To find the measure of angle \( A \) in the triangle using the Law of Cosines, you can use the formula:
\[ \cos(A) = \frac{b^2 + c^2 - a^2}{2bc} \]
In your case, let's assign the sides as follows:
- \( a = 13 , m \) (opposite angle \( A \))
- \( b = 6 , m \) (one side adjacent to angle \( A \))
- \( c = 9 , m \) (the other side adjacent to angle \( A \))
Now, plug in the values into the formula:
\[ \cos(A) = \frac{b^2 + c^2 - a^2}{2bc} = \frac{6^2 + 9^2 - 13^2}{2 \cdot 6 \cdot 9} \]
Calculating each term:
- \( 6^2 = 36 \)
- \( 9^2 = 81 \)
- \( 13^2 = 169 \)
Now substitute these values back into the equation:
\[ \cos(A) = \frac{36 + 81 - 169}{2 \cdot 6 \cdot 9} = \frac{117 - 169}{108} = \frac{-52}{108} \]
Simplifying the fraction:
\[ \cos(A) = \frac{-26}{54} = -\frac{13}{27} \]
Now, we need to find the angle \( A \):
\[ A = \cos^{-1}\left(-\frac{13}{27}\right) \]
Calculating the value:
Using a calculator to find the inverse cosine, you obtain:
\[ A \approx 119 \text{ degrees} \]
Thus, the measure of angle \( A \) is approximately 119 degrees.