Questions: If m∠angle TSB = 65°, calculate the measure of arc ST.

Solution Steps

The problem states that the measure of angle ∠TSE is 65°.

In the given diagram, ∠TSE is an angle formed by two tangents to the circle from point E. The measure of the angle formed by two tangents from a point outside a circle is equal to half the difference of the measures of the intercepted arcs.

The intercepted arcs in this case are arc ST and arc SRT. Since ∠TSE = 65°, we can set up the equation: \[ \angle TSE = \frac{1}{2} (\text{arc SRT} - \text{arc ST}) \]

Since arc SRT is a major arc and arc ST is a minor arc, we know that: \[ \text{arc SRT} = 360° - \text{arc ST} \] Substitute this into the equation: \[ 65° = \frac{1}{2} (360° - \text{arc ST} - \text{arc ST}) \] \[ 65° = \frac{1}{2} (360° - 2 \cdot \text{arc ST}) \] \[ 65° = 180° - \text{arc ST} \] \[ \text{arc ST} = 180° - 65° \] \[ \text{arc ST} = 115° \]

Final Answer