1. **State the problem:** We have a trapezium with a left vertical side of 20 cm, a bottom base of 13 cm, and a top-left angle of 129°. We need to find the perimeter of the trapezium. 2. **Identify known sides and angles:** - Left side (vertical): 20 cm - Bottom base (horizontal): 13 cm - Top-left angle: 129° 3. **Determine the shape and unknown sides:** The trapezium has two vertical sides (left and right). The right side is vertical, so the top-right angle is 90°. The top side is slanted. 4. **Find the length of the top side:** The top side connects the top of the left vertical side to the top of the right vertical side. 5. **Calculate the horizontal distance between the top points:** The bottom base is 13 cm. The top side is shorter because the left side is slanted inward due to the 129° angle. 6. **Use trigonometry to find the horizontal component of the left side:** The angle between the left side and the top side is 129°, so the angle between the left side and the vertical is $180^\circ - 129^\circ = 51^\circ$. The horizontal component (adjacent to 51°) of the left side is: $$20 \times \sin(51^\circ)$$ Calculate: $$20 \times \sin(51^\circ) \approx 20 \times 0.7771 = 15.542$$ cm 7. **Calculate the length of the top side:** The top side length is the bottom base minus the horizontal component of the left side: $$13 - 15.542 = -2.542$$ cm, which is negative, meaning the top side extends beyond the bottom base on the left side. This suggests the trapezium is not a simple shape with parallel sides as described, so instead, we calculate the length of the right side using the vertical difference. 8. **Calculate the vertical component of the left side:** $$20 \times \cos(51^\circ) \approx 20 \times 0.6293 = 12.586$$ cm 9. **Calculate the length of the right side:** Since the right side is vertical and the bottom is 13 cm, the right side length is the vertical height minus the vertical component of the left side: $$20 - 12.586 = 7.414$$ cm 10. **Calculate the length of the top side using Pythagoras:** The top side length is the hypotenuse of a right triangle with horizontal side 13 cm and vertical side 7.414 cm: $$\sqrt{13^2 + 7.414^2} = \sqrt{169 + 54.99} = \sqrt{223.99} \approx 14.97$$ cm 11. **Calculate the perimeter:** Sum of all sides: - Left side: 20 cm - Bottom base: 13 cm - Right side: 7.414 cm - Top side: 14.97 cm $$20 + 13 + 7.414 + 14.97 = 55.384$$ cm 12. **Round to 1 decimal place:** $$55.4$$ cm **Final answer:** The perimeter of the trapezium is **55.4 cm**.