What are the measures of angle BOS and angle CAS if UAB = 89° and UAC = 95°? Answer: The measure of angle BOS and angle

CAS can be found by using the fact that the sum of the angles in a triangle is equal to 180 degrees. Let"s start by drawing a diagram to visualize the problem.

\[
\begin{align*}
    B       O       C       A   \\
    \vert       \vert       \vert       \vert   \\
    \vert       \vert       \vert       \vert   \\
    \angle UAB       \angle BOS       \angle CAS       \angle UAC  
\end{align*}
\]

Given that angle UAB is 89 degrees and angle UAC is 95 degrees, we can use the fact that the sum of the angles in a triangle is equal to 180 degrees to find the measures of angle BOS and angle CAS.

In triangle OAB, we have two angles: angle UAB and angle BOS. The sum of these two angles must be equal to the remaining angle, which is angle BOA.

So, angle BOA = angle UAB + angle BOS = 89° + angle BOS.

Similarly, in triangle OAC, we have two angles: angle UAC and angle CAS. The sum of these two angles must be equal to the remaining angle, which is angle COA.

So, angle COA = angle UAC + angle CAS = 95° + angle CAS.

Since angle BOA = angle COA (they are opposite angles), we can set up an equation:

89° + angle BOS = 95° + angle CAS.

To solve for angle BOS and angle CAS, we need to isolate each angle. Let"s start with angle BOS:

angle BOS = 95° + angle CAS - 89°.

Next, let"s solve for angle CAS:

angle CAS = angle BOS + 89° - 95°.

Simplifying the expressions, we have:

angle BOS = angle CAS + 6°.

angle CAS = angle BOS - 6°.

So, the measures of angle BOS and angle CAS are interrelated, and we can express one in terms of the other.