To solve for $x$ in both triangles, we'll use the law of sines, which states:

$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$

Here is the process for each triangle:

First Triangle:

1. Given:

   • One side: $4.9 \, \text{cm}$
   • Angles: $85^\circ, 34^\circ, \text{and } 61^\circ$ (since $61^\circ = 180^\circ - 85^\circ - 34^\circ$).

2. Law of Sines: $\frac{x}{\sin 85^\circ} = \frac{4.9}{\sin 61^\circ}$

3. Solve for $x$: $x = \frac{4.9 \cdot \sin 85^\circ}{\sin 61^\circ}$

Second Triangle:

1. Given:

   • One side: $6.9 \, \text{cm}$
   • Angles: $79^\circ, 69^\circ, \text{and } 32^\circ$ (since $32^\circ = 180^\circ - 79^\circ - 69^\circ$).

2. Law of Sines: $\frac{x}{\sin 69^\circ} = \frac{6.9}{\sin 32^\circ}$

3. Solve for $x$: $x = \frac{6.9 \cdot \sin 69^\circ}{\sin 32^\circ}$

Let me calculate the exact values for you.import math

First Triangle

side1 = 4.9 angle1_opposite = 85 # in degrees angle1_given = 61 # in degrees

Calculate x for first triangle using the Law of Sines

x1 = (side1 * math.sin(math.radians(angle1_opposite))) / math.sin(math.radians(angle1_given))

Second Triangle

side2 = 6.9 angle2_opposite = 69 # in degrees angle2_given = 32 # in degrees

Calculate x for second triangle using the Law of Sines

x2 = (side2 * math.sin(math.radians(angle2_opposite))) / math.sin(math.radians(angle2_given))

x1, x2(5.581115976239453, 12.15601201468277)(5.581115976239453, 12.15601201468277)successThe solutions for the triangles are as follows:

1. For the first triangle, $x = 5.6 \, \text{cm}$ (rounded to 1 decimal place).
2. For the second triangle, $x = 12.2 \, \text{cm}$ (rounded to 1 decimal place).

Would you like further explanations or assistance? Here are some related questions to consider:

1. How does the Law of Sines apply to obtuse triangles?
2. Can you solve for angles using the Law of Sines if all sides are known?
3. What are the limitations of the Law of Sines compared to the Law of Cosines?
4. How would the solutions change if the angles were given in radians?
5. How can the Law of Sines be used in real-life applications, such as navigation?

Tip: Always verify the sum of the angles in a triangle to ensure calculations are accurate.