First notice that angle C is a right angle, then:
[tex]\measuredangle C=90[/tex]
Now, since this is a right triangle, we can find the measure of angle A using the trigonometric function sine:
[tex]\sin \theta=\frac{\text{opposite side}}{hypotenuse}[/tex]
In this case, we have the following:
[tex]\begin{gathered} a=9 \\ c=12 \\ \sin A=\frac{a}{c}=\frac{9}{12}=\frac{3}{4} \\ \Rightarrow A=\sin ^{-1}(\frac{3}{4})=48.6 \\ \measuredangle A=48.6\degree \end{gathered}[/tex]
Since the sum of the interior angles of a triangle equals 180, we can find the measure of angle B by solving the following equation:
[tex]\begin{gathered} \measuredangle A+\measuredangle B+\measuredangle C=180 \\ \measuredangle A=48.6 \\ \measuredangle C=90 \\ \Rightarrow48.6+\measuredangle B+90=180 \\ \Rightarrow138.6+\measuredangle B=180 \\ \Rightarrow\measuredangle B=180-138.6=41.4 \\ \measuredangle B=41.4\degree \end{gathered}[/tex]
therefore, angle A measures 48.6 degrees and angle B measures 41.4 degrees
