Transcript 2.5 - Postulates and Paragraph Proofs

Postulates and
Paragraph Proofs
Honors Geometry
Chapter 2, Section 5
Notes
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Postulate: a statement that is accepted to be true and describes a
fundamental relationship between the basic terms of geometry.
P2.1 Through any 2 points, there is exactly one line.
P2.2 Through any 3 points not on the same line, there is exactly
one plane.
P2.3 A line contains at least 2 points
P2.4 A plane contains at least 3 points NOT on the same line
P2.5 If two points lie in a plane, then the entire line containing those
points lies in that plane.
P2.6 If two lines intersect, then their intersection is exactly one point
P2.7 If two planes intersect, then their intersection is a line.
Always, Sometimes, Never
Practice!
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If points A, B, & C lie in plane M, then they
are collinear.
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There is exactly one plane that contains
noncollinear points P,Q, and R.
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There are at least two lines through points M
and N.
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Notes
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Theorem: a statement or conjecture that has been
proven true using logic.
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Theorems can be used just like postulates and definitions to
prove other statements.
Proof: a logical argument in which each step
towards the conclusion is supported by a definition,
postulate, or theorem i.e. supporting statements.
Paragraph Proof: a type of proof that gives the
steps and supporting statements in paragraph form
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Informal Proof: paragraph proofs are a type of informal
proof
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Example:
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Recall:
 The definition of a midpoint is the point half way between
the endpoints of a segment.
 AB means the measure or length of the segment AB
 while AB refers to the segment with endpoints A and B
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Given: M is the mid point of
Prove: PM  MQ
PQ
Q
M
P
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Proof: From the definition of midpoint of a segment, PM =
MQ. i.e. the measures of the segments are equal. This
means that PM and MQ have the same measure. By
definition of congruence, if two segments have the same
measure, then they are congruent. Thus, PM  MQ .
Notes
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The proof we just did was the proof of the Midpoint
Theorem. We can now use this theorem in other proofs.
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i.e. If M is the midpoint of segment PQ, then segment PM is
congruent to segment MQ
Steps to writing a good proof:
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List the given information.
State what is to be proven.
Draw a diagram of the given information.
Develop an argument that properly uses deductive
reasoning.
 Each step follows from the previous
 Each step is supported by accepted facts.
On Your Own:
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In the figure at the right, P is the midpoint of QR and ST ,
and QR  ST .
Write a paragraph proof to show that PQ = PT
 Given:
Q
 P is the midpoint of QR and ST
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and QR  ST
P
Prove: PQ = PT
S
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T
R
From the definition of midpoint of a segment, QP = PR. From the
segment addition postulate, QR = QP + PR. Substituting QP for
PR, QR = QP + QP. Simplifying gives QR = 2QP. Likewise, ST =
2PT. From the definition of congruent segments, QR = ST from
the given information. Substituting 2QP for QR and 2PT for ST
gives 2QP = 2PT. Dividing both sides by 2 gives QP = PT. Since
QP is another way of naming the segment PQ, we can rewrite
the last equation as PQ = PT and reach our desired conclusion.