Given: ΔABC is a right triangle and CD ⊥ AB.
Prove: AC2 + BC2 = AB2
An Image of a triangle CAB with C as a right angle, and CB as the Base. D is the Point between AB such that it forms a right angle for a triangle CAD with AD as base.
Proof:
Statements Reasons
1. ∠ACB ≅ ∠ADC ≅ ∠CDB Angles with the same measure are congruent
2. ∠CAB ≅ ∠DAC
∠ABC ≅ ∠CBD Reflexive property
3. ΔABC ~ ΔACD
ΔABC ~ ΔCBD AA similarity criteria
4.
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5. AC2 = (AB)(AD)
BC2 = (AB)(DB) Cross-multiplication
6. AC2 + BC2 = (AB)(AD) + (AB)(DB) Addition
7. AC2 + BC2 = AB(AD + DB) Distributive property
8. AB = AD + DB Segment addition
9. AC2 + BC2 = (AB)(AB) Substitution
10. AC2 + BC2 = AB2 Multiplication
Which reason completes the proof?