4.5 Segment and Angle Proofs. Basic geometry symbols you need to know Word(s)SymbolDefinition Point...

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Transcript of 4.5 Segment and Angle Proofs. Basic geometry symbols you need to know Word(s)SymbolDefinition Point...
4.5 Segment and Angle Proofs
Basic geometry symbols you need to knowWord(s) Symbol Definition
Point A
Line AB
Line Segment AB
Ray
Angle ABC
Measure of angle ABC
Congruent
Supplementary Angles
Complementary Angles
Congruent Angles
Substitution properties If you prove 2 parts are congruent, they can substitute
Reflexive properties The same part is equal to the same part. AB=AB
Transitive properties If part 1 = part 2, and part 2 = part 3, then 1 = 3
Symmetric properties If A = B then B = A.
Vocabulary Proof – a logical argument that shows a
statement is true
Two – column proof – numbered statements in one column, corresponding reason in other
Statement Reasons
Postulate – a rule that is accepted without proof. Write forwards and backwards – reverse it and it‘s still true
Segment Addition Postulate
if B is between A and C, then AB + BC = AC
converse….. Reverse it….
 if AB + BC = AC, then B is between A and C
Draw it….
Angle Addition Postulate –  If P is the interior (inside) of <RST then the measure
of < RST is equal to the sum of the measures of <RSP and <PST.
Draw it 
Theorem – a statement that can be proven Theorem 4.1 – Congruence of Segments Reflexive –
Symmetric –
Transitive –
Theorem 4.2 – Congruence of Angles Reflexive
Symmetric
Transitive
EXAMPLE 1 Write a twocolumn proof
Write a twocolumn proof for the situation in Example 4 from Lesson 2.5.
GIVEN:m∠1 = m∠3
PROVE:m∠EBA = m∠DBC
1.m∠1 = m∠32.m∠EBA = m∠3 + m∠23.m∠EBA = m∠1 + m∠2
1. Given2. Angle Addition Postulate
3. Substitution Property of Equality
STATEMENT REASONS
4.m∠1 + m∠2 = m∠DBC4. Angle Addition Postulate
5.m∠EBA = m∠DBC 5. Transitive Property of Equality
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
EXAMPLE 2 Name the property shown
Name the property illustrated by the statement.
a. If R T and T P, then R P.
b. If NK BD , then BD NK .
SOLUTION
Transitive Property of Angle Congruencea.
b. Symmetric Property of Segment Congruence
GUIDED PRACTICE for Example 2
2. CD CD
3. If Q V, then V Q.
Reflexive Property of Congruence
ANSWER
Symmetric Property of Congruence
ANSWER
Name the property illustrated by the statement.
Solving for x. Based on the properties learned, if you know 2
“parts” are congruent, you set them equal to each other and solve.
m<A=2x+15, m<B=4x3 2x+15=4x3 15=2x3 18=2x 9=x 2(9)+15=18+15=33 A B
A B∠ ∠
EXAMPLE 3 Use properties of equality
Prove this property of midpoints: If you know that M is the midpoint of AB ,prove that AB is two times AM and AM is one half of AB.
GIVEN: M is the midpoint of AB .
PROVE: a. AB = 2 AM
b.AM = AB21
STATEMENT REASONS
EXAMPLE 3 Use properties of equality
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
6. 2AM = ABa.
AM = AB217.b.
6. Distributive Property
7. Division Property of Equality
EXAMPLE 4 Solve a multistep problem
GIVEN: B is the midpoint of AC .C is the midpoint of BD .
PROVE: AB = CD
STATEMENT REASONS
1. B is the midpoint of AC .C is the midpoint of BD .
1. Given
2. Definition of midpoint2. AB BC
3. BC CD 3. Definition of midpoint
5. AB = CD
4. AB CD 4. Transitive Property of Congruence
5. Definition of congruent segments