Question

daca [AD este bisectoarea unghiului CAB si (AC)=(CD) , demonstrati ca CD paralel cu AB
FIG.10 dau coroana + 80 de puncte

Answer 1

daca [AD bisectoare ∡CAB⇒ ∡CAD≡∡DAB

daca [AC]≡[CD]⇒ΔCAD isoscel⇒ ∡CAD≡∡CDA

deci daca ∡CAD≡∡DAB   si                ∡CAD≡∡CDA              ⇒ ∡DAB≡∡CDA

 dar ∡A=∡CAD+∡DAB       si             ∡D=∡CDA+∡ADB           ⇒ ∡DAB≡∡ADB

∡A=∡CAD+∡DAB              si               ∡CAD≡∡DAB                 ⇒ ∡A=2∡CAD

 ∡D=∡CDA+∡ADB  dar ∡CDA≡∡ADB si  ∡CDA≡∡CAD           ⇒ ∡D=2∡CAD

∡ADB≡∡DAB

∡CAD≡∡DAB

daca ∡D=2∡CAD  si   ∡A=2∡CAD⇒                                           ∡D≡∡A

 

daca∡DAB≡∡ADB⇒ ca ΔABD are 2 unghiuri la baza egale deci este Δ isoscel⇒ ca are 2 laturi congruente⇒ [AB]≡[BD]

daca [AC]≡[CD]   (din ipoteza)  si    [AB]≡[BD] (demonstrate mai sus) ⇒ ABCD patrat

⇒ AB║CD  si   AC║BD

Answer 2

Am atasat rezolvarea!