Statement 1:
and
are vertical angles
Reason 1: Given
We basically just restate the given information word for word. This is true of any two column proof.
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Statement 2: ![m \angle 2 + m \angle 3 = 180](https://tex.z-dn.net/?f=m%20%5Cangle%202%20%2B%20m%20%5Cangle%203%20%3D%20180)
Reason 2:
and
are a linear pair
The term "linear pair" means the angles are adjacent and supplementary (they form a straight line), so this is why the two angles add to 180.
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Statement 3: ![m \angle 3 + m \angle 4 = 180](https://tex.z-dn.net/?f=m%20%5Cangle%203%20%2B%20m%20%5Cangle%204%20%3D%20180)
Reason 3:
and
are a linear pair
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Statement 4: ![m \angle 2 + m \angle 3 = m \angle 3 + m \angle 4](https://tex.z-dn.net/?f=m%20%5Cangle%202%20%2B%20m%20%5Cangle%203%20%3D%20m%20%5Cangle%203%20%2B%20m%20%5Cangle%204)
Reason 4: Substitution
Each of the equations formed in statements 2 and 3 above have 180 on the right side, so the left hand sides must be the same
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Statement 5: ![m \angle 2 = m \angle 4](https://tex.z-dn.net/?f=m%20%5Cangle%202%20%3D%20m%20%5Cangle%204)
Reason 5: Subtraction property of equality
We subtracted the quantity
from both sides (they go away)
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Statement 6: ![\angle 2 \cong \angle 4](https://tex.z-dn.net/?f=%5Cangle%202%20%5Ccong%20%5Cangle%204)
Reason 6: Definition of congruence
If two items are congruent, then they have the same measure. In other words, they are the same.