Given that RS and RT are tangent lines to the circle with center O, the radii OS and OT are perpendicular to the tangents at the points of tangency S and T, respectively.

Step 1: Identify the properties of tangents from an external point. Given that RS and RT are tangent lines to the circle with center O, the radii OS and OT are perpendicular to the tangents at the points of tangency S and T, respectively. Therefore, we have: m( OSR) = 90^ m( OTR) = 90^ Step 2: Find the measure of the central angle SOT. Consider the quadrilateral OSRT. The sum of the interior angles of a quadrilateral is 360^. We are given m( SRT) = 40^. So, we can write the equation: m( OSR) + m( SRT) + m( OTR) + m( SOT) = 360^ Substitute the known angle measures: 90^ + 40^ + 90^ + m( SOT) = 360^ 220^ + m( SOT) = 360^ Subtract 220^ from both sides: m( SOT) = 360^ - 220^ m( SOT) = 140^ Step 3: Determine the measure of arc ST. The measure of a central angle is equal to the measure of its intercepted arc. Since SOT is a central angle subtending arc ST: m(arc ST) = m( SOT) = 140^ Step 4: Find the measure of the inscribed angle TUS. The measure of an inscribed angle is half the measure of its intercepted arc. TUS is an inscribed angle that subtends arc ST. m( TUS) = (1)/(2) m(arc ST) Substitute the measure of arc ST: m( TUS) = (1)/(2) × 140^ m( TUS) = 70^ The measure of TUS is 70^. Send me the next one 📸