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Geometry Section 6-2A
Proofs with Parallelograms
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Proofs with Parallelograms:
We have been working on developing skills in
writing proofs. Each proof has become
increasingly difficult and you have been asked to
fill in more and more as time has gone by. You
must continue to build this skill so that you can
write a proof from scratch all by yourself.
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Proofs:
5 steps to writing a proof.
1.
2.
3.
4.
Rewrite
Draw
State (“Given” and “Prove”)
Plan
a. Think backwards.
b. Do you need to prove things about congruent angles,
parallel lines, triangles, etc?
5. Demonstrate (Write the proof)
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We have not spent as much time on the planning
steps as we have on the other steps. Today we will
focus on that as well as writing a proof from scratch.
We will be focusing on parallelograms because they
have many properties that you know well.
a. mPMN
135o
b. mMNO
45o
c. mOPM
45o
d. MP
7
e. OP
15
f. MQ
5.5
g. NQ
10.5
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Writing a Proof
Prove: The opposite angles of a parallelogram are congruent.
Rewrite: If a quadrilateral is a parallelogram,
then its opposite angles are congruent.
Draw:
A
B
D
C
State: Given: ABCD is a parallelogram
Prove: ABC @ CDA, DAB @ BCD
Plan: If we can divide this into 2 triangles and prove
that they are congruent, then we can use
CPCTC to match up congruent angles.
How do we divide this into 2 triangles? Draw an auxiliary
line.
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B
A
Given: ABCD is a parallelogram
Prove: ABC @ CDA, BAD @ BCD
D
ABCD is a parallelogram
Draw AC
AB DC
Given
Two pts. determine a line
Def. of parallelogram
AD BC
ACD @ CAB
DAC @ BCA
AC @ AC
DABC @ DCDA
ABC @ CDA
Draw BD
ABD @ BDC
ADB @ DBC
Def. of parallelogram
Alt. Int. ‘s are @.
Alt. Int. ‘s are @.
Reflexive Property
ASA
CPCTC
Two pts. determine a line
Alt. Int. ‘s are @.
Alt. Int. ‘s are @.
BD @ BD
DBAD @ DBCD
BAD @ BCD
Reflexive Property
ASA
CPCTC
C
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Properties of Parallelograms:
1. Opposite sides of a parallelogram are parallel.
2. Opposite angles of a parallelogram are congruent.
3. Opposite sides of a parallelogram are congruent.
4. Consecutive angles of a parallelogram are supplementary.
5. Diagonals of a parallelogram bisect each other.
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If I give you 3 dots
on a coordinate grid,
how many different
parallelograms could
we make?
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Homework: Practice 6-2A
Change #12 to Prove: AB @ CD and BC @ AD
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