Question

Fie ABCD un romb si AM bisectoarea unghiului CAD, M c (CD). Daca m (<AMD) = 54° , atunci m (<BCD) este:

A. 60°
B. 72°
C. 84°
D. 96°

Answer 1

ABCD- romb⇒∡BAD≡∡BCD ; ∡ABC≡∡ADC
AC si BD - diagonale⇒ bisectoarele unghiurilor
∡BAC≡∡DAC
AM - bisectoarea ∡CAD⇒ ∡CAM≡∡DAM= ∡CAD/2⇒
daca ∡AMD = 54°⇒ ∡AMC = 180-54=126°
In ΔAMC:∡CAM =  ∡CAD/2  = ∡ACD/2
∡CAM + ∡MCA+∡CMA=180°
∡ACM / 2 +∡ ACM + 126=180
∡ACM / 2 +∡ ACM = 180-126=54 /*2
∡ACM + 2∡ACM = 108
3∡ACM=108
∡ACM= 108:3=36°⇒ ∡ ACD = 36°⇒ ∡ BCD = 36 *2=72°