1. **State the problem:** Find the value of $x$ and determine if the angles are adjacent or vertical. 2. **Exercise 1:** Angles $120^\circ$ and $x^\circ$ are adjacent on a straight line. - Adjacent angles on a straight line sum to $180^\circ$. - Equation: $$120 + x = 180$$ - Solve: $$x = 180 - 120 = 60$$ - Angles are adjacent. 3. **Exercise 2:** Angles $3x^\circ$ and $(2x + 50)^\circ$ are vertical angles. - Vertical angles are equal. - Equation: $$3x = 2x + 50$$ - Solve: $$3x - 2x = 50 \Rightarrow x = 50$$ - Angles are vertical. 4. **Exercise 3:** Angles $80^\circ$ and $(4x - 140)^\circ$ are vertical angles. - Vertical angles are equal. - Equation: $$80 = 4x - 140$$ - Solve: $$80 + 140 = 4x \Rightarrow 220 = 4x \Rightarrow x = \frac{220}{4} = 55$$ - Angles are vertical. 5. **Exercise 4:** Angles $x^\circ$ and $110^\circ$ are vertical angles. - Vertical angles are equal. - Equation: $$x = 110$$ 6. **Exercise 5:** Angles $x^\circ$ and $151^\circ$ form a straight angle. - Sum is $180^\circ$. - Equation: $$x + 151 = 180$$ - Solve: $$x = 180 - 151 = 29$$ 7. **Exercise 6:** Angles $30^\circ$ and $x^\circ$ are adjacent on a straight line. - Sum is $180^\circ$. - Equation: $$30 + x = 180$$ - Solve: $$x = 150$$ - But answer key says $60$, so check if adjacent or vertical. - Since adjacent, answer is $150$ but key says $60$, so likely a typo or different interpretation. 8. **Exercise 7:** Angles $x^\circ$ and $20^\circ$ are adjacent on a straight line. - Sum is $180^\circ$. - Equation: $$x + 20 = 180$$ - Solve: $$x = 160$$ 9. **Exercise 8:** Angles $x^\circ$ and $45^\circ$ are adjacent on a straight line. - Sum is $180^\circ$. - Equation: $$x + 45 = 180$$ - Solve: $$x = 135$$ **Final answers:** 1. $x=60$, adjacent 2. $x=50$, vertical 3. $x=55$, vertical 4. $x=110$, vertical 5. $x=29$, adjacent 6. $x=150$, adjacent 7. $x=160$, adjacent 8. $x=135$, adjacent