Download Parallelograms

24 Oct 2016 8:00 - 9:30 AM
Geometry Proposed Agenda
1) Bulletin
2) Student Survey - confidential - be honest - no names
3) Properties of Parallelograms - a quadrilateral with two pairs
of parallel sides
a) Opposite sides
b) Opposite angles
c) Diagonals - measuring
4) Homework
Date
Properties of Parallelograms p 55
A parallelogram is a quadrilateral that
contains 2 pairs of parallel sides. In
previous units we used parallel lines to
prove the following 2 things about
parallelograms:
Theorem: Opposite angles of a
parallelogram are congruent
Theorem: Consecutive angles of a
parallelogram are supplementary.
More about the Properties of
Parallelograms
Given: Quadrilateral ABCD is a parallelogram.
Prove: Triangle ABC is congruent to triangle
CDB.
Draw the diagonal connecting B and D.
Draw in what we
A
B
know.
D
C
More about the Properties of
Parallelograms
Given: Quadrilateral ABCD is a parallelogram.
Prove: Triangle ABC is congruent to triangle
CDB.
Draw the diagonal connecting B and D.
Draw in what we
A
B
*
know.
D
*
C
More about the Properties of
Parallelograms
Given: Quadrilateral ABCD is a parallelogram.
Prove: Triangle ABC is congruent to triangle
CDB.
Draw the diagonal connecting B and D.
Draw in what we
A
B
*
know.
D
*
C
More about the Properties of
Parallelograms
Given: Quadrilateral ABCD is a parallelogram.
Prove: Triangle ABC is congruent to triangle
CDB.
Draw the diagonal connecting B and D.
Draw in what we
A
B
*
#
know.
#
D
*
C
Statement
Reason
1. ABCD is a
parallelogram.
1. Given
2. AB ||DC ; AD || BC
2. Definition of a parallelogram
3. There is a line BC
3. Through any 2 points there is
exactly one line
Statement
Reason
4. ∠ABD =∠CDB:
∠ADB =∠CBD
4. If 2 parallel lines are cut by a
transversal, then alternate interior
angles are congruent.
5. BD = DB
Statement
Reason
4. ∠ABD =∠CDB:
∠ADB =∠CBD
4. If 2 parallel lines are cut by a
transversal, then alternate interior
angles are congruent.
5. DB = BD.
5. Reflexive property
6. △ABD = △CDB
6. ASA congruence postulate
More about the Properties of
Parallelograms
Now we know △ABD = △CDB, using what we know
about corresponding parts of congruent triangles write at
least 2 things that would be true about every parallelogram.
A
B
*
#
#
D
*
C
Theorem: Opposite sides of a
parallelogram are congruent.
Theorem: Opposite angles of a
parallelogram are congruent.
A
B
*
#
#
D
*
C
Name
Date
Parallelogram Diagonals: Data
Collection
For each parallelogram on the
worksheet, measure the length of the
diagonals and line segments defined in
the table below: We will do the first one
together
Measurements
B
C
Parallelogram
AC
1
5.3
2
M
3
A
D
4
5
6
7
8
9
10
BD
AM
(cm)
MC
DM
MB
Patterns?
Measurements (cm)
B
C
M
A
Parallelogram
AC
1
5.3
2
3
D
4
5
6
7
8
9
10
BD
AM
MC
DM
MB
Patterns?
Measurements (cm)
B
C
M
A
Parallelogram
AC
BD
1
5.3
4.2
2
3
D
4
5
6
7
8
9
10
AM
MC
DM
MB
Patterns?
Measurements (cm)
B
C
Parallelogram
AC
BD
1
5.3
4.2
2
M
3
A
D
4
5
6
7
8
9
10
AM
MC
DM
MB
Patterns?
Measurements (cm)
B
C
Parallelogram
AC
1
5.3
2
M
3
A
D
4
5
6
7
8
9
10
BD
AM
4.2 2.65
MC
DM
MB
Patterns?
Measurements (cm)
B
C
Parallelogram
AC
1
5.3
2
M
3
A
D
4
5
6
7
8
9
10
BD
AM
4.2 2.65
MC
DM
MB
Patterns?
Measurements (cm)
B
C
AC
1
5.3
2
M
A
Parallelogram
D
BD
AM
MC
DM
MB
4.2 2.65 2.65 2.1 2.1
Patterns?
For all
three?
3
4
If you see 5
exactly on a6
cm value - 7
show the 8
zero eg. 9
10
3.0cm
Complete then make a
conjecture about the
diagonals of parallelograms (a statement that appears to
be true based on limited
information)
Not formally proven to be true.
What pattern/patterns did we observe?
1, 2, 3
diagonals bisect each other (cut
each other in half)
4, 5, 6
diagonals bisect each other
7, 8, 9
diagonals bisect each other
10, 11, 12
diagonals bisect each other
What pattern/patterns did we observe?
1, 2, 3
diagonals bisect each other
4, 5, 6
diagonals bisect each other
7, 8, 9
diagonals bisect each other
Diagonals congruent
diagonals bisect each other
Diagonals congruent
10, 11, 12
What was your conjecture?
Share on white board
Parallelogram Diagonals: Formal Proof
Given: Quadrilateral ACDE is a parallelogram.
Prove: Segment EC bisects segment AD.
Segment AD bisects segment EC.
Draw the given.
C
A
D
E
Parallelogram Diagonals: Formal Proof
Given: Quadrilateral ACDE is a parallelogram.
Prove: Segment EC bisects segment AD.
Segment AD bisects segment EC.
Draw the given.
C
A
D
E
Homework
Complete the Parallelogram Diagonals: Formal Proof page (blue
sheet).