Transitive Property of Equality Variation 3
Distance Property 5
Transitive Property of Equality Variation 2
Distance Property 6
Collinear Then 180
Subtract Both Sides
Add Term to Both Sides 6
Subtract Both Sides 2
Add Term to Both Sides 7
Transitive Property of Equality Variation 1
Vertical Angles
Collinear Angles Property 10
Angle Addition Theorem
Angle Symmetry Example 2
Commutative Property Variation 1
Substitution 2
Substitution 6
Rearrange Angles
Whole is Greater Than Parts
Vertical Angles 3
Collinear Angles Property 3
Collinear Angles Property 3 B
Collinear Angles Property 3 C
Exterior Angle Other

Proof: Exterior Angle Other

Let's prove the following theorem:

if m∠XZE = 180, then m∠YZE > m∠YXZ

Z M G E X Y F

Proof:

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Given
1 m∠XZE = 180
Assumptions
2 M is the midpoint of line XZ
3 M is the midpoint of line YG
4 m∠YZF = 180
5 m∠FZG > 0
Proof Table
# Claim Reason
1 distance XM = distance MZ if M is the midpoint of line XZ, then distance XM = distance MZ
2 distance ZM = distance XM if distance XM = distance MZ, then distance ZM = distance XM
3 distance YM = distance MG if M is the midpoint of line YG, then distance YM = distance MG
4 distance MG = distance MY if distance YM = distance MG, then distance MG = distance MY
5 m∠YMG = 180 if M is the midpoint of line YG, then m∠YMG = 180
6 m∠XMZ = 180 if M is the midpoint of line XZ, then m∠XMZ = 180
7 m∠ZMG = m∠XMY if m∠XMZ = 180 and m∠YMG = 180, then m∠ZMG = m∠XMY
8 △ZMG ≅ △XMY if distance ZM = distance XM and m∠ZMG = m∠XMY and distance MG = distance MY, then △ZMG ≅ △XMY
9 m∠GZM = m∠YXM if △ZMG ≅ △XMY, then m∠GZM = m∠YXM
10 m∠GZM = m∠GZX if m∠XMZ = 180, then m∠GZM = m∠GZX
11 point M is in segment XZ if M is the midpoint of line XZ, then point M is in segment XZ
12 point G lies in interior of ∠XZF if point M is in segment XZ and m∠YMG = 180 and m∠YZF = 180, then point G lies in interior of ∠XZF
13 m∠XZF = (m∠XZG) + (m∠GZF) if point G lies in interior of ∠XZF, then m∠XZF = (m∠XZG) + (m∠GZF)
14 m∠FZX = (m∠FZG) + (m∠GZX) if m∠XZF = (m∠XZG) + (m∠GZF), then m∠FZX = (m∠FZG) + (m∠GZX)
15 m∠FZX > m∠GZX if m∠FZX = (m∠FZG) + (m∠GZX) and m∠FZG > 0, then m∠FZX > m∠GZX
16 m∠FZX = m∠YZE if m∠YZF = 180 and m∠XZE = 180, then m∠FZX = m∠YZE
17 m∠YZE > m∠GZX if m∠FZX > m∠GZX and m∠FZX = m∠YZE, then m∠YZE > m∠GZX
18 m∠GZX = m∠YXM if m∠GZM = m∠GZX and m∠GZM = m∠YXM, then m∠GZX = m∠YXM
19 m∠YZE > m∠YXM if m∠YZE > m∠GZX and m∠GZX = m∠YXM, then m∠YZE > m∠YXM
20 m∠YXM = m∠YXZ if m∠XMZ = 180, then m∠YXM = m∠YXZ
21 m∠YZE > m∠YXZ if m∠YZE > m∠YXM and m∠YXM = m∠YXZ, then m∠YZE > m∠YXZ
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