Answer: AB2 = AC × AD Reason: cross multiplication
BC2 = AC × DC Reason: cross multiplication
AB2 + BC2 = AC × AD + AC × DC Reason: addition
AB2 + BC2 = AC(AD + DC) Reason: Distributive Property
AB2 + BC2 = AC2 Reason: segment addition
The completed proof is:
Draw ∆ABD ∼ ∆ABC (construction)
∠ABC ≅ ∠BDC (Angles with the same measure are congruent.)
∠BCA ≅ ∠DCB (Reflexive Property of Congruence)
∆ABC ∼ ∆DBC (AA criterion for similarity)
Corresponding sides of similar triangles are proportional.
BC2 = AC × DC (cross multiplication)
∠ABC ≅ ∠ADB (Angles with the same measure are congruent.)
∠BAC ≅ ∠DAB (Reflexive Property of Congruence)
∆ABD ∼ ∆ABC (AA criterion for similarity)
Corresponding sides of similar triangles are proportional.
AB2 = AC × AD (cross multiplication)
BC2 = AC × DC (cross multiplication)
AB2 + BC2 = AC × AD + AC × DC (addition)
AB2 + BC2 = AC(AD + DC) (Distributive Property)
AB2 + BC2 = AC2 (segment addition)
The final statement, AB2 + BC2 = AC2, is the Pythagorean theorem.
Step-by-step explanation: