Proof: Triangle Property
Let's prove the following theorem:
if distance XY > distance YZ, then m∠XZY > m∠YXZ
Proof:
Given
Additional Assumptions
| 1 | distance XY > distance YZ |
|---|
| 2 | points X E and Y are collinear |
|---|---|
| 3 | distance YZ = distance YE |
| 4 | m∠XZE > 0 |
| 5 | m∠XZY = (m∠EZY) + (m∠XZE) |
| # | Claim | Reason |
|---|---|---|
| 1 | distance XY > distance YE | if distance XY > distance YZ and distance YZ = distance YE, then distance XY > distance YE |
| 2 | distance XY > distance EY | if distance XY > distance YE, then distance XY > distance EY |
| 3 | m∠XEY = 180 | if points X E and Y are collinear and distance XY > distance EY, then m∠XEY = 180 |
| 4 | m∠ZEY = m∠EZY | if distance YZ = distance YE, then m∠ZEY = m∠EZY |
| 5 | m∠ZEY > m∠YXZ | if m∠XEY = 180, then m∠ZEY > m∠YXZ |
| 6 | m∠EZY > m∠YXZ | if m∠ZEY > m∠YXZ and m∠ZEY = m∠EZY, then m∠EZY > m∠YXZ |
| 7 | m∠XZY > m∠EZY | if m∠XZY = (m∠EZY) + (m∠XZE) and m∠XZE > 0, then m∠XZY > m∠EZY |
| 8 | m∠XZY > m∠YXZ | if m∠XZY > m∠EZY and m∠EZY > m∠YXZ, then m∠XZY > m∠YXZ |
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