2 5 proving statements about segments
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2.5 Proving Statements about Segments. Mrs. Spitz Geometry Fall 2004. Standards/Objectives:. Standard 3: Students will learn and apply geometric concepts. Objectives: Justify statements about congruent segments. Write reasons for steps in a proof. Assignment:. pp. 104-106 #1-16 all.

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2.5 Proving Statements about Segments

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2 5 proving statements about segments

2.5 Proving Statements about Segments

Mrs. Spitz

Geometry

Fall 2004


Standards objectives

Standards/Objectives:

Standard 3: Students will learn and apply geometric concepts.

Objectives:

  • Justify statements about congruent segments.

  • Write reasons for steps in a proof.


Assignment

Assignment:

  • pp. 104-106 #1-16 all


Definitions

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.


Note put in the definitions properties postulates theorems formulas portion of your notebook

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


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

Example 1: Symmetric Property of Segment Congruence


Paragraph proof

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.


Example 2 using congruence

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


2 5 proving statements about segments

________________

________________

LK = JK

LK ≅ JK

JK ≅ JL

________________

Given

Given

Transitive Property

_________________

Given

Transitive Property

Statements: Reasons:


2 5 proving statements about segments

LK = 5

JK = 5

LK = JK

LK ≅ JK

JK ≅ JL

LK ≅ JL

Given

Given

Transitive Property

Def. Congruent seg.

Given

Transitive Property

Statements: Reasons:


Example 3 using segment relationships

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.


2 5 proving statements about segments

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

Statements: Reasons:


Pg 104 activity copy a segment

Pg. 104 – Activity—Copy 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.


Plan for next week

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.


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