC ≅ BCA
5. AB║CD, AD ║CB.
6. ABCD is a
Reasons:
1. Given
C
D
B
A
Ex. 1: Proof of Theorem 6.6
Statements:
1. AB ≅ CD, AD ≅ CB.
2. AC ≅ AC
3. ∆ABC ≅ ∆CDA
4. BAC ≅ DCA, DAC ≅ BCA
5. AB║CD, AD ║CB.
6. ABCD is a
Reasons:
1. Given
2. Reflexive Prop. of Congruence
C
D
B
A
Ex. 1: Proof of Theorem 6.6
Statements:
1. AB ≅ CD, AD ≅ CB.
2. AC ≅ AC
3. ∆ABC ≅ ∆CDA
4. BAC ≅ DCA, DAC ≅ BCA
5. AB║CD, AD ║CB.
6. ABCD is a
Reasons:
1. Given
2. Reflexive Prop. of Congruence
3. SSS Congruence Postulate
C
D
B
A
Ex. 1: Proof of Theorem 6.6
Statements:
1. AB ≅ CD, AD ≅ CB.
2. AC ≅ AC
3. ∆ABC ≅ ∆CDA
4. BAC ≅ DCA, DAC ≅ BCA
5. AB║CD, AD ║CB.
6. ABCD is a
Reasons:
1. Given
2. Reflexive Prop. of Congruence
3. SSS Congruence Postulate
4. CPCTC
C
D
B
A
Ex. 1: Proof of Theorem 6.6
Statements:
1. AB ≅ CD, AD ≅ CB.
2. AC ≅ AC
3. ∆ABC ≅ ∆CDA
4. BAC ≅ DCA, DAC ≅ BCA
5. AB║CD, AD ║CB.
6. ABCD is a
Reasons:
1. Given
2. Reflexive Prop. of Congruence
3. SSS Congruence Postulate
4. CPCTC
5. Alternate Interior s Converse
C
D
B
A
Ex. 1: Proof of Theorem 6.6
Statements:
1. AB ≅ CD, AD ≅ CB.
2. AC ≅ AC
3. ∆ABC ≅ ∆CDA
4. BAC ≅ DCA, DAC ≅ BCA
5. AB║CD, AD ║CB.
6. ABCD is a
Reasons:
1. Given
2. Reflexive Prop. of Congruence
3. SSS Congruence Postulate
4. CPCTC
5. Alternate Interior s Converse
6. Def. of a parallelogram.
C
D
B
A
Ex. 2: Proving Quadrilaterals are
Parallelograms
• As the sewing box below is opened, the
trays are always parallel to each other.
Why?
2.75 in. 2.75 in.
2 in.
2 in.
Ex. 2: Proving Quadrilaterals are
Parallelograms
• Each pair of hinges are opposite sides of a quadrilateral. The 2.75 inch sides of the quadrilateral are opposite and congruent. The 2 inch sides are also opposite and congruent. Because opposite sides of the quadrilateral are congruent, it is a parallelogram. By the definition of a parallelogram, opposite sides are parallel, so the trays of the sewing box are always parallel.
2.75 in. 2.75 in.
2 in.
2 in.
Another Theorem ~
• Theorem 6.10—If one pair of opposite
sides of a quadrilateral are congruent and
parallel, then the quadrilateral is a
parallelogram.
• ABCD is a
parallelogram.
A
B C
D
Ex. 3: Proof of Theorem 6.10
Given: BC║DA, BC ≅ DA
Prove: ABCD is a
Statements:
1. BC ║DA
2. DAC ≅ BCA
3. AC ≅ AC
4. BC ≅ DA
5. ∆BAC ≅ ∆DCA
6. AB ≅ CD
7. ABCD is a
Reasons:
1. Given
C
D
B
A
Ex. 3: Proof of Theorem 6.10
Given: BC║DA, BC ≅ DA
Prove: ABCD is a
Statements:
1. BC ║DA
2. DAC ≅ BCA
3. AC ≅ AC
4. BC ≅ DA
5. ∆BAC ≅ ∆DCA
6. AB ≅ CD
7. ABCD is a
Reasons:
1. Given
2. Alt. Int. s Thm.
C
D
B
A
Ex. 3: Proof of Theorem 6.10
Given: BC║DA, BC ≅ DA
Prove: ABCD is a
Statements:
1. BC ║DA
2. DAC ≅ BCA
3. AC ≅ AC
4. BC ≅ DA
5. ∆BAC ≅ ∆DCA
6. AB ≅ CD
7. ABCD is a
Reasons:
