2'5 Proving Statements about Segments

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2.5 Proving Statements about Segments

• Mrs. Spitz
• Geometry
• Fall 2004

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Standards/Objectives

• Standard 3 Students will learn and apply
  geometric concepts.
• Objectives
• Justify statements about congruent segments.
• Write reasons for steps in a proof.

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Assignment

• pp. 104-106 1-16 all

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Definitions

• Theorem
• A true statement that follows as a result of
  other true statements.
• Two-column proof
• Most commonly used. Has numbered statements and
  reasons that show the logical order of an
  argument.

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NOTE Put in the Definitions/Properties/
Postulates/Theorems/Formulas portion of your
notebook

• Theorem 2.1
• Segment congruence is reflexive, symmetric, and
  transitive.
• Examples
• 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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Example 1 Symmetric Property of Segment
Congruence

• Given PQ ? XY
• Prove XY ? PQ
• 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. It is
  as follows
• You are given that PQ ? to XY. By the definition
  of congruent segments, PQ XY. By the symmetric
  property of equality, XY PQ. Therefore, by the
  definition of congruent segments, it follows that
  XY ? PQ.

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Example 2 Using Congruence

• 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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Statements Reasons

• ________________
• ________________
• LK JK
• LK ? JK
• JK ? JL
• ________________

• Given
• Given
• Transitive Property
• _________________
• Given
• Transitive Property

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Statements Reasons

• LK 5
• JK 5
• LK JK
• LK ? JK
• JK ? JL
• LK ? JL

• Given
• Given
• Transitive Property
• Def. Congruent seg.
• Given
• Transitive Property

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Example 3 Using Segment Relationships

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

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Statements Reasons

• Q is the midpoint of PR.
• PQ QR
• PQ QR PR
• PQ PQ PR
• 2 PQ PR
• PQ ½ PR
• QR ½ PR

• Given
• Definition of a midpoint
• Segment Addition Postulate
• Substitution Property
• Distributive property
• Division property
• Substitution

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Pg. 104 ActivityCopy a segment

• Use the following steps to construct a segment
  that is congruent to AB.
• Use a straightedge to draw a segment longer than
  AB. Label the point C on the new segment.
• Set your compass at the length of AB.
• Place the compass point at C and mark a second
  point D on the new segment. CD is congruent to
  AB.

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Place this in your Lab Work section.

• HOMEWORK ASSIGNMENT
• 2.5 pp. 104-106 1-18 all
• Test next week either Thursday or Friday
  depending on which class period you are in.

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Plan for next week

• Monday 2.6 Notes/ HW due following class
  meeting.
• Tues/Wed Review Chapter 2 Due at the end of
  class period
• Thur/Fri Chapter 2 Exam Chapter 3 definitions
  pg. 128. Copy the following into your
  definitions/postulates/theorems portion of your
  binder pg. 130, 137, 138, 143, 150, 157, 166,
  172 Look for the green boxes.