Proof: Exterior Angle is Greater
Let's prove the following theorem:
if m∠XZE = 180, then m∠ZYX < m∠YZE
Proof:
Given
Additional Assumptions
| 1 | m∠XZE = 180 |
|---|
| 2 | M is the midpoint of line YZ |
|---|---|
| 3 | M is the midpoint of line XP |
| 4 | m∠PZE > 0 |
| # | Claim | Reason |
|---|---|---|
| 1 | distance XM = distance MP | if M is the midpoint of line XP, then distance XM = distance MP |
| 2 | distance XM = distance PM | if distance XM = distance MP, then distance XM = distance PM |
| 3 | distance YM = distance MZ | if M is the midpoint of line YZ, then distance YM = distance MZ |
| 4 | distance MY = distance MZ | if distance YM = distance MZ, then distance MY = distance MZ |
| 5 | m∠YMZ = 180 | if M is the midpoint of line YZ, then m∠YMZ = 180 |
| 6 | m∠ZMY = 180 | if m∠YMZ = 180, then m∠ZMY = 180 |
| 7 | m∠XMP = 180 | if M is the midpoint of line XP, then m∠XMP = 180 |
| 8 | m∠PMX = 180 | if m∠XMP = 180, then m∠PMX = 180 |
| 9 | m∠XMY = m∠PMZ | if m∠PMX = 180 and m∠ZMY = 180, then m∠XMY = m∠PMZ |
| 10 | △XMY ≅ △PMZ | if distance XM = distance PM and m∠XMY = m∠PMZ and distance MY = distance MZ, then △XMY ≅ △PMZ |
| 11 | m∠MYX = m∠MZP | if △XMY ≅ △PMZ, then m∠MYX = m∠MZP |
| 12 | m∠MZP = m∠MYX | if m∠MYX = m∠MZP, then m∠MZP = m∠MYX |
| 13 | point M is in segment YZ | if M is the midpoint of line YZ, then point M is in segment YZ |
| 14 | point P lies in interior of ∠YZE | if point M is in segment YZ and m∠XMP = 180 and m∠XZE = 180, then point P lies in interior of ∠YZE |
| 15 | m∠YZE = (m∠YZP) + (m∠PZE) | if point P lies in interior of ∠YZE, then m∠YZE = (m∠YZP) + (m∠PZE) |
| 16 | m∠YZE > m∠YZP | if m∠YZE = (m∠YZP) + (m∠PZE) and m∠PZE > 0, then m∠YZE > m∠YZP |
| 17 | m∠YZP = m∠MZP | if m∠ZMY = 180, then m∠YZP = m∠MZP |
| 18 | m∠YZP = m∠MYX | if m∠YZP = m∠MZP and m∠MZP = m∠MYX, then m∠YZP = m∠MYX |
| 19 | m∠MYX = m∠ZYX | if m∠ZMY = 180, then m∠MYX = m∠ZYX |
| 20 | m∠YZP = m∠ZYX | if m∠YZP = m∠MYX and m∠MYX = m∠ZYX, then m∠YZP = m∠ZYX |
| 21 | m∠YZE > m∠ZYX | if m∠YZE > m∠YZP and m∠YZP = m∠ZYX, then m∠YZE > m∠ZYX |
| 22 | m∠ZYX < m∠YZE | if m∠YZE > m∠ZYX, then m∠ZYX < m∠YZE |
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