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Lesson: 6.3 Tests for Parallelograms
Objectives:
 To Identify the 5 CONDITIONS that GUARANTEE
that a QUADRILATERAL is a PARALLELOGRAM
 To Use the 5 CONDITIONS to SOLVE Problems
GEOMETRY 6.3
IF a Quadrilateral is a Parallelogram
THEN:
GEOMETRY 6.3
IF a Quadrilateral is a Parallelogram
THEN:
1. OPPOSITE SIDES are PARALLEL
GEOMETRY 6.3
IF a Quadrilateral is a Parallelogram
THEN:
1. OPPOSITE SIDES are PARALLEL
2. OPPOSITE SIDES are CONGRUENT.
GEOMETRY 6.3
IF a Quadrilateral is a Parallelogram
THEN:
1. OPPOSITE SIDES are PARALLEL
2. OPPOSITE SIDES are CONGRUENT.
3. OPPOSITE ANGLES are CONGRUENT.
GEOMETRY 6.3
IF a Quadrilateral is a Parallelogram
THEN:
1. OPPOSITE SIDES are PARALLEL
2. OPPOSITE SIDES are CONGRUENT.
3. OPPOSITE ANGLES are CONGRUENT.
4. CONSECUTIVE ANGLES are SUPPLEMENTARY.
GEOMETRY 6.3
IF a Quadrilateral is a Parallelogram
THEN:
1. OPPOSITE SIDES are PARALLEL
2. OPPOSITE SIDES are CONGRUENT.
3. OPPOSITE ANGLES are CONGRUENT.
4. CONSECUTIVE ANGLES are SUPPLEMENTARY.
5. DIAGONALS Bisect each other.
GEOMETRY 6.3
Which, if any, of the Properties of a Parallelogram
PROVE that a Quadrilateral IS a Parallelogram?
GEOMETRY 6.3
IF a QUADRILATERAL has OPPOSITE SIDES
that are PARALLEL
Is it a PARALLELOGRAM?
GEOMETRY 6.3
IF a QUADRILATERAL has OPPOSITE SIDES
that are PARALLEL
Is it a PARALLELOGRAM?
YES – the DEFINITION of a PARALLELOGRAM
is a Quadrilateral for which
OPPOSITE SIDES are Parallel!
GEOMETRY 6.3
IF a QUADRILATERAL has
ONE PAIR of OPPOSITE SIDES that are CONGRUENT,
Is it a PARALLELOGRAM?
GEOMETRY 6.3
IF a QUADRILATERAL has
ONE PAIR of OPPOSITE SIDES that are CONGRUENT,
Is it a PARALLELOGRAM?
Can you DRAW a COUNTEREXAMPLE?
GEOMETRY 6.3
IF a QUADRILATERAL has
BOTH PAIR of OPPOSITE SIDES that are CONGRUENT,
Is it a PARALLELOGRAM?
Is there a COUNTEREXAMPLE?
GEOMETRY 6.3
IF a QUADRILATERAL has
BOTH PAIR of OPPOSITE SIDES that are CONGRUENT,
Is it a PARALLELOGRAM?
Is there a COUNTEREXAMPLE?
Can you PROVE it?
GEOMETRY 6.3
IF a QUADRILATERAL has
ONE PAIR of OPPOSITE ANGLES that are CONGRUENT,
Is it a PARALLELOGRAM?
Is there a COUNTEREXAMPLE?
Can you PROVE it?
GEOMETRY 6.3
IF a QUADRILATERAL has
BOTH PAIRs of OPPOSITE ANGLES that are CONGRUENT,
Is it a PARALLELOGRAM?
Is there a COUNTEREXAMPLE?
Can you PROVE it?
GEOMETRY 6.3
Given: Angles T and R are Congruent
Angles Q and S are Congruent
Prove: QRST is a Parallelogram
GEOMETRY 6.3
IF a QUADRILATERAL has
DIAGONALS that Bisect each other,
Is it a PARALLELOGRAM?
Is there a COUNTEREXAMPLE?
Can you PROVE it?
GEOMETRY 6.3
GEOMETRY 6.3
IF a QUADRILATERAL has
ONE PAIR of OPPOSITE SIDES that is BOTH
PARALLEL and CONGRUENT,
Is it a PARALLELOGRAM?
Is there a COUNTEREXAMPLE?
Can you PROVE it?
GEOMETRY 6.3
GEOMETRY 6.3
GEOMETRY 6.3
GEOMETRY 6.3
GEOMETRY 6.3
GEOMETRY 6.3
GEOMETRY 6.3
GEOMETRY 6.3
GEOMETRY 6.3
COORDINATE GEOMETRY Determine whether the
figure with vertices A(–3, 0), B(–1, 3), C(3, 2), and
D(1, –1) is a parallelogram.
Three Methods:
1. SLOPE formula
2. DISTANCE formula
3. MIDPOINT formula
Geometry 6.3
You should be able to:
 Determine is a Quadrilateral is a PARALLEOGRAM
 Determine if a CONDITION defines a PARALLELOGRAM
Lesson: 6.4 Rectangles
Objectives:
 To Identify the PROPERTIES of RECTANGLES
 To Use the Rectangle Properties to SOLVE Problems
 To Identify the PROPERTIES of SQUARES and RHOMBI
 To use the Squares and Rhombi Properties to SOLVE
Problems
GEOMETRY 6.4
A RECTANGLE is:
GEOMETRY 6.4
A RECTANGLE is:
A QUADRILATERAL
GEOMETRY 6.4
A RECTANGLE is:
A QUADRILATERAL
A PARALLELOGRAM
GEOMETRY 6.4
A RECTANGLE is:
A QUADRILATERAL
A PARALLELOGRAM
with 4 Right Angles
GEOMETRY 6.4
PROPERTIES of a Rectangle:
 Same as a Parallelogram
GEOMETRY 6.4
PROPERTIES of a Rectangle:
 Same as a Parallelogram