1. Given
2. Alt. Int. s Thm.
3. Reflexive Property
C
D
B
A
Ex. 3: Proof of Theorem 6.10
Given: BC║DA, BC ≅ DA
Prove: ABCD is a
Statements:
1. BC ║DA
2. DAC ≅ BCA
3. AC ≅ AC
4. BC ≅ DA
5. ∆BAC ≅ ∆DCA
6. AB ≅ CD
7. ABCD is a
Reasons:
1. Given
2. Alt. Int. s Thm.
3. Reflexive Property
4. Given
C
D
B
A
Ex. 3: Proof of Theorem 6.10
Given: BC║DA, BC ≅ DA
Prove: ABCD is a
Statements:
1. BC ║DA
2. DAC ≅ BCA
3. AC ≅ AC
4. BC ≅ DA
5. ∆BAC ≅ ∆DCA
6. AB ≅ CD
7. ABCD is a
Reasons:
1. Given
2. Alt. Int. s Thm.
3. Reflexive Property
4. Given
5. SAS Congruence Post.
C
D
B
A
Ex. 3: Proof of Theorem 6.10
Given: BC║DA, BC ≅ DA
Prove: ABCD is a
Statements:
1. BC ║DA
2. DAC ≅ BCA
3. AC ≅ AC
4. BC ≅ DA
5. ∆BAC ≅ ∆DCA
6. AB ≅ CD
7. ABCD is a
Reasons:
1. Given
2. Alt. Int. s Thm.
3. Reflexive Property
4. Given
5. SAS Congruence Post.
6. CPCTC
C
D
B
A
Ex. 3: Proof of Theorem 6.10
Given: BC║DA, BC ≅ DA
Prove: ABCD is a
Statements:
1. BC ║DA
2. DAC ≅ BCA
3. AC ≅ AC
4. BC ≅ DA
5. ∆BAC ≅ ∆DCA
6. AB ≅ CD
7. ABCD is a
Reasons:
1. Given
2. Alt. Int. s Thm.
3. Reflexive Property
4. Given
5. SAS Congruence Post.
6. CPCTC
7. If opp. sides of a quad. are ≅, then it is a .
C
D
B
A
Objective 2: Using Coordinate Geometry
• When a figure is in the coordinate plane,
you can use the Distance Formula (see—it
never goes away) to prove that sides are
congruent and you can use the slope
formula (see how you use this again?) to
prove sides are parallel.
Ex. 4: Using properties of parallelograms
• Show that A(2, -1), B(1,
3), C(6, 5) and D(7,1)
are the vertices of a
parallelogram.
6
4
2
-2
-4
5 10 15
D(7, 1)
C(6, 5)
B(1, 3)
A(2, -1)
Ex. 4: Using properties of parallelograms • Method 1—Show that opposite
sides have the same slope, so they are parallel.
• Slope of AB. – 3-(-1) = - 4
1 - 2
• Slope of CD. – 1 – 5 = - 4
7 – 6
• Slope of BC. – 5 – 3 = 2
6 - 1 5
• Slope of DA. – - 1 – 1 = 2
2 - 7 5
• AB and CD have the same slope, so they are parallel. Similarly, BC ║ DA.
6
4
2
-2
-4
5 10 15
D(7, 1)
C(6, 5)
B(1, 3)
A(2, -1)
Because opposite sides are
parallel, ABCD is a
parallelogram.
Ex. 4: Using properties of parallelograms
• Method 2—Show that
opposite sides have the
same length.
• AB=√(1 – 2)2 + [3 – (- 1)2] = √17
• CD=√(7 – 6)2 + (1 - 5)2 = √17
• BC=√(6 – 1)2 + (5 - 3)2 = √29
• DA= √(2 – 7)2 + (-1 - 1)2 = √29
• AB ≅ CD and BC ≅ DA.
Because both pairs of opposites
sides are congruent, ABCD is a
parallelogram.
6
4
2
-2
-4
5 10 15
D(7, 1)
C(6, 5)
B(1, 3)
A(2, -1)
Ex. 4: Using properties of parallelograms
• Method 3—Show that
one pair of opposite
sides is congruent and
parallel.
• Slope of AB = Slope of CD
= -4
• AB=CD = √17
• AB and CD are congruent
and parallel, so ABCD is a
parallelogram.
