Proof: Triangles Adjacent to Isosceles

Let's prove the following theorem:

Proof:

Proof Table

#	Claim	Reason
1	m∠YXM = m∠XYM	if distance XM = distance YM, then m∠YXM = m∠XYM (Angles of an Isosceles Triangle)
2	m∠XYM = m∠MXY	if m∠YXM = m∠XYM, then m∠XYM = m∠MXY (Angle Symmetry 4)
3	∠WXM and ∠MXY are supplementary	if m∠WXY = 180, then ∠WXM and ∠MXY are supplementary (Supplementary Angles Theorem (Converse))
4	(m∠WXM) + (m∠MXY) = 180	if ∠WXM and ∠MXY are supplementary, then (m∠WXM) + (m∠MXY) = 180 (Supplementary Angles Property)
5	m∠WXM = 180 + ((m∠MXY) ⋅ (-1))	if (m∠WXM) + (m∠MXY) = 180, then m∠WXM = 180 + ((m∠MXY) ⋅ (-1)) (Add Term to Both Sides 6)
6	∠XYM and ∠MYZ are supplementary	if m∠XYZ = 180, then ∠XYM and ∠MYZ are supplementary (Supplementary Angles Theorem (Converse))
7	(m∠XYM) + (m∠MYZ) = 180	if ∠XYM and ∠MYZ are supplementary, then (m∠XYM) + (m∠MYZ) = 180 (Supplementary Angles Property)
8	(m∠MXY) + (m∠MYZ) = 180	if (m∠XYM) + (m∠MYZ) = 180 and m∠XYM = m∠MXY, then (m∠MXY) + (m∠MYZ) = 180 (Substitution 8)
9	m∠MYZ = 180 + ((m∠MXY) ⋅ (-1))	if (m∠MXY) + (m∠MYZ) = 180, then m∠MYZ = 180 + ((m∠MXY) ⋅ (-1)) (Add Term to Both Sides 7)
10	m∠WXM = m∠MYZ	if m∠WXM = 180 + ((m∠MXY) ⋅ (-1)) and m∠MYZ = 180 + ((m∠MXY) ⋅ (-1)), then m∠WXM = m∠MYZ (Transitive Property of Equality Variation 1)
11	m∠WXM = m∠ZYM	if m∠WXM = m∠MYZ, then m∠WXM = m∠ZYM (Angle Symmetry Example)
12	△WXM ≅ △ZYM	if distance WX = distance ZY and m∠WXM = m∠ZYM and distance XM = distance YM, then △WXM ≅ △ZYM (SAS Property)
13	distance WM = distance ZM	if △WXM ≅ △ZYM, then distance WM = distance ZM (Congruent Triangles to Distance 3)

Comments

Please log in to add comments