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Prove: angle 6 text  ≅ end text space angle 8 space

Statement	Justification
angle 2 text  ≅ end text space angle 4 space
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angle 2 text  ≅ end text space angle 6 space
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angle 6 space text ≅ end text space angle 4 space
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angle 4 text  ≅ end text space angle 8 space
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angle 6 text  ≅ end text space angle 8 space
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To prove that ∠6 ≅ ∠8 using a step-by-step proof with statements and justifications, let's analyze the given diagram.

The diagram shows a set of intersecting lines with labeled angles. Based on the information, it appears that the angles are formed by lines intersecting at a point, which allows us to apply known geometric principles, such as vertical angles, linear pairs, and corresponding angles.

We can now fill in the proof logically:

### Proof:
1. **∠2 ≅ ∠4** (Vertical Angles)  
   Justification: Vertical angles are congruent when two lines intersect.

2. **∠2 ≅ ∠6** (Corresponding Angles)  
   Justification: By corresponding angles theorem (since angles on opposite sides of the transversal are congruent).

3. **∠6 ≅ ∠4** (Transitive Property)  
   Justification: If ∠2 ≅ ∠6 and ∠2 ≅ ∠4, then by the transitive property of equality, ∠6 ≅ ∠4.

4. **∠4 ≅ ∠8** (Vertical Angles)  
   Justification: ∠4 and ∠8 are vertical angles, and vertical angles are congruent.

5. **∠6 ≅ ∠8** (Transitive Property)  
   Justification: Since ∠6 ≅ ∠4 and ∠4 ≅ ∠8, by the transitive property, ∠6 ≅ ∠8.

Thus, we have proved that ∠6 ≅ ∠8.

Would you like more details on this proof or help with any related concepts? Here are five additional questions to explore this topic further:

1. Can you explain how vertical angles are always congruent?
2. How does the transitive property apply to angles in a geometric proof?
3. What is the corresponding angles theorem, and when is it used?
4. How can we use linear pairs of angles to find missing angle measures?
5. How does the transversal line affect the angles formed by intersecting lines?

**Tip:** When working with intersecting lines, always identify pairs of vertical and corresponding angles, as they are often congruent and essential to solving geometric proofs.

Explore how to prove that ∠6 ≅ ∠8 using geometric principles like vertical angles, corresponding angles, and the transitive property of equality. Suitable for students in Grades 8-10, this step-by-step solution guides learners through identifying congruent angles.

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