Question

thus, ∠1 and ∠2 are (choose one) < vertical angles. so, m∠2 = □°. we see that ∠2 and ∠3 are (choose one) < and since the lines g and h are parallel, ∠2 and ∠3 are (choose one) < so, m∠3 = □°. therefore, ∠1 and ∠3 are (choose one) < we also see that ∠1 and ∠3 are (choose one) < the relationship between ∠1 and ∠3 is an example of the following rule. when parallel lines are cut by a transversal, (choose one) <

Explanation:

To solve this, we assume some typical angle relationships (since the original problem likely has given angle measures for $\angle1$, but let's assume a common case where $\angle1$ and $\angle2$ are vertical angles, so $m\angle1 = m\angle2$. Then $\angle2$ and $\angle3$ are corresponding angles (if lines $g$ and $h$ are parallel), so $m\angle3 = m\angle2 = m\angle1$. Let's say $m\angle1 = 50^\circ$ (a common example).

Step 1: Vertical Angles Property

$\angle1$ and $\angle2$ are vertical angles, so $m\angle2 = m\angle1$. If $m\angle1 = 50^\circ$, then $m\angle2 = 50^\circ$.

Step 2: Corresponding Angles (Parallel Lines)

Lines $g$ and $h$ are parallel, so $\angle2$ and $\angle3$ are corresponding angles. Corresponding angles are equal, so $m\angle3 = m\angle2 = 50^\circ$.

Step 3: Transitive Property

$\angle1$ and $\angle3$ are equal (since $m\angle1 = m\angle2$ and $m\angle2 = m\angle3$), so $\angle1 \cong \angle3$. This is the transitive property of equality (or the "corresponding angles postulate" leading to vertical angles implying equal angles).

Answer:

(Filled in Blanks, Assuming $m\angle1 = 50^\circ$):

• $\angle1$ and $\angle2$ are vertical angles.
• $m\angle2 = \boldsymbol{50^\circ}$.
• $\angle2$ and $\angle3$ are corresponding angles.
• $m\angle3 = \boldsymbol{50^\circ}$.
• $\angle1$ and $\angle3$ are equal (or "congruent").
• The rule is: When parallel lines are cut by a transversal, corresponding angles are equal (and vertical angles are equal, so this implies $\angle1 \cong \angle3$ via transitivity).

(Note: The exact angle measure depends on the original problem’s given $\angle1$ measure. If $\angle1$ was, say, $70^\circ$, then $m\angle2 = 70^\circ$ and $m\angle3 = 70^\circ$.)