To determine the correct statement, let's break down each option based on the properties of parallel lines cut by a transversal.

Basic Principles

1. Corresponding Angles: Angles in the same relative position at each intersection are congruent (equal).
2. Alternate Interior Angles: Angles on opposite sides of the transversal but inside the parallel lines are congruent.
3. Alternate Exterior Angles: Angles on opposite sides of the transversal but outside the parallel lines are congruent.
4. Same-Side Interior Angles: Angles on the same side of the transversal and inside the parallel lines are supplementary (sum to 180°).

Now let's evaluate each statement:

Option A:

"∠4 ≅ ∠8 and ∠5 ≅ ∠2, so ∠4 + ∠5 = 180°."

• Analysis:
  • ∠4 and ∠8 are corresponding angles, so ∠4 ≅ ∠8. This is true.
  • ∠5 and ∠2 are also corresponding angles, so ∠5 ≅ ∠2. This is true.
  • However, ∠4 and ∠5 are not supplementary (they are not on the same side of the transversal and inside the parallel lines). This statement is false.

Option B:

"∠4 ≅ ∠8 and ∠3 ≅ ∠7, so ∠1 ≅ ∠5."

• Analysis:

  • ∠4 and ∠8 are corresponding angles, so ∠4 ≅ ∠8. This is true.
  • ∠3 and ∠7 are alternate exterior angles, so ∠3 ≅ ∠7. This is true.
  • ∠1 and ∠5 are corresponding angles, so ∠1 ≅ ∠5. This is true.

  Thus, this statement is true.

Option C:

"∠4 ≅ ∠8 and ∠6 ≅ ∠8, so ∠4 ≅ ∠6 = 180°."

• Analysis:
  • ∠4 and ∠8 are corresponding angles, so ∠4 ≅ ∠8. This is true.
  • However, ∠6 and ∠8 are not congruent. In fact, they are consecutive interior angles, meaning that they are supplementary (sum to 180°), but not congruent.
  • ∠4 and ∠6 cannot be both congruent and sum to 180°. This statement is false.

Option D:

"∠4 and ∠3 are complementary, ∠6 ≅ ∠3, so ∠4 + ∠6 = 90°."

• Analysis:
  • ∠4 and ∠3 are not complementary (their sum is not 90°), as they are on opposite sides of the transversal.
  • ∠6 and ∠3 are alternate interior angles, so ∠6 ≅ ∠3, but this does not mean that their sum is 90°.
  • This statement is false.

Conclusion:

The correct answer is Option B: ∠4 ≅ ∠8 and ∠3 ≅ ∠7, so ∠1 ≅ ∠5.

Would you like further clarification or details on any of these points?

Here are 5 related questions:

1. What is the relationship between alternate interior angles?
2. How do corresponding angles behave when two lines are parallel?
3. What is the sum of angles on the same side of a transversal and inside parallel lines?
4. Can complementary angles exist when parallel lines are cut by a transversal?
5. What is the difference between supplementary and complementary angles?

Tip: Always remember that corresponding angles are congruent when parallel lines are cut by a transversal!