Question

Given: <2 and <4 are vertical angles.

Answer 1

Statement 1: $\angle 2$ and $\angle 4$ 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:  $m \angle 2 + m \angle 3 = 180$

Reason 2: $\angle 2$ and $\angle 3$ 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: $m \angle 3 + m \angle 4 = 180$

Reason 3: $\angle 3$ and $\angle 4$ are a linear pair

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Statement 4: $m \angle 2 + m \angle 3 = m \angle 3 + m \angle 4$

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: $m \angle 2 = m \angle 4$

Reason 5: Subtraction property of equality

We subtracted the quantity $m \angle 3$ from both sides (they go away)

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Statement 6: $\angle 2 \cong \angle 4$

Reason 6: Definition of congruence

If two items are congruent, then they have the same measure. In other words, they are the same.