Based on the information you've provided, it seems that you are forming a proof in geometry using various theorems related to linear pairs, supplementary angles, and congruent angles. Here's how you could fill in the steps:
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Given: ∠q and ∠4 form a linear pair; ∠2 and ∠2 are supplementary.
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Linear Pair (Supplement) Theorem: A linear pair of angles is supplementary. Thus, ∠q + ∠4 = 180°.
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Given: ∠1 and ∠3 form a linear pair.
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Congruent Supplements Theorem: If two angles are supplementary to the same angle (or congruent angles), then those two angles are congruent.
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∠2 = ∠5: Since ∠2 and ∠2 are supplementary, we can express that if ∠4 is supplementary to ∠2, and since ∠4 is also supplementary to ∠5 (a congruent angle from the previous argument), then ∠2 must also equal ∠5.
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Transitive Property: If ∠2 = ∠5 and ∠5 is congruent to ∠6, then ∠2 must also be congruent to ∠6; therefore, we can apply the transitive property.
Here’s how the proof might look completely filled out:
- Given: ∠q and ∠4 form a linear pair; ∠2 and ∠2 are supplementary.
- Linear Pair (Supplement) Theorem: ∠q + ∠4 = 180°.
- Given: ∠1 and ∠3 form a linear pair.
- Congruent Supplements Theorem: Since ∠q and ∠4 are supplementary, and ∠1 is a supplement of ∠3, then ∠3 = ∠4.
- Let ∠5 = ∠4 (as the angle that is congruent).
- Transitive Property: If ∠2 + ∠5 = 180° and also ∠2 + ∠2 = 180°, then ∠2 = ∠5.
Make sure to adjust the specific angle names and properties to fit your particular problem as needed!