Let's analyze the given diagram and solve for the measures of the angles.

Information Given:

• The figure shows two parallel lines cut by a transversal.
• $m\angle 8 = 23^\circ$.

Using Angle Relationships:

1. Corresponding Angles: Angles that are on the same side of the transversal and in corresponding positions are equal.
2. Alternate Interior Angles: Angles on opposite sides of the transversal but inside the parallel lines are congruent.
3. Vertical Angles: Angles opposite each other when two lines intersect are congruent.
4. Linear Pair: Adjacent angles on a straight line add up to $180^\circ$.

Solving Step by Step:

• m∠8 = 23° (Given)

• m∠6: This is a vertical angle with ∠8, so: $m\angle 6 = m\angle 8 = 23^\circ$

• m∠2: ∠2 and ∠8 are alternate interior angles, so: $m\angle 2 = m\angle 8 = 23^\circ$

• m∠4: ∠4 and ∠8 form a linear pair, so: $m\angle 4 = 180^\circ - m\angle 8 = 180^\circ - 23^\circ = 157^\circ$

• m∠3: ∠3 and ∠4 are vertical angles, so: $m\angle 3 = m\angle 4 = 157^\circ$

• m∠5: ∠5 and ∠4 are corresponding angles, so: $m\angle 5 = m\angle 4 = 157^\circ$

Final Answers:

• $m\angle 2 = 23^\circ$
• $m\angle 3 = 157^\circ$
• $m\angle 4 = 157^\circ$
• $m\angle 5 = 157^\circ$
• $m\angle 6 = 23^\circ$

Would you like more details or have any questions?

Here are 5 related questions for deeper understanding:

1. What is the sum of the interior angles on the same side of the transversal?
2. How would the angle measures change if $m\angle 8$ was doubled?
3. What is the relationship between alternate exterior angles?
4. How are same-side interior angles related in terms of angle measures?
5. How would the solution change if the lines weren't parallel?

Tip: Remember that parallel lines cut by a transversal always create pairs of congruent and supplementary angles!