6
4
2
-2
-4
5 10 15
D(7, 1)
C(6, 5)
B(1, 3)
A(2, -1)
Practice!!
Pg. 307-308
#1-17 all
#26-29 all
#32-34 all
Geometry
6.4 Special Parallelograms
Rectangle
• A quadrilateral with four right angles.
Why is a rectangle a parallelogram?
Both Pairs of Opp. Angles are Congruent
Rhombus
• A quadrilateral with four congruent sides.
Why is a rhombus a parallelogram?
Both Pairs of Opp. Sides are Congruent
Square
• A quadrilateral with four congruent sides
and four right angles.
Why is a rhombus a parallelogram?
Both Pairs of Opp. Sides are Congruent
Both Pairs of Opp. Angles are Congruent
Review: Rectangles, Rhombuses, and
Squares all Share these Properties of a
Parallelogram… 1) Opp. Sides are //
2) Opp. Angles are congruent
3) Opp. Sides are congruent
4) Diagonals Bisect Each Other
In addition, rectangles, rhombuses, and squares all have their own special properties. These are the focus of this lesson.
Theorem: The diagonals of a
Rectangle are Congruent
Draw two congruent intersecting lines that bisect each
other.
Connect the corners. You drew a rectangle.
Theorem: The diagonals of a
rhombus are perpendicular.
Draw two lines that bisect each other & are perpendicular.
Connect the corners. You have drawn a rhombus.
Theorem: Each diagonal of a rhombus
bisects two angles of the rhombus.
Draw a rhombus and its diagonals.
You bisected all four angles.
Theorem: The midpoint of the hypotenuse of a
right triangle is equidistant from all three vertices.
Draw a right triangle and put a point at the midpoint of the
hypotenuse.
Draw a line from that point to the vertex of the right angle.
All three distances are equal.
.
Theorem: If an angle of a parallelogram is a right
angle, then the parallelogram is a rectangle.
Draw one right angle.
Draw the two other sides parallel to the opposite side.
You have drawn a rectangle.
Why is it a rectangle?
Opp. Angles of a Parallelogram
Are congruent
Parallel lines imply SS Int. angles
are supplementary.
Theorem: If two consecutive sides of a
parallelogram are congruent, then the
parallelogram is a rhombus. Draw two congruent sides of an angle.
Draw the two other sides parallel
to the opposite sides.
You have drawn a rhombus.
Why is it a rhombus?
Opp. Sides of a Parallelogram are congruent.
Given Quad. WXYZ is a rectangle. Complete the statements with numbers.
Make sure your + and – are clear!
3. If TX = 4.5, then WY = _____.
4. If WY = 3a + 16 and ZX = 5a – 18, then a = _____,
WY = _____ and ZX = _____.
5. If m<TWZ = 70, then m<TZW = _____ and
m<WTZ = _____.
Z Y
X W
T
7. If m<4 = 25, then m<5 = _____.
8. If m<DAB = 130, then m<ADC = _____.
9. If m<4 = 3x – 2 and m<5 = 2x + 7,
then x = ____, m<4 = ____, and m<5 =____.
11. If m<2 = 3y + 9 and m<4 = 2y – 4,
then y = _____, m<2 = _____, and m<4 = ____.
Given Quad. ABCD is a rhombus. Complete the statements with numbers.
D C
B A
5
4
3
2
1
Given Quad. JKLM is a square. Complete the statements with numbers.
14. If JL =18, then MK = _____, JX = _____, and XK = _____.
15. m<MJK = _____, m<MXJ = _____ and m<KLJ = _____.
M
L
K
J
x
M L
J K
X
Property Parallelogram Rectangle Rhombus Square
5) Diags. Bisect
each other X X X X
6) Diags. Are
conguent X X
X
X X
X 7) Diags. Are
Perpendicular
8) A diagonal
Bisects 2 angles
Review
Practice!!
Pg. 315-317
#1-9 all
#10,12,14
#22 &24
Lesson 6-5 Trapezoids and Kites
Definition
Kite – a quadrilateral that has two pairs of consecutive congruent sides, but opposite sides are not congruent.
Perpendicular Diagonals of a Kite
If a quadrilateral is a kite, then its diagonals are perpendicular.
D
C
A
B
BDAC
Non-Vertex Angles of a Kite
If a quadrilateral is a kite, then non-vertex angles are congruent
D
C
A
B
A C, B D
Vertex diagonals bisect vertex angles
D
C
A
B
If a quadrilateral is a kite then the vertex
diagonal bisects the vertex angles.
