Statements 
Reasons 
1) ∠ABC = 90^{0}

1) Given 
2) BD ⊥ AC 
2) By construction 
3) ∠ADB = 90^{0}

3) By definition of perpendicular 
4) ∠ADB = ∠ABC 
4) Each of 9090^{0}

5) ∠A = ∠A 
5) Reflexive 
6) ΔADB ~ ΔABC 
6) By AA postulate 
7) AD/AB = AB/AC 
7) By basic proportionality theorem 
8) AB^{2} = AD x AC

8) By cross multiplication 
9) ∠CDB = ∠ABC 
9) Each 90^{0}

10) ∠C = ∠C 
10) Reflexive 
11) ΔBDC ~ ΔABC 
11) By AA postulate 
12) DC/BC = BC/AC 
12) By basic proportionality theorem 
13) BC^{2} = DC x AC

13) By cross multiplication 
14) AB^{2} + BC^{2} = AD x AC + AC x DC

14) By adding (8) and (13) 
15) AB^{2} + BC^{2} = AC(AD + DC)

15) By distributive property 
16) AB^{2} + BC^{2}= AC^{2}

16) As AC = AD + DC and by substitution 