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In the figure provided, you are asked to explain how all four pairs of corresponding angles are congruent, given that ∠3 ≅ ∠6.

To understand why all four pairs of corresponding angles are congruent, let’s break down the reasoning:

1. **Corresponding Angles Theorem**: When two parallel lines are cut by a transversal (the lines that cross through the parallel lines), the angles that are in the same relative position at each intersection are called *corresponding angles*. According to the Corresponding Angles Theorem, these angles are congruent.

2. **Given Information**: You are told that ∠3 ≅ ∠6. These are a pair of corresponding angles, confirming that the lines are parallel (since corresponding angles are congruent).

3. **Other Corresponding Angles**: 
   - ∠1 corresponds to ∠5,
   - ∠2 corresponds to ∠6,
   - ∠4 corresponds to ∠8,
   - ∠3 corresponds to ∠7.

Since ∠3 ≅ ∠6, it follows from the Corresponding Angles Theorem that the other corresponding angle pairs must also be congruent:
   - ∠1 ≅ ∠5,
   - ∠2 ≅ ∠6,
   - ∠4 ≅ ∠8.

Thus, all four pairs of corresponding angles are congruent.

Would you like more details or have any questions?

Here are 5 related questions to explore further:
1. How do alternate interior angles behave when lines are parallel?
2. Can you explain how to identify corresponding angles from any given transversal diagram?
3. What is the difference between corresponding and alternate interior angles?
4. How does the congruence of corresponding angles prove that two lines are parallel?
5. In what other scenarios do corresponding angles appear, besides parallel lines?

**Tip**: Always remember that for corresponding angles to be congruent, the lines being cut by the transversal must be parallel.

In the figure, ∠3 ≅ ∠6. Explain how you know that all four pairs of corresponding angles are congruent.

Explore how to determine that all four pairs of corresponding angles are congruent when two parallel lines are cut by a transversal, with a focus on the Corresponding Angles Theorem. Suitable for students in Grades 6-8.

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