Geometry (Beta) / Chapter 3: Inequality / Outer Angles

Proof: Exterior Angle is Greater

Let's prove the following theorem:

if m∠XZE = 180, then m∠ZYX < m∠YZE

Z E X Y M P

Proof:

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Given
1 m∠XZE = 180
Additional Assumptions
2 M is the midpoint of line YZ
3 M is the midpoint of line XP
4 m∠PZE > 0
Proof Table
# 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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