Question

1. write a two - column proof.

given: \\( \angle 1 \cong \angle 2, m \angle 4 = 30 ^ { \circ } \\)
prove: \\( m \angle 2 = 30 ^ { \circ } \\)

Explanation:

Step1: Identify Vertical Angles

Vertical angles are equal. $\angle 1$ and $\angle 4$ are vertical angles? Wait, no, $\angle 2$ and $\angle 4$? Wait, let's look at the diagram. The straight line and the intersecting lines. Wait, $\angle 1$ and $\angle 2$ are given congruent. Also, $\angle 1$ and $\angle 4$? Wait, no, let's recall that $\angle 1$ and $\angle 4$: Wait, actually, $\angle 1$ and $\angle 4$? Wait, no, the vertical angles: $\angle 1$ and $\angle 4$? Wait, no, let's see. The angles around the point: $\angle 1$, $\angle 2$, $\angle 3$, $\angle 4$, $\angle 5$, $\angle 6$. The horizontal line is straight, so $\angle 1 + \angle 2 + \angle 3 = 180^\circ$? Wait, no, the vertical line? Wait, no, the diagram has two intersecting lines: one horizontal, one at an angle, and another? Wait, the given is $\angle 1 \cong \angle 2$, and $m\angle 4 = 30^\circ$. Also, $\angle 1$ and $\angle 4$: Wait, maybe $\angle 1$ and $\angle 4$ are vertical angles? Wait, no, $\angle 2$ and $\angle 4$? Wait, let's correct. Actually, $\angle 1$ and $\angle 4$: Wait, no, the vertical angles: $\angle 1$ and $\angle 4$? Wait, no, looking at the diagram, $\angle 1$ and $\angle 4$: Wait, maybe $\angle 1$ and $\angle 4$ are equal because they are vertical angles? Wait, no, let's start over.

Given: $\angle 1 \cong \angle 2$, $m\angle 4 = 30^\circ$. We need to prove $m\angle 2 = 30^\circ$.

First, $\angle 1$ and $\angle 4$: Wait, no, $\angle 1$ and $\angle 4$ are vertical angles? Wait, no, $\angle 1$ and $\angle 4$: Wait, in the diagram, the angle $\angle 1$ and $\angle 4$: Wait, maybe $\angle 1$ and $\angle 4$ are equal because they are vertical angles? Wait, no, let's see the lines. The horizontal line and the other lines. Wait, $\angle 1$ and $\angle 4$: Wait, actually, $\angle 1$ and $\angle 4$ are vertical angles? Wait, no, $\angle 2$ and $\angle 4$? Wait, maybe I made a mistake. Wait, the key is that $\angle 1$ and $\angle 4$ are vertical angles? Wait, no, let's check the given. $\angle 1 \cong \angle 2$, and $m\angle 4 = 30^\circ$. Also, $\angle 1$ and $\angle 4$: Wait, maybe $\angle 1$ and $\angle 4$ are equal because they are vertical angles. Wait, vertical angles are congruent. So if $\angle 1$ and $\angle 4$ are vertical angles, then $\angle 1 \cong \angle 4$. Then, since $\angle 1 \cong \angle 2$, by transitive property, $\angle 2 \cong \angle 4$, so $m\angle 2 = m\angle 4 = 30^\circ$.

Let's structure the two - column proof:

Statements	Reasons
2. $\angle 1 \cong \angle 4$	Vertical Angles Theorem (Vertical angles are congruent)
3. $\angle 2 \cong \angle 4$	Transitive Property of Congruence (If $\angle 1 \cong \angle 2$ and $\angle 1 \cong \angle 4$, then $\angle 2 \cong \angle 4$)
4. $m\angle 4 = 30^\circ$	Given
5. $m\angle 2 = m\angle 4$	Definition of Congruent Angles (If two angles are congruent, their measures are equal)
6. $m\angle 2 = 30^\circ$	Substitution (Substitute $m\angle 4 = 30^\circ$ into $m\angle 2 = m\angle 4$)

Answer:

The two - column proof is as follows:

Statements	Reasons
2. $\angle 1 \cong \angle 4$	Vertical Angles Theorem
3. $\angle 2 \cong \angle 4$	Transitive Property of Congruence
4. $m\angle 4 = 30^\circ$	Given
5. $m\angle 2 = m\angle 4$	Definition of Congruent Angles
6. $m\angle 2 = 30^\circ$	Substitution

And the measure of $\angle 2$ is $\boldsymbol{30^\circ}$