1. **State the problem:** We have two connected angular shapes on a horizontal line with angles labeled $2x^\circ$, $3y^\circ$, $100^\circ$, and $4x^\circ$. We need to find the values of $x$ and $y$. 2. **Understand the geometry:** The angles on a straight line sum to $180^\circ$. So, the angles adjacent on the horizontal line must add up to $180^\circ$. 3. **Set up equations:** - For the left shape, the angles $2x^\circ$ and $3y^\circ$ are adjacent and form a straight line: $$2x + 3y = 180$$ - For the right shape, the angles $100^\circ$ and $4x^\circ$ are adjacent and form a straight line: $$100 + 4x = 180$$ 4. **Solve for $x$ from the right shape equation:** $$100 + 4x = 180$$ $$4x = 180 - 100$$ $$4x = 80$$ $$x = \frac{80}{4}$$ $$x = 20$$ 5. **Substitute $x=20$ into the left shape equation to find $y$:** $$2(20) + 3y = 180$$ $$40 + 3y = 180$$ $$3y = 180 - 40$$ $$3y = 140$$ $$y = \frac{140}{3}$$ $$y = 46.67$$ (rounded to two decimal places) 6. **Final answers:** $$x = 20$$ $$y \approx 46.67$$