Question

obtuse triangle ABC with obtuse angle B. Point D on side AC such that angle BDC is also obtuse. Angle DAB is 2/3 of angle ABD. Angle BCD is 1/5 of angle CBD is 4 more than angle BAD. Find the measure of angle ABC.

Answer 1

The wording of " Angle BCD is 1/5 of angle CBD is 4 more than angle BAD." is rather ambiguous.

Did you mean
" Angle BCD is 1/5 of angle CBD, AND IT is 4 more than angle BAD." ?

Answer 2

yes

Answer 3

Ok, let me use < for angle

Let <ABD = 3x (I can avoid fractions this way)
then <DAB = 2x
by exterior angle theorem <BDC = 5x

<BCD = 2x + 4 (it said so)
<CBD is 5 times <BCD (it said so)
so <CBD = 5(2x+4) = 10x + 20

so by exterior angle theorem :
< BDC = 12x + 24

But CDA is a straight line, so
5x + 12x+24 = 180
x = 216/7 or 9.1765

then <ABC = 10x+20 + 3x
= 13x + 20 = 139.294°

let's check the rest
<ABD = 27.529
<BAD = 18.353
and 18.535/27.529 = .6666678 or 2/3
< BDA = 134.118
for a total sum of 134.118+27.529+18.353 = 180

In triangle BDC
<C = 4 more than <BAD = 22.353
which is 1/5 of <CBD making
<CBD = 111.765
Also we knew that <BDC = 5x = 45.882

check: is CDA a straight line?
is 17x+24 = 180 ?
17x + 24 = 17(9.1765) + 24 = 180.0005 , OK

what about the sum of angles in that triangle?
what is 111.765+45.882+22.353 = 180

So again
<b> <ABC = 10x+20 + 3x
= 13x + 20 = 139.294° </b>