1. **Problem Statement:** Find the length of side JK in each triangle given angles and some side lengths. 2. **Key Formula:** Use the Law of Sines: $$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$ where $a,b,c$ are sides opposite angles $A,B,C$ respectively. 3. **Important Rules:** - The sum of angles in a triangle is always 180°. - Identify the known sides and angles. - Use the Law of Sines to find unknown sides. --- ### a) Triangle JNK - Given: $\angle N = 40^\circ$, $\angle K = 20^\circ$, side $JM = 5.0$ cm (assuming $JM$ corresponds to side opposite $N$ or $K$; clarify if $JM$ is side JK or another side). - Step 1: Find $\angle J = 180^\circ - 40^\circ - 20^\circ = 120^\circ$. - Step 2: Use Law of Sines to find $JK$ (side opposite $N$ or $K$ depending on labeling). ### b) Triangle JAK - Given: side $AK = 3.0$ cm, $\angle A = 15^\circ$, $\angle B = 60^\circ$ (assuming $B$ is $J$ or $K$; clarify). - Step 1: Find $\angle K = 180^\circ - 15^\circ - 60^\circ = 105^\circ$. - Step 2: Use Law of Sines to find $JK$. ### c) Triangle JCDK - Given: side $CD = 3.0$ cm, $\angle C = 35^\circ$, $\angle D = 30^\circ$. - Step 1: Find $\angle J = 180^\circ - 35^\circ - 30^\circ = 115^\circ$. - Step 2: Use Law of Sines to find $JK$. ### d) Triangle FKJE - Given: side $FE = 4.2$ cm, $\angle F = 35^\circ$, $\angle J = 60^\circ$. - Step 1: Find $\angle K = 180^\circ - 35^\circ - 60^\circ = 85^\circ$. - Step 2: Use Law of Sines to find $JK$. --- **Example detailed solution for a):** 1. Calculate $\angle J = 180^\circ - 40^\circ - 20^\circ = 120^\circ$. 2. Assume side $JM = 5.0$ cm is opposite $\angle K = 20^\circ$. 3. Apply Law of Sines: $$\frac{JK}{\sin 40^\circ} = \frac{5.0}{\sin 20^\circ}$$ 4. Solve for $JK$: $$JK = \frac{5.0 \times \sin 40^\circ}{\sin 20^\circ}$$ 5. Calculate values: $$\sin 40^\circ \approx 0.6428, \quad \sin 20^\circ \approx 0.3420$$ $$JK = \frac{5.0 \times 0.6428}{0.3420} \approx 9.4 \text{ cm}$$ **Answer:** Length of $JK$ is approximately 9.4 cm. --- Repeat similar steps for b), c), and d) using the Law of Sines and angle sum rule.