1. **Stating the problem:** We have a right-angled quadrilateral with given side lengths and an angle of 130° at point C. We need to find all missing measures, including side lengths and angles. 2. **Known values:** - AB = 3 m - BD = unknown (vertical segment) - CD = 13.7 m - AE = 55 m - CE = 16 m - Angle at C = 130° 3. **Step 1: Identify right angles and use Pythagoras where applicable.** Since B is a right angle between AB and BD, triangle ABD is right-angled at B. 4. **Step 2: Calculate BD using Pythagoras in triangle ABD:** $$AB^2 + BD^2 = AD^2$$ We don't know AD yet, so let's find AD first. 5. **Step 3: Use triangle CDE with angle 130° at C and sides CD=13.7 m, CE=16 m to find DE using Law of Cosines:** $$DE^2 = CD^2 + CE^2 - 2 \times CD \times CE \times \cos(130^\circ)$$ Calculate: $$DE^2 = 13.7^2 + 16^2 - 2 \times 13.7 \times 16 \times \cos(130^\circ)$$ $$= 187.69 + 256 - 2 \times 13.7 \times 16 \times (-0.6428)$$ $$= 443.69 + 281.5 = 725.19$$ $$DE = \sqrt{725.19} \approx 26.93\,m$$ 6. **Step 4: Use triangle ADE with sides AE=55 m, DE=26.93 m, and AD unknown to find AD using Law of Cosines:** We need angle at D or A, but since we don't have it, let's consider triangle ABD first. 7. **Step 5: Calculate AD using triangle ABD:** Since AB=3 m and BD is vertical, AD is hypotenuse: $$AD = \sqrt{AB^2 + BD^2}$$ We don't know BD yet, so let's find BD. 8. **Step 6: Use triangle BDC (right angle at B) to find BD:** Since BD is vertical and CD=13.7 m horizontal, and angle at B is 90°, triangle BDC is right-angled. 9. **Step 7: Calculate BD using Pythagoras in triangle BDC:** $$BD = \sqrt{CE^2 - CD^2} = \sqrt{16^2 - 13.7^2} = \sqrt{256 - 187.69} = \sqrt{68.31} \approx 8.27\,m$$ 10. **Step 8: Calculate AD using AB=3 m and BD=8.27 m:** $$AD = \sqrt{3^2 + 8.27^2} = \sqrt{9 + 68.39} = \sqrt{77.39} \approx 8.8\,m$$ 11. **Step 9: Verify AE using triangle ADE with sides AD=8.8 m, DE=26.93 m, and AE=55 m:** Check if triangle inequality holds: $$AD + DE = 8.8 + 26.93 = 35.73 < AE = 55$$ This suggests AE is the longest side, consistent. 12. **Step 10: Calculate angle at D in triangle ADE using Law of Cosines:** $$\cos(\angle D) = \frac{AD^2 + DE^2 - AE^2}{2 \times AD \times DE}$$ $$= \frac{8.8^2 + 26.93^2 - 55^2}{2 \times 8.8 \times 26.93} = \frac{77.44 + 725.19 - 3025}{2 \times 8.8 \times 26.93} = \frac{-2222.37}{474.37} \approx -4.69$$ Since cosine cannot be less than -1, this indicates inconsistency in the given data or assumptions. **Conclusion:** Given the data, the missing measures BD and AD are approximately 8.27 m and 8.8 m respectively, and DE is approximately 26.93 m. However, the large length AE=55 m compared to other sides suggests the figure may not be planar or data may be inconsistent. **Final answers:** - BD \approx 8.27 m - AD \approx 8.8 m - DE \approx 26.93 m