Find the Angle #3
If,
∠ C A D = ∠ B A C = 1 0 ∘ ,
∠ D B A = 6 0 ∘ , and
∠ C B D = 8 0 ∘
Find ∠ D C A
This problem is a part of the set Find the Angle!
The answer is 20.
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2 solutions
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level 3 problem man
L e t x = ∠ D C A . S o ∠ C D O = 7 0 − x . ( ∗ ) i n d i c a t e A p p l i c a t i o n o f S i n L a w . I n Δ D A B A O i s t h e a n g l e b i s e c t o r o f ∠ D A B . S o O D B O = A D B A = S i n 6 0 S i n 1 0 0 . . ( ∗ ) . . . . . . . . . . . . . ( A ) I n Δ D O C , O C = O D ∗ S i n ( x ) S i n ( 7 0 − x ) . . ( ∗ ) . . . . . . . . ( B ) I n Δ C O B , O C = O B ∗ S i n 3 0 S i n 8 0 . . ( ∗ ) . . . . . . . . ( C ) F r o m ( B ) a n d ( C ) O D B O = S i n 8 0 S i n 3 0 ∗ S i n ( x ) S i n ( 7 0 − x ) = S i n 6 0 S i n 1 0 0 . . b y ( A ) E x p a n d i n g S i n a n d s i m p l i f y i n g , S i n 7 0 ∗ C o t ( x ) − C o s 7 0 = S i n 6 0 S i n 1 0 0 ∗ S i n 3 0 S i n 8 0 . ∴ x = T a n − 1 ⎝ ⎜ ⎛ S i n 6 0 ∗ S i n 3 0 S i n 1 0 0 ∗ S i n 8 0 + C o s 7 0 S i n 7 0 . ⎠ ⎟ ⎞ ∠ D C A = 2 0 .
I used calculator when needed . I can't type the answer , instead I uploaded a photo of the ANSWER .
Let angle DAC=x ,,,,,