Vertex diagonal bisects the non-vertex diagonal
D
C
A
B
If a quadrilateral is a kite then the vertex diagonal
bisects the non-vertex diagonal
Definition-a quadrilateral with exactly one pair of parallel sides.
Leg Leg
Base
Base A B
C D ›
›
Trapezoid
<A + <C = 180
<B + <D = 180
A B
C D ›
›
Leg Angles are Supplementary
Property of a Trapezoid
Isosceles Trapezoid
Definition - A trapezoid with congruent legs.
Isosceles Trapezoid - Properties
1) Base Angles Are Congruent
2) Diagonals Are Congruent
Example
PQRS is an isosceles trapezoid. Find m P, m Q and mR.
50S R
P Q
m R = 50 since base angles are congruent
mP = 130 and mQ = 130 (consecutive angles
of parallel lines cut by a transversal are )
Find the measures of the angles in trapezoid
48
m< A = 132
m< B = 132
m< D = 48
Find BE
AC = 17.5, AE = 9.6
E
Example
Find the side lengths of the kite.
20
12
1212
UW
Z
Y
X
Example Continued
WX = 4 34
likewise WZ = 4 34
XY =12 2
likewise ZY =12 2
20
12
1212
UW
Z
Y
X
We can use the Pythagorean Theorem to
find the side lengths.
122 + 202 = (WX)2
144 + 400 = (WX)2
544 = (WX)2
122 + 122 = (XY)2
144 + 144 = (XY)2
288 = (XY)2
Find the lengths of the sides of the kite
W
X
Y
Z
4
5 5
8
Find the lengths of the sides of kite to the nearest tenth
4
2
2
7
Example 3
Find mG and mJ.
60132
J
G
HK
Since GHJK is a kite G J
So 2(mG) + 132 + 60 = 360
2(mG) =168
mG = 84 and mJ = 84
Try This!
RSTU is a kite. Find mR, mS and mT.
x
125
x+30
S
U
R T
x +30 + 125 + 125 + x = 360
2x + 280 = 360
2x = 80
x = 40
So mR = 70, mT = 40 and mS = 125
Try These
base
base
legleg
A B
D C base
base
legleg
A B
D C
1. If <A = 134, find m<D 2. m<C = x +12 and
m<B = 3x – 2, find x and the
measures of the 2 angles
m<D = 46
x = 42.5
m<C = 54.5
m<B = 125.5
Using Properties of Trapezoids
Find the area of this trapezoid.
When working with a trapezoid, the height may be measured
anywhere between the two bases. Also, beware of "extra"
information. The 35 and 28 are not needed to compute this area.
Area of trapezoid = 212
1bbh
A = ½ * 26 * (20 + 42)
A = 806
Using Properties of Trapezoids
Find the area of a trapezoid
with bases of 10 in and 14
in, and a height of 5 in.
Example 2
Using Properties of Kites
D
A
B
C
Area Kite = one-half product of diagonals
212
1ddA
BDACArea 2
1
Using Properties of Kites
D
A
B
C
Example 6
E
2
4 4
4
ABCD is a Kite.
a) Find the lengths of all
the sides.
b) Find the area of the Kite.
Venn Diagram:
http://teachers2.wcs.edu/high/rhs/staceyh/Geometry/Chapter%206%20Notes.ppt#435,22,6.2 – Properties of Parallelograms
Flow Chart:
Practice!!
Pg. 322-324
#1-16 all
#20-25 odd
#40 for 5 extra credit points
6-6
Placing Figures in the
Coordinate Plane
ABCD is a rectangle. A(0,0), B(5,0),
C(x,y) and D(0,4) give coordinates for
point C .
ABCD is a rectangle. A(0,0), B(m,0),
C(x,y) and D(0,n) give coordinates for
point C in terms of m and n..
Give coordinates for points W and Z
without using any new variables.
Give coordinates for points W and Z
without using any new variables.
ABCD is a square. A(3,3) and B(-3,3) find the
coordinates for C and D in the third and fourth
quadrants. Can you find the other set of coordinates
in the first and second quadrants?
Plot the points to make a trapezoid.
A(0,0), B(6,0), C(4,5) and D(-2,5)
Give coordinates for points W and Z
without using any new variables.
To Recap the main ideas…
Practice!!
Pg. 328-39
#1-13 all
#33 &34
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