1. **Problem 4:** In a right triangle, one acute angle is 9 times the other. Find each acute angle. 2. The sum of acute angles in a right triangle is 90° because the right angle is 90°. 3. Let the smaller acute angle be $x$. Then the larger is $9x$. 4. Equation: $$x + 9x = 90$$ 5. Simplify: $$10x = 90$$ 6. Divide both sides by 10: $$\cancel{10}x = \cancel{10}9$$ 7. Solve for $x$: $$x = 9$$ 8. Larger angle: $$9x = 9 \times 9 = 81$$ 9. **Answer:** The acute angles are $9^\circ$ and $81^\circ$. --- 1. **Problem 5:** In $\triangle DEC$, find length $DE$ given expressions for sides and total angle sum. 2. Sum of angles in triangle: $$12x + 5 + 7x + 25 + 3x + 18 = 180$$ 3. Combine like terms: $$12x + 7x + 3x + 5 + 25 + 18 = 180$$ 4. Simplify: $$22x + 48 = 180$$ 5. Subtract 48 from both sides: $$22x + \cancel{48} - \cancel{48} = 180 - 48$$ 6. Simplify: $$22x = 132$$ 7. Divide both sides by 22: $$\cancel{22}x = \frac{132}{\cancel{22}}$$ 8. Solve for $x$: $$x = 6$$ 9. Find $DE$: $$DE = 12x + 5 = 12 \times 6 + 5 = 72 + 5 = 77$$ 10. **Answer:** $DE = 77$ units. --- 1. **Problem 6:** Find $x$ and the measure of the exterior angle. 2. Given angles: $(3x + 5)^\circ$, $(9x + 7)^\circ$, and $44^\circ$ sum to 180°. 3. Equation: $$(3x + 5) + (9x + 7) + 44 = 180$$ 4. Combine like terms: $$3x + 9x + 5 + 7 + 44 = 180$$ 5. Simplify: $$12x + 56 = 180$$ 6. Subtract 56 from both sides: $$12x + \cancel{56} - \cancel{56} = 180 - 56$$ 7. Simplify: $$12x = 124$$ 8. Divide both sides by 12: $$\cancel{12}x = \frac{124}{\cancel{12}}$$ 9. Solve for $x$: $$x \approx 10.33$$ 10. Find exterior angle: $$(9x + 7) = 9 \times 10.33 + 7 = 93 + 7 = 100$$ 11. **Answer:** $x \approx 10.33$, exterior angle $\approx 100^\circ$.