Geometry (Beta) / Chapter 5: Quadrilaterals / The Rhombus

Proof: Rhombus Diagonal Equilateral Triangles

Let's prove the following theorem:

if WXYZ is a rhombus and m∠WXY = 60, then △YZW is an equilateral triangle

Y Z W X

Proof:

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Given
1 WXYZ is a rhombus
2 m∠WXY = 60
Proof Table
# Claim Reason
1 distance WX = distance XY if WXYZ is a rhombus, then distance WX = distance XY
2 distance WX = distance YX if distance WX = distance XY, then distance WX = distance YX
3 m∠XWY = m∠XYW if distance WX = distance YX, then m∠XWY = m∠XYW
4 m∠YWX = m∠XYW if m∠XWY = m∠XYW, then m∠YWX = m∠XYW
5 ((m∠WXY) + (m∠XYW)) + (m∠YWX) = 180 ((m∠WXY) + (m∠XYW)) + (m∠YWX) = 180
6 ((m∠WXY) + (m∠XYW)) + (m∠XYW) = 180 if ((m∠WXY) + (m∠XYW)) + (m∠YWX) = 180 and m∠YWX = m∠XYW, then ((m∠WXY) + (m∠XYW)) + (m∠XYW) = 180
7 (m∠WXY) + ((m∠XYW) ⋅ 2) = 180 if ((m∠WXY) + (m∠XYW)) + (m∠XYW) = 180, then (m∠WXY) + ((m∠XYW) ⋅ 2) = 180
8 60 + ((m∠XYW) ⋅ 2) = 180 if (m∠WXY) + ((m∠XYW) ⋅ 2) = 180 and m∠WXY = 60, then 60 + ((m∠XYW) ⋅ 2) = 180
9 m∠XYW = 60 if 60 + ((m∠XYW) ⋅ 2) = 180, then m∠XYW = 60
10 m∠WXY = m∠XYW if m∠WXY = 60 and m∠XYW = 60, then m∠WXY = m∠XYW
11 m∠XYW = m∠YWX if m∠YWX = m∠XYW, then m∠XYW = m∠YWX
12 △WXY is an equilateral triangle if m∠WXY = m∠XYW and m∠XYW = m∠YWX, then △WXY is an equilateral triangle
13 distance ZW = distance WX if WXYZ is a rhombus, then distance ZW = distance WX
14 distance YZ = distance ZW if WXYZ is a rhombus, then distance YZ = distance ZW
15 distance YW = distance WX if △WXY is an equilateral triangle, then distance YW = distance WX
16 distance ZW = distance YW if distance ZW = distance WX and distance YW = distance WX, then distance ZW = distance YW
17 distance ZW = distance WY if distance ZW = distance YW, then distance ZW = distance WY
18 △YZW is an equilateral triangle if distance YZ = distance ZW and distance ZW = distance WY, then △YZW is an equilateral triangle
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