Let Y be the common vertex for <1 and <2. Let <1 = XYZ and <2 = ZYW. The common ray for these angles is ray YZ.
<1 and <2 are a linear pair -- GIVEN
Ray XY and Ray YW are opposite rays. -- By the definition of a linear pair.
<XYW had measure of 180° -- By the definition of a straight angle (opposite rays)
measure of <1 + measure of <2 = measure of <XYW -- Definition of addition of angle measures
measure of <1 + measure of <2 = 180 -- By substitution
<1 and <2 are supplementary -- By definition of supplementary angles
The logic is <1 and <2 are a linear pair, therefore their non-adjacent sides form opposite rays. Opposite rays have a measure of 180 degrees. The measure of the angle formed by the opposite rays is just the sum of the measures of <1 and <2. The sum of the measures of <1 and <2 are therefore 180 degress. Thus <1 and <2 are supplementary.