To prove that the bisectors of two adjacent supplementary angles include a right angle, let's consider the following scenario:


- Two adjacent supplementary angles, say angle AOB and angle BOC.
- The bisector of angle AOB is AD.
- The bisector of angle BOC is BE.

We need to prove that angle ADB is a right angle.



Step 1:
Draw the line segment OC extending from point O towards C, such that angle AOC is a straight line (180 degrees).

Step 2:
Since angle AOB and angle BOC are adjacent supplementary angles, they add up to 180 degrees.

Step 3:
The bisector of angle AOB, AD, divides angle AOB into two equal halves, creating angle ADB and angle BDA.

Step 4:
Similarly, the bisector of angle BOC, BE, divides angle BOC into two equal halves, creating angle BDE and angle EDC.

Step 5:
Since angle AOB and angle BOC are supplementary, angle AOC is a straight line.

Step 6:
From step 3, we know that angle ADB and angle BDA are equal in measure.

Step 7:
From step 4, we know that angle BDE and angle EDC are equal in measure.

Step 8:
From step 6 and step 7, we can conclude that angle BDA + angle BDE + angle EDC = 180 degrees.

Step 9:
Since angle BDA + angle BDE + angle EDC is equal to the measure of angle AOC (180 degrees), we can conclude that angle BDA + angle BDE + angle EDC = angle AOC.

Step 10:
From step 5 and step 9, we can conclude that angle BDA + angle BDE + angle EDC = 180 degrees = angle AOC.

Step 11:
Since angle ADB and angle BDA are equal in measure (from step 3), and angle BDE and angle EDC are equal in measure (from step 4), we can conclude that angle ADB + angle BDE + angle EDC = 180 degrees.

Step 12:
From step 11, we know that angle ADB + angle BDE + angle EDC = 180 degrees = angle AOC.

Step 13:
From step 12, we can conclude that angle ADB + angle BDE + angle EDC = angle AOC, which implies that angle ADB is equal to angle AOC.

Step 14:
Since angle AOC is a straight line (180 degrees), angle ADB must