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# 2.5 Proving Statements about Segments - PowerPoint PPT Presentation

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

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
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
• 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
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
• 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
• 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
________________

________________

LK = JK

LK ≅ JK

JK ≅ JL

________________

Given

Given

Transitive Property

_________________

Given

Transitive Property

Statements: Reasons:

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
• 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.
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

Substitution Property

Distributive property

Division property

Substitution

Statements: Reasons:

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

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.