Question: Question 5 3 Marks 5.1. Determine the exact value of $\theta$ in the diagram. $\frac{\pi}{7}$ $\frac{5\pi}{14}$ $\theta$ (3 Marks) Graph: a triangle in the center-right of the page with a horizontal base, a vertical right side, and a slanted left side; the top angle is labeled $\frac{\pi}{7}$, the bottom-left angle is labeled $\frac{5\pi}{14}$, and the bottom-right angle is labeled $\theta$. position_hint=center

1. **State the problem:** We are given a triangle with three angles: $\frac{\pi}{7}$, $\frac{5\pi}{14}$, and $\theta$. We need to find the exact value of $\theta$. 2. **Recall the triangle angle sum rule:** The sum of the interior angles of any triangle is always $\pi$ radians. 3. **Write the equation for the sum of angles:** $$\frac{\pi}{7} + \frac{5\pi}{14} + \theta = \pi$$ 4. **Find a common denominator to add the fractions:** The denominators are 7 and 14. The least common denominator is 14. Convert $\frac{\pi}{7}$ to $\frac{2\pi}{14}$: $$\frac{\pi}{7} = \frac{2\pi}{14}$$ 5. **Add the known angles:** $$\frac{2\pi}{14} + \frac{5\pi}{14} = \frac{7\pi}{14}$$ 6. **Substitute back into the equation:** $$\frac{7\pi}{14} + \theta = \pi$$ 7. **Isolate $\theta$:** $$\theta = \pi - \frac{7\pi}{14}$$ 8. **Rewrite $\pi$ as $\frac{14\pi}{14}$ to subtract:** $$\theta = \frac{14\pi}{14} - \frac{7\pi}{14}$$ 9. **Subtract the fractions:** $$\theta = \frac{14\pi - 7\pi}{14} = \frac{7\pi}{14}$$ 10. **Simplify the fraction:** $$\theta = \frac{7\pi}{14} = \frac{\cancel{7}\pi}{2\cancel{14}} = \frac{\pi}{2}$$ **Final answer:** $$\boxed{\theta = \frac{\pi}{2}}$$