Simple Geometric Proofs

Are the figures congruent? (What type of transformation is shown?)

• NOT Congruent (dilation)

• NOT Congruent (translation)

• Congruent (dilation)

• Congruent (translation)

Which statement is true about the transformation?

• The transformation preserves congruency and maintains orientation.

• The transformation preserves congruency and does not maintains orientation.

• The transformation does not preserves congruency but maintains orientation.

• The transformation does not preserves congruency or maintains orientation.

Are the figures congruent?

(What type of transformation is shown?)

• Congruent (rotation)

• Congruent (reflection)

• NOT Congruent (rotation)

• NOT Congruent (reflection)

Are the figures congruent?

(What type of transformation is shown?)

• Congruent (rotation)

• Congruent (reflection)

• NOT Congruent (rotation)

• NOT Congruent (reflection)

Given: AC = AB + AB
Prove: AB = BC
Reason 1: Given
Reason 2: Segment addition postulate
Reason 3: Transitive property of equality
Reason 4: Subtraction property of equality
What is Step 4?

• 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
What is Step 1?

• ∠1 is a complement of ∠2

• 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
What is Step 3?

• ∠1 is a complement of ∠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
What is Step 5?

• ∠2 ≅ ∠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 
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
What is the justification for Step 6?

• Subtraction property of equality

• Substitution

• Addition property of equality

• 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
Step 5: 15⁰ + m∠2 = 30⁰
Step 6: m∠2 = 15⁰
Step 7: m∠1 = m∠2
Step 8: ∠1 ≅ ∠2
What is the justification for Step 8?

• Symmetric property of congruence

• Transitive property of equality

• Addition property of equality

• Definition of congruent angles