1. **Problem 1: Find angle PRQ in the triangle with angles expressed in terms of $x$.** Given angles: - Angle at P: $8x + 2$° - Angle at Q: $10x - 19$° - Exterior angle at R: $16x - 7$° 2. **Recall the exterior angle theorem:** The exterior angle at a vertex of a triangle equals the sum of the two opposite interior angles. So, exterior angle at R = angle P + angle Q 3. **Set up the equation:** $$16x - 7 = (8x + 2) + (10x - 19)$$ 4. **Simplify the right side:** $$16x - 7 = 8x + 2 + 10x - 19$$ $$16x - 7 = 18x - 17$$ 5. **Bring all terms to one side:** $$16x - 7 - 18x + 17 = 0$$ $$\cancel{16x} - 7 - \cancel{18x} + 17 = 0$$ $$-2x + 10 = 0$$ 6. **Solve for $x$:** $$-2x = -10$$ $$x = \frac{-10}{-2}$$ $$x = 5$$ 7. **Find angle PRQ, which is angle at R inside the triangle.** Since the exterior angle at R is $16x - 7$, the interior angle at R is supplementary: $$\text{angle } PRQ = 180° - (16x - 7)$$ Substitute $x=5$: $$\text{angle } PRQ = 180 - (16 \times 5 - 7) = 180 - (80 - 7) = 180 - 73 = 107°$$ --- 8. **Problem 2: Determine the type of triangle based on side lengths.** We classify triangles by side lengths: - Equilateral: all sides equal - Isosceles: two sides equal - Scalene: all sides different Also, check if right-angled using Pythagoras theorem: If $a^2 + b^2 = c^2$ for sides $a,b,c$ with $c$ largest, triangle is right-angled. **a) Sides: 16, 30, 34** - Largest side: 34 - Check $16^2 + 30^2 = 256 + 900 = 1156$ - Check $34^2 = 1156$ - Since equal, triangle is right-angled scalene. **b) Sides: 12, 19, 23** - Largest side: 23 - Check $12^2 + 19^2 = 144 + 361 = 505$ - Check $23^2 = 529$ - Not equal, scalene triangle. **c) Sides: 8, 20, 22** - Largest side: 22 - Check $8^2 + 20^2 = 64 + 400 = 464$ - Check $22^2 = 484$ - Not equal, scalene triangle. **d) Sides: 10.5, 14, 17.5** - Largest side: 17.5 - Check $10.5^2 + 14^2 = 110.25 + 196 = 306.25$ - Check $17.5^2 = 306.25$ - Equal, right-angled scalene triangle. --- **Final answers:** - Angle PRQ = $107°$ - Triangle types: a) Right-angled scalene b) Scalene c) Scalene d) Right-angled scalene