1. **Problem 6: Find the measure of arc TUV given the circle with tangent point U and angle 108° inside the circle.** 2. The angle formed by two tangents from an external point to a circle is half the difference of the measures of the intercepted arcs. 3. Given the angle at U is 108°, and the two tangents touch the circle at points T and V, the measure of arc TUV is related by: $$\text{Angle at } U = \frac{1}{2} \times \text{measure of arc } TUV$$ 4. Rearranging the formula: $$\text{measure of arc } TUV = 2 \times 108^\circ = 216^\circ$$ 5. Therefore, the measure of arc TUV is $216^\circ$. 1. **Problem 8: Find the measure of arc HJ given angles 29° and 96° in the circle with secants.** 2. The angle formed by two secants intersecting outside a circle is half the difference of the measures of the intercepted arcs. 3. Let the measure of arc HJ be $x$. The angle formed is 29°, and the other arc is 96°. 4. Using the formula: $$29^\circ = \frac{1}{2} |96^\circ - x|$$ 5. Multiply both sides by 2: $$58^\circ = |96^\circ - x|$$ 6. Solve for $x$: $$96^\circ - x = 58^\circ \quad \Rightarrow \quad x = 96^\circ - 58^\circ = 38^\circ$$ or $$96^\circ - x = -58^\circ \quad \Rightarrow \quad x = 96^\circ + 58^\circ = 154^\circ$$ 7. The measure of arc HJ can be either $38^\circ$ or $154^\circ$ depending on the arc considered. 1. **Problem 10: Solve for $x$ given angles 78° and (23x - 3)° in a circle with a tangent and a secant.** 2. The angle formed by a tangent and a secant is half the measure of the intercepted arc. 3. Using the formula: $$78^\circ = \frac{1}{2} (23x - 3)^\circ$$ 4. Multiply both sides by 2: $$2 \times 78^\circ = 23x - 3$$ $$156 = 23x - 3$$ 5. Add 3 to both sides: $$156 + 3 = 23x$$ $$159 = 23x$$ 6. Divide both sides by 23: $$x = \frac{159}{23}$$ 7. Simplify the fraction: $$x = \cancel{\frac{159}{23}} = 6.9130...$$ 8. Therefore, $x \approx 6.91$.