2.5 Proving Statements about Segments Geometry
Dec 22, 2015
Standards/Objectives:
Students will learn and apply geometric concepts.
Objectives:
• Justify statements about congruent segments.
• Write reasons for steps in a proof.
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≅
Example 1: Symmetric Property of Segment Congruence
Given: PQ ≅ XYProve XY ≅ PQ
Statements:
1. PQ ≅ XY2. PQ = XY
3. XY = PQ
4. XY ≅ PQ
Reasons:
1. Given2. Definition of congruent
segments3. Symmetric Property of
Equality4. Definition of congruent
segments
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≅
1. _______________
2. _______________
3. LK = JK
4. LK ≅ JK5. JK ≅ JL6. ________________
1. Given
2. Given
3. Transitive Property
4. _______________
5. Given
6. Transitive Property
Statements: Reasons:
Example 3: Using Segment Relationships
• GIVEN: Q is the midpoint of PR.
• PROVE: PQ = ½ PR and QR = ½ PR.
1. Q is the midpoint of PR.2. PQ = QR3. PQ + QR = PR4. PQ + PQ = PR5. 2 ∙ PQ = PR6. PQ = ½ PR7. QR = ½ PR
1. Given2. Definition of a midpoint3. Segment Addition
Postulate
4. Substitution Property5. Distributive property6. Division property7. Substitution
Statements: Reasons:
GUIDED PRACTICE for Example 1
GIVEN : AC = AB + AB
PROVE : AB = BC
1. Four steps of a proof are shown. Give the reasons for the last two steps.
1. AC = AB + AB
2. AB + BC = AC
3. AB + AB = AB + BC
4. AB = BC
1. Given
2. Segment Addition Postulate
STATEMENT REASONS
3. ?
4. ?
GUIDED PRACTICE for Example 1
GIVEN : AC = AB + AB
PROVE : AB = BC
ANSWER
1. AC = AB + AB
2. AB + BC = AC
3. AB + AB = AB + BC
4. AB = BC
1. Given
2. Segment Addition Postulate
3. Transitive Property of Equality
4. Subtraction Property of Equality
STATEMENT REASONS
Ex. Writing a proof:
Given: 2AB = AC
Prove: AB = BCA B C
Copy or draw diagrams and label given info to help develop proofs
Statements Reasons
1. 2AB = AC
2. AC = AB + BC
3. 2AB = AB + BC
4. AB = BC
1. Given
2. Segment addition postulate
3. Transitive
4. Subtraction Prop.
EXAMPLE 3 Use properties of equality
GIVEN: M is the midpoint of AB .
PROVE: a. AB = 2 AM
b.AM = AB21
STATEMENT REASONS
EXAMPLE 3
1. M is the midpoint of AB.
2. AM MB
3. AM = MB
4. AM + MB = AB
1. Given
2. Definition of midpoint
3. Definition of congruent segments
4. Segment Addition Postulate
5. AM + AM = AB 5. Substitution Property of Equality
PROVE: a. AB = 2 AM
b. AM = AB21
6. 2AM = ABa.
AM = AB217.b.
6. Addition Property
7. Division Property of Equality
EXAMPLE 1Write a two-column proof
Write a two-column proof for this situation
GIVEN:m∠1 = m∠3
PROVE:m∠EBA = m∠DBC
1.m∠1 = m∠3
2.m∠EBA = m∠3 + m∠2
3.m∠EBA = m∠1 + m∠2
1. Given
2. Angle Addition Postulate
3. Substitution Property of Equality
STATEMENT REASONS
5.m∠EBA = m∠DBC
4.m∠1 + m∠2 = m∠DBC4. Angle Addition Postulate
5. Transitive Property of Equality