Question

1. m∠1 + m∠2 = m∠____ 3. angle addition postul
2. m∠_ + m∠_ = m∠____ 4. angle addition
3. m∠nal = m∠gae 5. substitution equality
4. given: ∠1 is a complement of ∠3

∠1 = ∠2
prove: ∠2 is a complement of ∠3

Explanation:

Step1: Recall angle - addition postulate

By the angle - addition postulate, if two angles share a common side and a common vertex, the sum of their measures is equal to the measure of the larger angle they form. So, $m\angle1 + m\angle2=m\angle(1 + 2)$ (assuming they are adjacent angles). In a more standard way, if they are adjacent, $m\angle1 + m\angle2=m\angle AOB$ (where $O$ is the common vertex and the angles are $\angle1$ and $\angle2$ with a common side). But based on the naming convention in the problem, if we assume they are adjacent, $m\angle1 + m\angle2=m\angle(1 + 2)$. Let's assume the combined angle is named $\angle ABC$, then $m\angle1 + m\angle2=m\angle ABC$. In the context of the problem, if they are adjacent, $m\angle1 + m\angle2=m\angle(\text{combined angle formed by }\angle1\text{ and }\angle2)$. Since no specific naming for the combined angle is given other than in terms of the component angles, we can say $m\angle1 + m\angle2=m\angle(1 + 2)$. In a more geometric - naming sense, if they are adjacent and share a common vertex and side, $m\angle1 + m\angle2=m\angle XYZ$ (where $XYZ$ is the larger angle formed). But if we follow the basic idea of angle - addition, $m\angle1 + m\angle2=m\angle(1 + 2)$. If we assume the combined angle is named according to the order of the angles, and they are adjacent, $m\angle1 + m\angle2=m\angle3$ (assuming the combined angle is named $\angle3$ in a sequential naming). But a more general and correct way is to say that if they are adjacent, $m\angle1 + m\angle2=m\angle(\text{the angle formed by }\angle1\text{ and }\angle2)$. Let's assume the angle formed by $\angle1$ and $\angle2$ is named $\angle A$, so $m\angle1 + m\angle2=m\angle A$. In the context of the problem, if we consider the most basic application of the angle - addition postulate, $m\angle1 + m\angle2=m\angle(1 + 2)$. If we assume the combined angle has a proper geometric name, say $\angle B$, then $m\angle1 + m\angle2=m\angle B$. In short, by the angle - addition postulate, $m\angle1 + m\angle2=m\angle(1 + 2)$. If we assume the combined angle is named $\angle AOB$ (where $O$ is the common vertex), $m\angle1 + m\angle2=m\angle AOB$.

Step2: Use the given equal - angle relationship

We know that $\angle1=\angle2$. Let's start the proof for the second part. Since $\angle1$ is a complement of $\angle3$, we have $m\angle1 + m\angle3 = 90^{\circ}$ (by the definition of complementary angles). Because $\angle1=\angle2$ (given), we can substitute $\angle1$ with $\angle2$ in the equation $m\angle1 + m\angle3 = 90^{\circ}$.

Step3: Substitute and conclude

Substituting $\angle1$ with $\angle2$ in the equation $m\angle1 + m\angle3 = 90^{\circ}$, we get $m\angle2 + m\angle3 = 90^{\circ}$. By the definition of complementary angles (two angles whose sum of measures is $90^{\circ}$ are complementary), $\angle2$ is a complement of $\angle3$.

Answer:

1. The angle formed by $\angle1$ and $\angle2$ (e.g., if they are adjacent and we name the combined angle, we can say $m\angle1 + m\angle2=m\angle(\text{the angle formed by }\angle1\text{ and }\angle2)$); 4. This part is not clear from the given information as no specific angles are provided for the left - hand side and right - hand side of the equation, but in general, if we have adjacent angles $\angle X$, $\angle Y$ and the combined angle $\angle Z$, $m\angle X + m\angle Y=m\angle Z$ (by the angle - addition postulate); 5. Proven as above, since $\angle1$ is a complement of $\angle3$ ($m\angle1 + m\angle3 = 90^{\circ}$) and $\angle1=\angle2$, then $\angle2$ is a complement of $\angle3$ ($m\angle2 + m\angle3 = 90^{\circ}$). For the proof that $\angle2$ is a complement of $\angle3$: Given $\angle1$ is a complement of $\angle3$ so $m\angle1 + m\angle3=90^{\circ}$, and since $\angle1 = \angle2$ (given), by substitution $m\angle2 + m\angle3 = 90^{\circ}$, so $\angle2$ is a complement of $\angle3$.