Opposite Sides are Parallel
Opposite Sides are Congruent
Opposite Angles are Congruent
Consecutive Sides are Supplementary
Diagonals BISECT each other.
GEOMETRY 6.4
PROPERTIES of a Rectangle:
 Same as a Parallelogram





Opposite Sides are Parallel
Opposite Sides are Congruent
Opposite Angles are Congruent
Consecutive Sides are Supplementary
Diagonals BISECT each other.
 All ANGLES are CONGRUENT
GEOMETRY 6.4
PROPERTIES of a Rectangle:
 Same as a Parallelogram





Opposite Sides are Parallel
Opposite Sides are Congruent
Opposite Angles are Congruent
Consecutive Sides are Supplementary
Diagonals BISECT each other.
 All ANGLES are CONGRUENT
 DIAGONALS are
GEOMETRY 6.4
GIVEN:
A
ABCD is a RECTANGLE
PROOF
B
AC and BD are diagonals
PROVE:
AC  BD
D
 DIAGONALS are
C
M
N
GEOMETRY 6.4
MO  2x  8
NP  23
Find X
C
P
O
M
N
GEOMETRY 6.4
CN  x  1
CO  3x  11
2
C
P
Find X
O
M
N
GEOMETRY 6.4
MO  4x  13
PC  x  7
C
P
Find X
O
GEOMETRY 6.4
TRUE or FALSE?
If a QUADRILATERAL has OPPOSITE SIDES
that are CONGRUENT,
then it is a RECTANGLE.
GEOMETRY 6.4
m 1  70
Find: m2 
m5 
m6 
L
K
7
8
1
2
C
9 10
3
6
5
N
4
M
GEOMETRY 6.4
m 9  128
Find: m6 
m7 
m8 
L
K
7
8
1
2
C
9 10
3
6
5
N
4
M
L
K
GEOMETRY 6.4
7
m 5  36
Find m2 
m3 
8
1
2
C
9 10
3
6
5
N
4
M
Kyle is building a barn for his horse. He measures the
diagonals of the door opening to make sure that they bisect
each other and they are congruent. How does he know that
the measure of each corner is 90?
Quadrilateral ABCD has vertices A(–2, 1), B(4, 3),
C(5, 0), and D(–1, –2). Determine whether ABCD is a
rectangle.
Methods:
1. Slope
2. Distance