To solve for the measures of angles 1 through 7 in the given diagram, let's apply the rules for parallel lines cut by a transversal.

• Lines m and n are parallel.
• Line t is the transversal.

Angle relationships:

1. Corresponding angles: Angles in matching corners when two lines are crossed by another line (transversal). For example, angle 1 and angle 5 are corresponding.
2. Alternate interior angles: Angles on opposite sides of the transversal but inside the parallel lines. For example, angle 3 and angle 5 are alternate interior angles.
3. Vertical angles: Opposite angles formed by two intersecting lines, and they are always equal. For example, angle 1 and angle 3 are vertical angles.
4. Supplementary angles: Angles that add up to 180° because they form a straight line.

Given:

• m∠3 = 57°, as shown in the diagram.

Now let's find the measures of all the angles:

• m∠1 = m∠3 = 57° (Vertical angles)
• m∠5 = m∠1 = 57° (Corresponding angles)
• m∠7 = m∠5 = 57° (Vertical angles)
• m∠2 = 180° - m∠1 = 180° - 57° = 123° (Supplementary to angle 1)
• m∠4 = m∠2 = 123° (Vertical angles)
• m∠6 = m∠2 = 123° (Corresponding angles)

Final answers:

• m∠1 = 57°
• m∠2 = 123°
• m∠3 = 57°
• m∠4 = 123°
• m∠5 = 57°
• m∠6 = 123°
• m∠7 = 57°

Let me know if you'd like further clarifications or have any questions!

Relative questions:

1. What are the conditions for lines to be considered parallel?
2. How can you determine if two angles are corresponding in a transversal?
3. What is the relationship between alternate interior angles in a transversal situation?
4. How can vertical angles be identified and calculated?
5. What types of angles are supplementary in parallel lines cut by a transversal?

Tip:

Always look for vertical, corresponding, and alternate interior angles when working with parallel lines and a transversal!