Proof: If Diagonals Perpendicular Then Rhombus

Let's prove the following theorem:

Proof:

Proof Table

#	Claim	Reason
1	△PYZ ≅ △PWX	if WXYZ is a parallelogram and m∠WPY = 180 and m∠XPZ = 180, then △PYZ ≅ △PWX (If Parallelogram Inner Congruent)
2	distance YP = distance WP	if △PYZ ≅ △PWX, then distance YP = distance WP (Congruent Triangles to Distance)
3	distance WP = distance YP	if distance YP = distance WP, then distance WP = distance YP (Symmetric Property of Equality)
4	∠WPZ and ∠ZPY are supplementary	if m∠WPY = 180, then ∠WPZ and ∠ZPY are supplementary (Supplementary Angles Theorem (Converse))
5	(m∠WPZ) + (m∠YPZ) = 180	if ∠WPZ and ∠ZPY are supplementary, then (m∠WPZ) + (m∠YPZ) = 180 (Supplementary Then 180)
6	m∠YPZ = 90	if ∠YPZ is a right angle, then m∠YPZ = 90 (Right Angle Property)
7	(m∠WPZ) + 90 = 180	if (m∠WPZ) + (m∠YPZ) = 180 and m∠YPZ = 90, then (m∠WPZ) + 90 = 180 (Substitution Example 10)
8	m∠WPZ = 90	if (m∠WPZ) + 90 = 180, then m∠WPZ = 90 (Add Number to Both Sides)
9	m∠WPZ = m∠YPZ	if m∠WPZ = 90 and m∠YPZ = 90, then m∠WPZ = m∠YPZ (Transitive Property of Equality Variation 1)
10	distance PZ = distance PZ	distance PZ = distance PZ (Reflexive Property of Equality)
11	△WPZ ≅ △YPZ	if distance WP = distance YP and m∠WPZ = m∠YPZ and distance PZ = distance PZ, then △WPZ ≅ △YPZ (SAS Property)
12	distance WZ = distance YZ	if △WPZ ≅ △YPZ, then distance WZ = distance YZ (Congruent Triangles to Distance 3)
13	distance YZ = distance ZW	if distance WZ = distance YZ, then distance YZ = distance ZW (Distance Property 6)
14	WXYZ is a rhombus	if WXYZ is a parallelogram and distance YZ = distance ZW, then WXYZ is a rhombus (Rhombus Property)

Comments

Please log in to add comments