From the first figure
Since LR and JV are 2 chords intersected at a point inside the circle, then
[tex]180-m\angle1=\frac{1}{2}\lbrack56+146\rbrack[/tex]
The angle next to <1 and form a line JV with it
[tex]\begin{gathered} 180-m\angle1=\frac{1}{2}\lbrack202\rbrack \\ 180-m\angle1=101 \end{gathered}[/tex]
Add m<1 to both sides and subtract 101 from both sides
[tex]\begin{gathered} 180-m\angle1+m\angle1-101=101-101+m\angle1 \\ 79^{\circ}=m\angle1 \end{gathered}[/tex]
In the second figure
Since TU is a tangent to the circle at point T
Since ST is a chord in the circle
Then angle of tangency subtended by the major arc ST, Its measure is half the measure of the subtended arc.
Since the major arc ST = the measure of the circle - the measure of the minor arc ST
[tex]m\angle2=\frac{1}{2}\lbrack360-arcST\rbrack[/tex]
Since the measure of the minor arc ST is 134 degrees, then
[tex]\begin{gathered} m\angle2=\frac{1}{2}\lbrack360-134\rbrack \\ m\angle2=\frac{1}{2}\lbrack226\rbrack \\ m\angle2=113^{\circ} \end{gathered}[/tex]
