Let's analyze the diagram and complete the statements based on the given figure of two parallel lines, $m$ and $n$, cut by a transversal $d$. The angles 1, 2, and 3 are formed at the intersections.

Step-by-step explanation:

1. Angles 1 and 2:

   • $\angle 1$ and $\angle 2$ are corresponding angles because they are on the same side of the transversal and on the same side of lines $m$ and $n$.
   • Since lines $m$ and $n$ are parallel, corresponding angles are equal.
   • Therefore, $m \angle 1 = m \angle 2$.

2. Angles 2 and 3:

   • $\angle 2$ and $\angle 3$ are alternate interior angles because they lie on opposite sides of the transversal $d$, between the two parallel lines.
   • Alternate interior angles are also equal when the lines are parallel.
   • So, $m \angle 2 = m \angle 3$.

3. Angles 1 and 3:

   • $\angle 1$ and $\angle 3$ are corresponding angles, meaning they are on the same side of the transversal but at different intersections.
   • Corresponding angles are equal when the lines are parallel, so $m \angle 1 = m \angle 3$.

4. General Rule:

   • The relationship between $\angle 1$ and $\angle 3$ is an example of corresponding angles formed by a transversal cutting parallel lines.
   • The rule is: "When parallel lines are cut by a transversal, corresponding angles are congruent (equal)."

Now, let's fill in the blanks from the statements:

• $\angle 1$ and $\angle 2$ are corresponding angles.

• Thus, $\angle 1$ and $\angle 2$ are equal (congruent).

• $m \angle 2 =$ the same measure as $\angle 1$.

• $\angle 2$ and $\angle 3$ are alternate interior angles.

• Since lines $m$ and $n$ are parallel, $\angle 2$ and $\angle 3$ are equal (congruent).

• $m \angle 3 =$ the same measure as $\angle 2$.

• Therefore, $\angle 1$ and $\angle 3$ are equal (congruent).

• The relationship between $\angle 1$ and $\angle 3$ is an example of the rule: "When parallel lines are cut by a transversal, corresponding angles are congruent."

Would you like more details or have any questions? Here are 5 related questions to further explore:

1. What happens to the angles if the lines $m$ and $n$ are not parallel?
2. Can alternate interior angles be supplementary in some cases?
3. What is the relationship between angles on the same side of the transversal?
4. How can you prove that two lines are parallel using angle relationships?
5. What is the sum of the interior angles on the same side of the transversal when lines are parallel?

Tip: Remember, when working with parallel lines and a transversal, always look for corresponding, alternate interior, or consecutive interior angles to quickly find angle measures.