Download 28 Oct 2015 9:50 - 11:20 Geometry Agenda

28 Oct 2015 9:50 - 11:20
Geometry Agenda
1) Properties of Parallelograms (p.25 your
worksheets)
2) G-CO Is this a parallelogram? Group task.
3) Homework - revised
4) Don’t forget the Geometry Dictionary project - 4
pictures per week was the plan - catch up you
have a long weekend
Objectives
Individual :
Students more responsible for learning correcting - reflecting - teaching - editing
Academic:
Properties of parallelograms
Properties of Parallelograms (p.25)
A parallelogram is a quadrilateral that contains two
pairs of parallel sides. In previous units we used
parallel lines to prove the following two things
about parallelograms:
Theorem: Opposite angles of a parallelogram
are congruent
Theorem: Consecutive angles of a
parallelogram are supplementary
Do a sketch in
space below box
cont.
In this unit, we are going to use what we know
about the properties of congruent triangles to find
some more properties of parallelograms. Let’s look
at the parallelogram below.
A
D
B
C
cont.
Like we did before, draw a diagonal of the
parallelogram connecting B and D, in other words
draw line segment BD.
A
D
B
C
Now use the space below to prove triangle
ABD is congruent to triangle CDB
Statement
Reason
1. ABCD is a parallelogram 1. Given
2. Line segment AB is
2. Definition of a
parallel to line segment DC; parallelogram
line segment AD is parallel
to line segment BC
3. There is a line BD
3. Through any 2 points
there is exactly one line
Now use the space below to prove triangle
ABD is congruent to triangle CDB
Statement
1.
ABCD is a parallelogram
Reason
1.
Given
2. Line segment AB is parallel to line segment
DC; line segment AD is parallel to line segment
BC
2. Definition of a parallelogram
3. There is a line BD
3. Through any 2 points there is exactly one line
4. Line segment BD is
congruent to line segment
DB
4.
Now use the space below to prove triangle
ABD is congruent to triangle CDB
Statement
1.
ABCD is a parallelogram
Reason
1.
Given
2. Line segment AB is parallel to line segment
DC; line segment AD is parallel to line segment
BC
2. Definition of a parallelogram
3. There is a line BD
3. Through any 2 points there is exactly one line
4. Line segment BD is
congruent to line segment
DB
4. Reflexive property
Now use the space below to prove triangle
ABD is congruent to triangle CDB
Statement
1.
ABCD is a parallelogram
Reason
1.
Given
2. Line segment AB is parallel to line segment DC; line
segment AD is parallel to line segment BC
2. Definition of a parallelogram
3. There is a line BD
3. Through any 2 points there is exactly one line
4. Line segment BD is congruent to line segment DB
4. Reflexive property
5. Angle ABD is congruent 5. For 2 parallel lines cut by
to angle CDB; angle ADB is a transversal alternate
congruent to CBD
interior angles are
congruent.
Now use the space below to prove triangle
ABD is congruent to triangle CDB
Statement
1.
Reason
ABCD is a parallelogram
1.
Given
2. Line segment AB is parallel to line segment DC; line segment AD
is parallel to line segment BC
2. Definition of a parallelogram
3. There is a line BD
3. Through any 2 points there is exactly one
line
4. Line segment BD is congruent to line segment DB
4. Reflexive property
5. Angle ABD is congruent to angle CDB; angle ABD is
congruent to CDB
5. For 2 parallel lines cut by a
transversal alternate interior angles
are congruent.
6. Triangle ABC is
congruent to triangle CDB
\
ASA
Two angles and the included side of one
triangle are congruent to two angles and the
included side of the other triangle
Now use the space below to prove triangle
ABD is congruent to triangle CDB
Statement
1.
ABCD is a parallelogram
Reason
1.
Given
2. Line segment AB is parallel to line segment DC; line
segment AD is parallel to line segment BC
2. Definition of a parallelogram
3. There is a line BD
3. Through any 2 points there is exactly one line
4. Line segment BD is congruent to line segment DB
4. Reflexive property
5. Angle ABD is congruent to angle CDB; angle
ABD is congruent to CDB
5. For 2 parallel lines cut by a transversal
alternate interior angles are congruent.
6. Triangle ABD is congruent to triangle CDB
6. Angle-Side-Angle
Now
that you know that triangle ABD is congruent to
triangle CDB, use what you know about
corresponding parts of congruent triangles (that
they are congruent) to write at least two things
that would be true about every parallelogram.
angle BAD is congruent to angle DCB
line segment AD and CD are congruent; line
segments DC and BA are congruent
Theorem: Opposite angles of a
parallelogram are congruent.
Theorem: Opposite sides of a
parallelogram are congruent.
Page 2
Theorem: Diagonals of a parallelogram
bisect each other. Going to prove this.
Below is a parallelogram with two diagonals that
intersect at point M.
Use the space below to give a 2 column proof of
how you know that triangle WMZ is congruent to
triangle YMX. You may use the definition of a
parallelogram and the theorems we did on the
previous page.
6 mins - I think a 4 or 5 line proof is possible
Statement
1.
Quadrilateral WXYZ is a
parallelogram
Reason
1.
Given
2. line segment WZ || line segment YX
and line segement WX|| line segment ZY
2. Definition of a parallelogram
3. Angle ZWM is congruent to angle XYM
3. If 2 parallel lines are cut by a transversal,
then alternate interior angles are congruent.
4. line segment WZ is congruent to line
segment XY; line segment WX is congruent
to line segment ZY
4. Definition of a parallelogram
5. Angle WMZ is congruent to angle YMX
5. Vertical angles are congruent
Now that we know
that triangle WMZ is congruent to triangle YMX,
what can we say about line segment MX and
line segment MZ?
How do we know this?
Now that we know
that triangle WMZ is congruent to triangle YMX,
what can we say about line segment MX and
line segment MZ? They are congruent.
How do we know this?
Corresponding parts of congruent triangles are
congruent.
Since we have shown that MX = MZ,
we know that M is the midpoint of line segment
XZ.
Is there any reason that the reasoning that we
used above could not be used for triangle WMZ
and triangle YMX (not YMZ).
As long as we are looking at the diagram we
have here where XMZ are colinear we can use
this reasoning.
Since we can say that M is the midpoint
of XZ and WY,
we have proved the following:
Theorem: Diagonals of a parallelogram bisect
each other.
p. 334 textbook for further reference
Homework - Due Monday
Problems Adjusted from what I said on Monday.
4-4 Proving Congruence SSS, SAS
Exercises starting on page 230
7, 8, 14 - 18, 22 and Extra Credit 23.
Do a two column proof for question 14, please.