Proof: Vertical Angles
Let's prove the following theorem:
if m∠XPW = 180 and m∠YPZ = 180, then m∠WPZ = m∠XPY
Since XPW and YPZ are collinear, we can make the following 2 claims:
m∠XPY + m∠YPW = 180
m∠YPW + m∠WPZ = 180
Notice that both claims contain m∠YPW.
Next, we subtract m∠YPW from both sides:
m∠XPY = 180 - m∠YPW
m∠WPZ = 180 - m∠YPW
Proof:
Proof Table
| # | Claim | Reason |
|---|---|---|
| 1 | (m∠XPY) + (m∠YPW) = 180 | if m∠XPW = 180, then (m∠XPY) + (m∠YPW) = 180 |
| 2 | m∠XPY = 180 + ((m∠YPW) ⋅ (-1)) | if (m∠XPY) + (m∠YPW) = 180, then m∠XPY = 180 + ((m∠YPW) ⋅ (-1)) |
| 3 | (m∠YPW) + (m∠WPZ) = 180 | if m∠YPZ = 180, then (m∠YPW) + (m∠WPZ) = 180 |
| 4 | m∠WPZ = 180 + ((m∠YPW) ⋅ (-1)) | if (m∠YPW) + (m∠WPZ) = 180, then m∠WPZ = 180 + ((m∠YPW) ⋅ (-1)) |
| 5 | m∠WPZ = m∠XPY | if m∠WPZ = 180 + ((m∠YPW) ⋅ (-1)) and m∠XPY = 180 + ((m∠YPW) ⋅ (-1)), then m∠WPZ = m∠XPY |
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