whats the proof to this?

Question

Given:  ∠5 and ∠6 are complementary; ∠7 and ∠8 are complementary; ∠5 ≅ ∠7

 

Prove:  ∠6 ≅ ∠8

Answer 1

1.  ∠5 and ∠6 are complementary;     Given

 ∠7 and ∠8 are complementary 

 

2.  m∠5 + m∠6 = 90;                        Def. of complementary ∠s

m∠7 + m∠8 = 90

 

3.  m∠5 + m∠6 = m∠7 + m∠8            Transitive prop. of equality

 

4.  ∠5 ≅ ∠7                                       Given

 

5.  m∠5 = m∠7                                  Def. congruent ∠s

 

6. m∠7 + m∠6 = m∠7 + m∠8              Substitution prop. of equality

 

7. m∠7 = m∠7                                   Reflexive

 

8. m∠6 = m∠8                                   Subtraction prop. of equality

 

9. ∠6 ≅ ∠8                                         Def. of congruent ∠s

Answer 2

Since they are complementary:

 

∠5 + ∠6 = 90^o

So ∠6 = 90^o - ∠5

 

∠7 + ∠8 = 90^o

So ∠8 = 90^o - ∠7

 

Since ∠5 = ∠7:

 

∠8 = 90^o - ∠7 = 90^o - ∠5

Since ∠6 = 90^o - ∠5, we have:

 

∠6 = ∠8