1. **State the problem:** We have a circle with center G and points I, V, E, N on the circumference. Given arcs: - Arc VE = $3x - 5$ - Arc VI = $4x - 1$ - Arc IN = $4x$ - Arc NE = $5x + 4$ We need to find the value of $x$, the measure of Arc IN, Arc VE, Angle IGN, and Angle IGV. 2. **Use the property of circle arcs:** The total circumference is 360 degrees. The arcs around the circle add up to 360 degrees: $$ (3x - 5) + (4x - 1) + 4x + (5x + 4) = 360 $$ 3. **Simplify the equation:** $$ 3x - 5 + 4x - 1 + 4x + 5x + 4 = 360 $$ $$ (3x + 4x + 4x + 5x) + (-5 -1 + 4) = 360 $$ $$ 16x - 2 = 360 $$ 4. **Solve for $x$:** $$ 16x - 2 = 360 $$ $$ 16x = 360 + 2 $$ $$ 16x = 362 $$ $$ x = \frac{362}{16} $$ $$ x = \frac{\cancel{362}}{\cancel{16}} = 22.625 $$ 5. **Find Arc IN:** $$ \text{Arc IN} = 4x = 4 \times 22.625 = 90.5^\circ $$ 6. **Find Arc VE:** $$ \text{Arc VE} = 3x - 5 = 3 \times 22.625 - 5 = 67.875 - 5 = 62.875^\circ $$ 7. **Find Angle IGN:** Angle at center G subtended by Arc IN and Arc NE. Since IGN is formed by radii GI and GN, its measure equals the measure of the intercepted arc IN + NE: $$ \text{Arc IN} + \text{Arc NE} = 4x + (5x + 4) = 9x + 4 $$ $$ = 9 \times 22.625 + 4 = 203.625 + 4 = 207.625^\circ $$ 8. **Find Angle IGV:** Angle IGV is formed by radii GI and GV, subtending arcs IV and VE. The measure of angle IGV equals the measure of arc IV + VE: $$ (4x - 1) + (3x - 5) = 7x - 6 $$ $$ = 7 \times 22.625 - 6 = 158.375 - 6 = 152.375^\circ $$ **Final answers:** - $x = 22.625$ - Arc IN = $90.5^\circ$ - Arc VE = $62.875^\circ$ - Angle IGN = $207.625^\circ$ - Angle IGV = $152.375^\circ$