created by chris markstrum © 2005 6.3 proving quadrilaterals are parallelograms california state...
TRANSCRIPT
created by chris markstrum © 2005
6.3 Proving Quadrilaterals are Parallelograms
California State Standards for Geometry4: Prove basic theorems involving congruence & similarity.7: Prove & use theorems involving parallel lines and properties of quadrilaterals.12: Find & use side and angle measures in triangles and polygons17: Prove theorems using coordinate geometry
created by chris markstrum 2005
If a quadrilateral has both pairs of opposite sides
are congruent,
AB DCAD BC
||AB DC
||AD BC
theorem
A B
CD
parallelogra
m
then it is a parallelogram
created by chris markstrum 2005
theorem
If both pairs of opposite angles of a quadrilateral
are congruent,
A B
CD
A C B D
||AB DC
||AD BCparallelogra
m
then it is a parallelogram
created by chris markstrum 2005
180m A m B 180m A m D
||AB DC
||AD BC
theorem
If a quadrilateral has one angle that issupplementary to both of its consecutive
angles,
A B
CD
parallelogra
m
then it is a parallelogram.
created by chris markstrum 2005
AM CM||AB DC
theorem
if a quadrilateral’s diagonals bisect each other,
A B
CD
M
BM DM||AD BC
parallelogra
m
then it is a parallelogram
created by chris markstrum 2005
AD BC
||AB DC||AD BC
A B
CD
theorem
If a quadrilateral has one pair of opposite sides
parallel and congruent,
AB DCparallelogra
m
then it is a parallelogram
||AD BC
created by chris markstrum 2005
Example
Decide whether you are given enough information to determine that thequadrilateral is a parallelogram. Explain your reasoning.
If a quadrilateral has one angle that is supplementary to both of its consecutive
angles,then it is a parallelogram.
YES
created by chris markstrum 2005
Example
Decide whether you are given enough information to determine that thequadrilateral is a parallelogram. Explain your reasoning.
If both pairs of opposite angles
of a quadrilateral are congruent,
then it is a parallelogram.
YES
created by chris markstrum 2005
Example
Decide whether you are given enough information to determine that thequadrilateral is a parallelogram. Explain your reasoning.
Definition of a parallelogram
YES
created by chris markstrum 2005
Example
Describe how you would prove that ABCD is a parallelogram.
If a quadrilateral has one pair of opposite sides
parallel and congruent, then it is a parallelogram.
Congruent alternate interior angles
make parallel lines.||BC AD
created by chris markstrum 2005
Example
Definition of parallelogram
Congruent alternate interior angles
make parallel lines.||BC AD
Describe how you would prove that ABCD is a parallelogram.
||AB CD
created by chris markstrum 2005
Example
Describe how you would prove that ABCD is a parallelogram.
Definition of parallelogram
Congruent alternate interior angles
make parallel lines.
||BC AD
||AB CD
Congruent corresponding angles
make parallel lines.
created by chris markstrum 2005Given:
Prove: PQRS is a parallelogram P Q S is supplementary to and
3. is supplementary to P S
1. is supplementary to P Q
Statement Reason
2. ||PS QR
4. ||PQ SR
5. is a parallelogramPQRS
2. Supplementary consecutive interior
angles means the lines are parallel.
1. Given
3. Given
4. Supplementary consecutive interior
angles means the lines are parallel.
5. Definition of a parallelogram
created by chris markstrum 2005Given:
Prove: ABCD is a parallelogram
, ||AB CD AB CD A B
CD
3. CAB ACD
1. Given1. , ||AB CD AB CD
Statement Reason
2. Reflexive2. AC AC
3. || . . 'a i s
4. SAS 4. CAB ACD 5. ACB CAD 5. C.P.C.T.C.
7. is a parallelogramABCD
6. . . ' ||a i s 6. ||AD BC
7. Def parallelogram
created by chris markstrum 2005
is a parallelogram
PQT RSTPQRS
Given:Prove:
P Q
RS
T
3. ||PQ SR
1. Given1. PQT RST Statement Reason
2. C.P.C.T.C.2. ,PQ RS PQT RST
3. . . ' ||a i s
4. opp. sides and ||4. is a parallelogramPQRS
created by chris markstrum 2005
To get a parallelogram Show opposite sides || Show opposite sides Show opposite angles Show consecutive
angles are supplementary Show diagonals bisect each other Show one pair of
opposite sides are both || and
( – 1, 2) A
C (1, – 2)
x
y
( 3, 2)B
D( – 3, – 2)
Show that quadrilateral ABCD is a parallelogram
Which of
the
theorems
can be
used?
1
1
–3
–4
Not on the coordinate
plane.
created by chris markstrum 2005
x
y
202 22 4 223 1 2 2
2 22 2 1 3 2 24 22 520
( – 1, 2) A
C (1, – 2)
( 3, 2)B
D( – 3, – 2)
Show that quadrilateral ABCD is a parallelogram
1
1
–3
–4
ABm Show opposite sides ||
DCm
ADm
BCm
00
22
||AB DC ||AD BC
AB 4
4
2 5
AB DC
AD BC
Show opposite sides
DC
AD
BC
created by chris markstrum 2005
Summary
Six ways to show a quadrilateral is a parallelogram1) Show two pairs of opposite parallel sides.
2) Show two pairs of opposite congruent sides.
3) Show two pairs of opposite congruent angles.
4) Show one angle is supplementary to both consecutive angles.
5) Show the diagonals bisect each other.
6) Show that one pair of opposite sides are both congruent and parallel.