Algebraic properties of equality are used in

Geometry.–Will help you solve problems and justify each step.

In Geometry, you accept postulates and properties as true.

–Some of the properties you accept as true are the properties of equality

from Algebra.

Properties of EqualityLet a, b, and c be any real numbers.

Addition Property: If a = b, then a + c = b + c.

Subtraction Property: If a = b, then a - c = b - c.

Multiplication Property: If a = b, then a * c = b * c.

Division Property: If a = b and c ≠ 0, then a/c = b/c.

Reflexive Property: a = a.

Symmetric Property: If a = b, then b = a.

Transitive Property: If a = b and b = c, then a = c.

Substitution Property: If a = b, then b can replace a in any expression.

Distributive PropertyUse multiplication to distribute a to each term of the sum or difference

within the parentheses.

Sum: a (b + c) = ab + ac

Difference: a (b – c) = ab – ac

Properties of CongruenceThe following are the properties of congruence. Some

textbooks list just a few of them, others list them all. These are

analogous to the properties of equality for real numbers. Here

we show congruence of angles, but the properties apply just as

well for congruent segments, triangles, or any other geometric

object.

Writing Two Column Proofs

A two column proof is a proof in which has to be written using two-columns,

obviously. In one column you have to have a statement and in the other

column you have to have a reason. This is the structure you use in order to

do a 2 column proof.

What I mean by information given, is that they will give you a "given"

statement and a "prove" statement.

Given is what you are starting with and what your first statement be. Prove

is what you have to prove throughout the proof, this should be the last part

of the 2 column-proof.

Statement: Is the problem you conclude from the proof. Is what

you have to give a name to. It's the what part of the proof.

Reason: Is the theorem or postulate you give in order to give a

name for the statement. It's the why part of the proof.

You write a 2 column-proof by drawing 2 columns. The first column

with a statement and the other with a reason. This is the structure you

have to follow in order to draw a nice 2 column proof. You have to

name the theorems and the postulates to give a reason.

EXAMPLES:Given: <1 congruence <4

Prove: <2 congruence <3

Statements: Reason:

1. <1 congruence <4 1. Given

2. <1 congruence <2 and <3 congruence <4 2. Vert. <s theorem.

3. <2 congruence <4 3. Transitive.

Property of congruence

4. <2 congruence <3 4. Transitive.

Property of congruenceGiven: <LXN is a right angle

Prove: <1 and <2 are complementary

Statement: Reason:

1. <LXN= 90 degrees 1. Given

2. m<LXN=90 2. Def. of right angles

3. m<1 + m<2=m<LXN 3. Angle Addition Postulate.

(AAP)

4.) m<1 + m<2=90 4. Substitution

5. <1 and <2= complementary 5. definition of complementary

Given: BD bisects

Prove: 2m<1 = m<ABC

Statement: Reason:

1. BD bisects <ABC 1. Given

2. <1 cong. <2 2. Def. Bisect

3. m<1+m<2=m<ABC 3. Angle Addition Bisects

4. m<1 cong. m<2 4. Def. of Congruent

5. m<1 + m<1= m<ABC 5. Substitution

6. 2m<1=m<ABC 6. Simplify

6. 2 m<1 = m<ABC

Given : <1 and <2 form a linear pair

Prove: <1 and <2 are supplementary

1. <1 and <2 form a linear pair. 1. Given

2. -> BA and -> BC form a line. 2. Def. of. linear pair

3. m<ABC = 180* 3. Def. of straight angle

4. m<AB + m<BC = m<ABC 4. Angle addition postulate

5. <1 + <2 = 180* 5. Substitution

6. <1 and <2 are supplementary 6. Def. of supplementary

Given: m<LAN = 30*, m<1 = 15*

Prove: -> AM bisects <LAN

1) m<LAN = 30*, m<1 = 15* 1. Given

2) m<1 + m<2 = m<LAN 2. Angle addition postulate

3) m<1 + m<2 = 30*, 15* + m<2 = 30* 3. Substitution

4) m<2 = 15* 4. Subtraction

5) m<2 = m<1 5. Transitive

6) m<2 =~ m<1 6. Def. of Congruence

7) AM bisects <LAN 7. Def. of bisect

Given: <2 =~ <3

Prove: <1 and <3 are supplementary

1) <2 =~ <3 1. Given

2) m<2 = m<3 2. Congruent supp. theorem

3) <1 and <2 form a linear pair 3. Linear pair theorem

4) m<1 + m<2 = 180* 4. Def. of a supp. angle

5) m<1 + m<3 = 180* 5. Def. of. supplementary

6) <1 and <3 are supplementary 6. Def. of. supplementary

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