1. This is a postulate, not a theorem
congruent
(This was Euclid's 4th postulate)
Given
∠K and ∠L are right angles
Right Angles Congruence Postulate

2. congruent
∠1 ≅ ∠3
congruent
∠4 ≅ ∠6

3. Given
∠2 ≅ ∠4
Definition of congruent angles
Given
m∠4 = 45°

4. No, you cannot prove that ∠K and ∠L are right angles because the converse of the Right Angles Congruence Postulate is not always true.
No, ∠B and ∠C are complements by the Congruent Complements Theorem, so they cannot be supplements.

5. supplementary
180°
congruent

6. Given
m∠4 = 90°
Vertical Angles Congruence Theorem
m∠2 = m∠4
Substitution Property of Equality
∠2 and ∠4 are supplementary

7. m∠1 + m∠4 = 180°
m∠1 + 63° = 180°
m∠1 = 117°
m∠2 = m∠4
m∠2 = 63°
m∠1 = m∠3
m∠1 = 121°
m∠2 + m∠3 = 180°
m∠ ° = 180°
m∠2 = 59°
m∠2 = m∠4
59° = m∠4

8. 180°
180° 17 17 68
67°

9. m∠AEB = m∠DEC
4x - 18 = 3x + 4
x - 18 = 4
x = 22
m∠AEB = 4x - 18
m∠AEB = 4(22) - 18
m∠AEB = 70°