2'5 Proving Statements about Segments

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2.5 Proving Statements about Segments
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Goals

• Goal 1
• Justify statements about congruent segments.
• Goal 2
• Write reasons for steps in a proof.

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Theorem

• A true statement that follows as result of other
  true statements is called a theorem.
• You can prove a theorem using a two-column proof.
• A two-column proof has numbered statements and
  reasons that show the logical order of an
  argument.

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Theorem 2.1 Properties of Segment Congruence

• Reflexive
• For any segment AB, AB ? AB.
• Symmetric
• If AB ? CD, then CD ? AB.
• Transitive
• If AB ? CD, and CD ? EF, then AB ? EF.

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Symmetric Property of Segment Congruence

• You can prove the Symmetric Property of Segment
  Congruence as follows
• Given PQ ? XY
• Prove XY ? PQ

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• Statements
• PQ ? XY
• PQ XY
• XY PQ
• XY ? PQ

• Reasons
• Given
• Definition of Congruent Segments
• Symmetric Property of Equality
• Definition of Congruent Segments

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Paragraph Proof

• A proof can be written in paragraph form, called
  paragraph proof.
• Paragraph Proof for the Symmetric Property of
  Segment Congruence.
• Paragraph Proof- You are given that PQ ? XY. By
  the definition of congruent segments, PQ XY.
  By the symmetric property of equality, XY PQ.
  Therefore, by definition of congruent segments,
  it follows that XY ? PQ.

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Using Congruence of Segments

• Use the diagram and the given information to
  complete the missing steps and reasons in the
  proof.
• Given LK 5, JK 5, JK ? JL
• Prove LK ? JL

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Two-Column Proof

• Statements
• __________
• __________
• LK JK
• LK ? JK
• JK ? JL
• __________

• Reasons
• Given
• Given
• Transitive property of equality
• ____________
• Given
• Transitive Property of Congruence

LK 5
JK 5
Definition of congruent segments
LK ? JL
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Using Segment Relationships

• In the diagram, Q is the midpoint of PR. Show
  that PQ and QR are each equal to ½ PR.
• Given Q is the midpoint of PR.
• Prove PQ ½ PR and QR ½PR

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Two Column Proof

• Statement

• Reasons

Q is the midpoint of PR
Given
PQ QR
Definition of midpoint
Segment Addition Postulate
PQ QR PR
PQ PQ PR
Substitution Property of Equality
2 x PQ PR
Distributive Property
PQ 1/2PR
Division property of Equality
QR 1/2PR
Substitution Property of Equality
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Homework

• Page 105-107
• 6-11, 16-26