1. **State the problem:** We have parallelogram KLMN with sides LM = 5 cm, MN = 3 cm, and diagonal LN = 3.2 cm. We need to find the measure of angle LKN to the nearest degree (Part A). 2. **Identify the triangle and angle:** Angle LKN is at vertex K, formed by points L, K, and N. Since KLMN is a parallelogram, points K, L, and N form a triangle with sides KL, KN, and LN. 3. **Find lengths KL and KN:** Since LM = 5 cm and MN = 3 cm, and KLMN is a parallelogram, KL = MN = 3 cm and KN = LM = 5 cm. 4. **Use the Law of Cosines to find angle LKN:** $$\text{Law of Cosines: } c^2 = a^2 + b^2 - 2ab\cos(C)$$ Here, side opposite angle LKN is LN = 3.2 cm, and sides adjacent to angle LKN are KL = 3 cm and KN = 5 cm. So, $$3.2^2 = 3^2 + 5^2 - 2 \times 3 \times 5 \times \cos(\angle LKN)$$ 5. **Calculate:** $$10.24 = 9 + 25 - 30 \cos(\angle LKN)$$ $$10.24 = 34 - 30 \cos(\angle LKN)$$ 6. **Isolate cosine:** $$30 \cos(\angle LKN) = 34 - 10.24$$ $$30 \cos(\angle LKN) = 23.76$$ $$\cos(\angle LKN) = \frac{23.76}{30}$$ $$\cos(\angle LKN) = 0.792$$ 7. **Find angle:** $$\angle LKN = \cos^{-1}(0.792) \approx 37.5^\circ$$ Rounded to the nearest degree, angle LKN is **38º**. --- **Part B:** 8. **Calculate the area of the shaded region:** The shaded region is inside the parallelogram. The area of parallelogram KLMN is base times height. Base LM = 5 cm. Height can be found using the sine of angle LKN (since LN is diagonal, we use the sine of angle between sides KL and KN): $$\text{height} = KN \times \sin(\angle LKN) = 3 \times \sin(38^\circ)$$ $$\sin(38^\circ) \approx 0.6157$$ $$\text{height} = 3 \times 0.6157 = 1.847$$ Area: $$\text{Area} = \text{base} \times \text{height} = 5 \times 1.847 = 9.235$$ Rounded to nearest whole number, area is **9 cm²**. 9. **Calculate the perimeter of the shaded region:** The shaded region is the parallelogram, so perimeter is sum of all sides: $$2( LM + MN ) = 2(5 + 3) = 2 \times 8 = 16$$ Perimeter rounded to nearest whole number is **16 cm**. --- **Final answers:** - Part A: Angle LKN = **38º** - Part B: Area = **9 cm²**, Perimeter = **16 cm**