Question

topic 9: angle proofs
given: ∠1 and ∠2 are complementary; ∠3 and ∠4 are complementary
prove: ∠3 and ∠4 are complementary
statements

1. ∠1 and ∠2 are complementary
2. m∠1 + m∠2 = 90°
3. ∠1 ≅ ∠4
4. m∠1 = m∠4
5. ∠2 ≅ ∠3
6. m∠2 = m∠3
7. m∠4 + m∠3 = 90°
8. ∠3 and ∠4 are complementary

reasons
1.
2.
3.
4.
5.
6.
7.
8.
know the following:
definition of complementary angles
definition of supplementary angles
definition of angle bisector
definition of congruence
definition of perpendicular
definition of right angle
angle addition postulate
vertical angles theorem
complement theorem
congruent complements theorem
congruent supplements theorem
linear pair (supplementary) theorem

Explanation:

Step1: Given information

Given that $\angle1$ and $\angle2$ are complementary, so $m\angle1 + m\angle2=90^{\circ}$ (by the definition of complementary angles).

Step2: Given equality

Given $\angle1\cong\angle4$, which means $m\angle1 = m\angle4$ (by the definition of congruent angles).

Step3: Substitution

Substitute $m\angle1$ with $m\angle4$ in $m\angle1 + m\angle2 = 90^{\circ}$, we get $m\angle4 + m\angle2=90^{\circ}$.

Step4: Given equality

Given $\angle2\cong\angle3$, so $m\angle2 = m\angle3$ (by the definition of congruent angles).

Step5: Substitution

Substitute $m\angle2$ with $m\angle3$ in $m\angle4 + m\angle2 = 90^{\circ}$, we get $m\angle4 + m\angle3=90^{\circ}$.

Step6: Definition of complementary

Since $m\angle4 + m\angle3 = 90^{\circ}$, $\angle3$ and $\angle4$ are complementary (by the definition of complementary angles).

Answer:

The proof is completed as above, showing that if $\angle1$ and $\angle2$ are complementary, $\angle1\cong\angle4$ and $\angle2\cong\angle3$, then $\angle3$ and $\angle4$ are complementary.