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To find the area of a triangle when two sides and the included angle are given, we can use the formula:

\[ \text{Area} = \frac{1}{2}ab \sin(C) \]

where \(a\) and \(b\) are the lengths of the two sides, and \(C\) is the included angle.

Given:
- \(a = 6 \, \text{yd}\)
- \(b = 7 \, \text{yd}\)
- \(C = 156^\circ\)

Now, we need to calculate the area:

\[ \text{Area} = \frac{1}{2} \times 6 \times 7 \times \sin(156^\circ) \]

First, we find \(\sin(156^\circ)\):
\(\sin(156^\circ) = \sin(180^\circ - 24^\circ) = \sin(24^\circ)\)

Using a calculator:

\[
\sin(24^\circ) \approx 0.4067
\]

Now, we can plug this value back into our formula:

\[
\text{Area} = \frac{1}{2} \times 6 \times 7 \times 0.4067 \approx \frac{1}{2} \times 42 \times 0.4067 \approx 8.5407 \, \text{yd}^2
\]

Thus, the area of the triangle is approximately \(8.54 \, \text{yd}^2\).

Would you like more details or have any questions?

Here are 5 questions you can ask next:
1. How do you find the area of a triangle with different given conditions?
2. Can you explain the Law of Sines?
3. What is the Law of Cosines and how is it used?
4. How do you find the height of a triangle given its area and base?
5. What are the different types of triangles and their properties?

**Tip:** Remember to always check your angle mode (degrees or radians) on your calculator to ensure accurate trigonometric calculations.

Learn how to find the area of a triangle when given two sides and the included angle using trigonometric principles. This problem includes detailed calculations using the formula Area = (1/2)ab sin(C), where a = 6 yd, b = 7 yd, and C = 156 degrees. Suitable for high school students in Grades 10-12.

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