Let's break down the proofs in the image one by one.

Problem 29:

Given: ∠1 and ∠2 are complementary; ∠1 ≅ ∠4
Prove: ∠3 and ∠4 are complementary

Proof structure:

1. Given: ∠1 and ∠2 are complementary.
   • Reason: Given.
2. m∠1 + m∠2 = 90°
   • Reason: Definition of complementary angles.
3. ∠1 ≅ ∠4
   • Reason: Given.
4. m∠1 = m∠4
   • Reason: Definition of congruent angles.
5. ∠2 ≅ ∠3
   • Reason: Vertical angles are congruent.
6. m∠2 = m∠3
   • Reason: Definition of congruent angles.
7. m∠4 + m∠3 = 90°
   • Reason: Substitution (replace m∠1 with m∠4 and m∠2 with m∠3 in step 2).
8. ∠3 and ∠4 are complementary.
   • Reason: Definition of complementary angles.

Problem 30:

Given: ∠JKM and ∠MKL form a linear pair; ∠JKM and ∠LKN are supplementary.
Prove: ∠MKL ≅ ∠LKN

Proof structure:

1. ∠JKM and ∠MKL form a linear pair.
   • Reason: Given.
2. ∠JKM and ∠MKL are supplementary.
   • Reason: Linear pairs are supplementary.
3. ∠JKM and ∠LKN are supplementary.
   • Reason: Given.
4. ∠MKL ≅ ∠LKN.
   • Reason: Angles supplementary to the same angle (∠JKM) are congruent.

Would you like further details or have any questions about the process? Here are some follow-up questions:

1. What does it mean for two angles to be congruent?
2. Why are vertical angles always congruent?
3. Can supplementary angles ever be congruent?
4. How does the definition of linear pairs help prove that angles are supplementary?
5. What role does substitution play in solving angle proofs?

Tip: When dealing with angle proofs, always keep in mind key properties such as congruence and the definitions of complementary, supplementary, and vertical angles.