Question 257494
This is also called the exterior triangle theorem. With angle 4 vertical to angle 3, you cannot prove 
angle 4 = angle 1 + angle 2. For this to work, angle 4 must be supplementary to angle 3. Here is the proof for that.
STATEMENT 1: triangle ABC 
REASON 1: given
STATEMENT 2: angle 1 + angle 2 + angle 3 = 180 degrees
REASON 2: definition of sum of angles of  triangle 
STATEMENT 3: angle 3 + angle 4 = 180
REASON 3: definition of supplementary angles 
STATEMENT 4: angle 1 + angle 2 + angle 3 = angle 3 + angle 4
REASON 4: substitution [2,3]
STATEMENT 5: angle 1 + angle 2 = angle 4
REASON 5: subtraction property of equality