It looks like you're working through a geometric proof involving angles and their relationships. Let's break down the statements you've provided and clarify their meanings and implications:
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∡A + ∡B = 180: This establishes that angles A and B are linear pairs, which means they form a straight line together and are supplementary.
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∡A = ∡E: Here you're stating that angle A is equal to angle E. This could imply that angles A and E are corresponding angles, alternate interior angles, or simply two angles that have been shown to be equal through some prior reasoning.
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∡E + ∡C = 180: This statement indicates that angles E and C are also supplementary, which means they too form a linear pair.
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Substitution: The substitution here refers to using the fact that angle A is equal to angle E to replace one angle with the other in an equation or statement.
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∠E & ∠C are supplementary: This conclusion reiterates that angles E and C, when added together, equal 180 degrees.
Putting this together in a more structured geometric proof format might look like this:
Proof
- Given: ∡A + ∡B = 180 (Linear Pair)
- Given: ∡A = ∡E
- Substituting: From step 2 into step 1, we can state ∡E + ∡B = 180.
- Given: ∡E + ∡C = 180 (Also a Linear Pair)
- By comparing the equations, since both ∡E + ∡B and ∡E + ∡C equal 180, we can conclude that ∡B = ∡C (if ∡E is common in both).
Conclusion
In essence, if two angles (A and C) are equal because they both lead to a common angle (E) being part of their linear pairs, we conclude that ∡B = ∡C. This forms a basis for established relationships between these angles.
If you have any specific aspect of this proof you want to explore further or any particular question, feel free to ask!