By proposition 3.21 is ether acute, right or obtuse

Question

Let A, B, C be points on a line with A ∗ B ∗ C. Let D be a point not on that line such

that angle ABD is obtuse. Show that angle CBD is acute. Do NOT use the Measurement Theorem. (Hint: By Proposition 3.21, angle CBD is either acute, right, or obtuse. Consider the cases when it’s right and obtuse separately, obtaining a contradiction with the hypothesis that angle ABD is obtuse in each case.)

for axioms http://math.gcsu.edu/~ryan/4510/notes/congruence.pdf

proposition 3.21

a. Exactly one of the following conditions holds (trichotomy): ∠P < ∠Q, ∠P ∼= ∠Q, or ∠P > ∠Q.

b. If ∠P < ∠Q and ∠Q ∼= ∠R, then ∠P < ∠R.

c. If ∠P > ∠Q and ∠Q ∼= ∠R, then ∠P > ∠R.

d. If ∠P < ∠Q and ∠Q < ∠R, then ∠P < ∠R.

can someone help me put this in context

Answer 1

Hint: $∠ABC=180^o=∠ABD+∠CBD$

Since $∠ABD$ is obtuse, $∠ABD>90^o$

Case I: $∠CBD>90^o$

$∠ABC=∠ABD+∠CBD>90^o+90^o=180^o$

which is a contradictiona as $∠ABC$ is a straight line angle.

Case II: $∠CBD=90^o$

$∠ABC=∠ABD+∠CBD=∠ABD+90^o>90^o+90^o=180^o$

which is a contradictiona as $∠ABC$ is a straight line angle.

You could alse see that in the photo below