Proof: Two Angles Equal Then Isosceles

Let's prove the following theorem:

Proof:

Proof Table

#	Claim	Reason
1	m∠PXZ = m∠YXZ	if m∠XPY = 180, then m∠PXZ = m∠YXZ (Collinear Angles Property 9)
2	m∠PYZ = m∠XYZ	if m∠XPY = 180, then m∠PYZ = m∠XYZ (Collinear Points Property)
3	m∠PXZ = m∠XYZ	if m∠PXZ = m∠YXZ and m∠YXZ = m∠XYZ, then m∠PXZ = m∠XYZ (Transitive Property of Equality)
4	m∠PXZ = m∠PYZ	if m∠PXZ = m∠XYZ and m∠PYZ = m∠XYZ, then m∠PXZ = m∠PYZ (Transitive Property of Equality Variation 1)
5	m∠XZP = m∠PZY	if ray ZP bisects ∠XZY, then m∠XZP = m∠PZY (Angle Bisector Property)
6	m∠XZP = m∠YZP	if m∠XZP = m∠PZY, then m∠XZP = m∠YZP (Angle Symmetry Example)
7	distance ZP = distance ZP	distance ZP = distance ZP (Reflexive Property of Equality)
8	△XZP ≅ △YZP	if m∠PXZ = m∠PYZ and m∠XZP = m∠YZP and distance ZP = distance ZP, then △XZP ≅ △YZP (Angle Angle Side Triangle)
9	distance ZX = distance ZY	if △XZP ≅ △YZP, then distance ZX = distance ZY (Congruent Triangles to Distance)

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