Simple Geometric Proofs

Given: m∠LAN = 30⁰ , m∠1 = 15⁰
Prove: AM bisects ∠LAN
Step 1: m∠LAN = 30⁰ 
Step 2: m∠1 + m∠2 = m∠LAN
Step 3: m∠1 + m∠2 = 30⁰
Step 4: m∠1 = 15⁰
Step 5: 15⁰ + m∠2 = 30⁰
Step 6: m∠2 = 15⁰
Step 7: m∠1 = m∠2
Step 8: ∠1 ≅ ∠2
Step 9: AM bisects ∠LAN
What is the justification for Step 9?

• Symmetric property of congruence

• Definition of angle bisector

• Addition property of equality

• Definition of congruent angles

Given: AC = AB + AB
Prove: AB = BC
Reason 1: Given
Reason 2: Segment addition postulate
What is Step 2?

• AC = AB + AB

• AB = BC

• AB + BC = AC

• AB + AB = AB + BC

Given: AC = AB + AB
Prove: AB = BC
Reason 1: Given
Reason 2: Segment addition postulate
Reason 3: Transitive property of equality
What is Step 3?

• AC = AB + AB

• AB = BC

• AB + BC = AC

• AB + AB = AB + BC

Given: ∠1 is a complement of ∠2  and  ∠2 ≅ ∠3
Prove: ∠1 is a complement of ∠3
Reason 1: Given
Reason 2: Given
What is Step 2?

• ∠2 ≅ ∠3

• m∠1 + m∠2 = 90⁰

• m∠2 = m∠3

• m∠1 + m∠3 = 90⁰

Given: ∠1 is a complement of ∠2  and  ∠2 ≅ ∠3
Prove: ∠1 is a complement of ∠3
Reason 1: Given
Reason 2: Given
Reason 3: Definition of complementary angles
Reason 4: Definition of congruent angles
What is Step 4?

• ∠2 ≅ ∠3

• m∠1 + m∠2 = 90⁰

• m∠2 = m∠3

• m∠1 + m∠3 = 90⁰

Given: ∠1 is a complement of ∠2  and  ∠2 ≅ ∠3
Prove: ∠1 is a complement of ∠3
Reason 1: Given
Reason 2: Given
Reason 3: Definition of complementary angles
Reason 4: Definition of congruent angles
Reason 5: Substitution property of equality
Reason 6: Definition of complement
What is Step 6?

• ∠1 is a complement of ∠3

• m∠1 + m∠2 = 90⁰

• m∠2 = m∠3

• m∠1 + m∠3 = 90⁰

Given: m∠LAN = 30⁰ , m∠1 = 15⁰
Prove: AM bisects ∠LAN
Step 1: m∠LAN = 30⁰ 
What is the justification for Step 1?

• Given

• Substitution

• Angle addition postulate

• Definition of congruent angles

Given: m∠LAN = 30⁰ , m∠1 = 15⁰
Prove: AM bisects ∠LAN
Step 1: m∠LAN = 30⁰ 
Step 2: m∠1 + m∠2 = m∠LAN
What is the justification for Step 2?

• Given

• Substitution

• Angle addition postulate

• Definition of congruent angles

Given: m∠LAN = 30⁰ , m∠1 = 15⁰
Prove: AM bisects ∠LAN
Step 1: m∠LAN = 30⁰ 
Step 2: m∠1 + m∠2 = m∠LAN
Step 3: m∠1 + m∠2 = 30⁰
What is the justification for Step 3?

• Given

• Substitution

• Angle addition postulate

• Definition of congruent angles

Given: m∠LAN = 30⁰ , m∠1 = 15⁰
Prove: AM bisects ∠LAN
Step 1: m∠LAN = 30⁰ 
Step 2: m∠1 + m∠2 = m∠LAN
Step 3: m∠1 + m∠2 = 30⁰
Step 4: m∠1 = 15⁰
What is the justification for Step 4?

• Given

• Substitution

• Angle addition postulate

• Definition of congruent angles