Let's go through the proof step-by-step to understand where the justification that angles with a combined degree measure of \(90^{\circ}\) are complementary is used.
1. Statement 1: \( m \angle 1 = 40^{\circ} \)
- Reason 1: This information is given directly in the problem.
2. Statement 2: \( m \angle 2 = 50^{\circ} \)
- Reason 2: This information is also given directly in the problem.
3. Statement 3: \( \angle 1 \) is complementary to \( \angle 2 \)
- Reason 3: According to the definition of complementary angles, two angles are complementary if the sum of their degree measures is \( 90^{\circ} \). Given \( m \angle 1 = 40^{\circ} \) and \( m \angle 2 = 50^{\circ} \), their sum is \( 40^{\circ} + 50^{\circ} = 90^{\circ} \). Therefore, \( \angle 1 \) and \( \angle 2 \) are complementary. This step uses the definition of complementary angles directly, so it is where the justification is applied.
4. Statement 4: \( \angle 2 \) is complementary to \( \angle 3 \)
- Reason 4: This information is given directly in the problem.
5. Statement 5: \( \angle 1 \cong \angle 3 \)
- Reason 5: According to the congruent complements theorem, if two angles are complementary to the same angle (in this case, \( \angle 2 \)), then those two angles are congruent. Since \( \angle 1 \) and \( \angle 3 \) are both complementary to \( \angle 2 \), they are congruent to each other.
So, the part of the proof that directly uses the justification that angles with a combined degree measure of [tex]\( 90^{\circ} \)[/tex] are complementary is Statement 3.
