1. **Problem statement:** Given the circle diagrams and angle/arc measures, find the values of $x$ and the measures of the angles. 2. **Step 1: Solve for $x$ using the equation from the first diagram:** Given: $$166 = 9x + 40$$ Subtract 40 from both sides: $$166 - 40 = 9x + \cancel{40} - \cancel{40}$$ $$126 = 9x$$ Divide both sides by 9: $$\frac{126}{9} = \frac{9x}{9}$$ $$14 = x$$ 3. **Step 2: Find $m\angle ABC$ using the expressions given:** Given: $$m\angle ABC = (16x - 26)^\circ$$ $$m\angle ABC = (5x + 11)^\circ$$ Since these represent the same angle, set equal: $$16x - 26 = 5x + 11$$ Subtract $5x$ from both sides: $$16x - 5x - 26 = 5x - 5x + 11$$ $$11x - 26 = 11$$ Add 26 to both sides: $$11x - 26 + 26 = 11 + 26$$ $$11x = 37$$ Divide both sides by 11: $$x = \frac{37}{11} \approx 3.36$$ Calculate $m\angle ABC$: $$m\angle ABC = 16x - 26 = 16 \times 3.36 - 26 = 53.76 - 26 = 27.76^\circ$$ 4. **Step 3: Find $m\angle WX$ using the expressions given:** Given: $$m\angle WX = (6x - 7)^\circ$$ $$m\angle WX = (10x - 47)^\circ$$ Set equal: $$6x - 7 = 10x - 47$$ Subtract $6x$ from both sides: $$6x - 6x - 7 = 10x - 6x - 47$$ $$-7 = 4x - 47$$ Add 47 to both sides: $$-7 + 47 = 4x - 47 + 47$$ $$40 = 4x$$ Divide both sides by 4: $$x = 10$$ Calculate $m\angle WX$: $$m\angle WX = 6x - 7 = 6 \times 10 - 7 = 60 - 7 = 53^\circ$$ 5. **Step 4: Find $m\angle DGF$ using the expressions given:** Given: $$m\angle DGF = (4x + 7)^\circ$$ $$m\angle DGF = (8x - 31)^\circ$$ Set equal: $$4x + 7 = 8x - 31$$ Subtract $4x$ from both sides: $$4x + 7 - 4x = 8x - 4x - 31$$ $$7 = 4x - 31$$ Add 31 to both sides: $$7 + 31 = 4x - 31 + 31$$ $$38 = 4x$$ Divide both sides by 4: $$x = 9.5$$ Calculate $m\angle DGF$: $$m\angle DGF = 4x + 7 = 4 \times 9.5 + 7 = 38 + 7 = 45^\circ$$ 6. **Step 5: Given values from the problem:** - $mTS = 102^\circ$ - $mQT = 54^\circ$ These are given and do not require calculation. **Final answers:** - $x = 14$ (from first equation) - $m\angle ABC \approx 27.76^\circ$ - $m\angle WX = 53^\circ$ - $m\angle DGF = 45^\circ$ - $mTS = 102^\circ$ - $mQT = 54^\circ$