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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

Standard 3: Students will learn and apply geometric concepts.

Objectives:

- Justify statements about congruent segments.
- Write reasons for steps in a proof.

- pp. 104-106 #1-16 all

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.

- 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

- 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.

- 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:

- 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

Segment Addition Postulate

Substitution Property

Distributive property

Division property

Substitution

Statements: Reasons:

- 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.

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.