1. **Problem 8:** \(\angle A\) is obtuse and \(\angle A = (x + 20)^\circ\). Find the limits of \(x\). - An obtuse angle is greater than 90° and less than 180°. - So, \(90 < x + 20 < 180\). - Subtract 20 from all parts: \(90 - 20 < x < 180 - 20\). - Simplify: \(70 < x < 160\). **Answer:** \(x\) must satisfy \(70 < x < 160\). 2. **Problem 9:** Find the complement and supplement of each angle. - Complement of an angle \(\theta\) is \(90^\circ - \theta\). - Supplement of an angle \(\theta\) is \(180^\circ - \theta\). **a) For 60°:** - Complement: \(90 - 60 = 30^\circ\) - Supplement: \(180 - 60 = 120^\circ\) **b) For \((x + 10)^\circ\):** - Complement: \(90 - (x + 10) = 80 - x\) - Supplement: \(180 - (x + 10) = 170 - x\) 3. **Problem 10:** The supplement of an angle is 40 more than six times the complement. Find the angle. - Let the angle be \(\theta\). - Complement: \(90 - \theta\) - Supplement: \(180 - \theta\) Given: \(180 - \theta = 6(90 - \theta) + 40\) Step 1: Expand right side: $$180 - \theta = 540 - 6\theta + 40$$ Step 2: Simplify right side: $$180 - \theta = 580 - 6\theta$$ Step 3: Add \(6\theta\) to both sides: $$180 + 5\theta = 580$$ Step 4: Subtract 180 from both sides: $$5\theta = 400$$ Step 5: Divide both sides by 5: $$\theta = 80$$ **Answer:** The angle is \(80^\circ\).