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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.

Supplementary angles add up to 180 degrees. When placed next to one another, two supplementary angles form a 180 degree angle.

A 180 degree angle is a line.

If two angles are linear, then by definition, they are two angles placed next to one another, that form a straight line. That is, the two angles sum to 180 degrees, so they are therefore supplementary.