The document provides instructions and examples for proving geometric theorems through deductive reasoning using two-column geometric proofs. It defines geometric proofs as proofs which use geometric properties and definitions. It explains that a two-column geometric proof begins with the given statement(s) and ends with the statement to be proven, with steps listed in the left column and justifications in the right column using definitions, postulates, and previously proven theorems. The document also provides examples of completed two-column geometric proofs.

1. Obj. 11 Geometric Proof
Objective
The student will be able to (I can):
• Prove geometric theorems by deductive reasoning

2. geometric
proof
A proof which uses geometric properties
and definitions
• A two-column geometric proof begins
with the GivenGivenGivenGiven statement and ends with
the ProveProveProveProve statement.
• List the steps of the proof in the left
column and the justifications (reasons)
in the right column.
• You may use definitions, postulates, and
previously proven theorems as reasons.

3. Line segments with equal lengths are
congruent, and angles with equal measures
are also congruent. Therefore, the reflexive,
symmetric, and transitive properties of
equality have corresponding properties ofproperties ofproperties ofproperties of
congruencecongruencecongruencecongruence.
Reflexive Property of Congruence
fig. A ≅ fig. A
Symmetric Property of Congruence
If fig. A ≅ fig. B, then fig. B ≅ fig. A.
Transitive Property of Congruence
If fig. A ≅ fig. B and fig. B ≅ fig. C,
then fig. A ≅ fig. C.

4. Given:Given:Given:Given: ∠BAC is a right angle;
∠2 ≅ ∠3
Prove:Prove:Prove:Prove: ∠1 and ∠3 are comp.
1
2
3
•
•
B
A C
StatementsStatementsStatementsStatements ReasonsReasonsReasonsReasons
1. ∠BAC is a right angle 1. Given
2. m∠BAC = 90° 2. _______________
3. _______________________ 3. ∠ Add. post.
4. m∠1 + m∠2 = 90° 4. Subst. prop. =
5. ∠2 ≅ ∠3 5. Given
6. _______________________ 6. Def. ≅ ∠s
7. m∠1 + m∠3 = 90° 7. _______________
8. _______________________ 8. Def. comp. ∠s
Def. rightDef. rightDef. rightDef. right ∠∠∠∠
mmmm∠∠∠∠1 + m1 + m1 + m1 + m∠∠∠∠2 = m2 = m2 = m2 = m∠∠∠∠BACBACBACBAC
Subst. prop. =Subst. prop. =Subst. prop. =Subst. prop. =
mmmm∠∠∠∠2 = m2 = m2 = m2 = m∠∠∠∠3333
∠∠∠∠1 and1 and1 and1 and ∠∠∠∠3 are comp.3 are comp.3 are comp.3 are comp.

5. Example: GivenGivenGivenGiven ∠1 and ∠2 are supplementary, and
∠2 and ∠3 are supplementary
ProveProveProveProve ∠1 ≅ ∠3
1. ∠1 and ∠2 are supp. 1. Given
∠2 and ∠3 are supp.
2. m∠1 + m∠2 = 180° 2. Def. supp. ∠
m∠2 + m∠3 = 180°
3. 180° = m∠2 + m∠3 3. Sym. prop =
4. m∠1+m∠2=m∠2+m∠3 4. Trans. prop =
5. m∠1 = m∠3 5. Subtr. prop.=
6. ∠1 ≅ ∠3 6. Def. ≅ ∠s

6. Given:Given:Given:Given: ∠2 ≅ ∠3
Prove:Prove:Prove:Prove: ∠1 and ∠3 are supplementary
1. ∠2 ≅ ∠3 1. Given
2. ∠1 and ∠2 are supp. 2. Def. linear pair
3. m∠1 + m∠2 = 180° 3. Def. supp. ∠s
4. m∠2 = m∠3 4. Def. ≅ ∠s
5. m∠1 + m∠3 = 180° 5. Subst. prop. =
6. ∠1 and ∠3 are supp. 6. Def. supp. ∠s
1 2 3