To find the measures of each angle in the triangle, we will use the Triangle Angle Sum Theorem, which states that the sum of the angles in a triangle equals 180 degrees.
The angles of the triangle are given as:
- \( (5x - 1)^\circ \)
- \( (8x + 5)^\circ \)
- \( (4x + 6)^\circ \)
We can set up the equation:
\[ (5x - 1) + (8x + 5) + (4x + 6) = 180 \]
Now let's combine like terms:
\[ 5x + 8x + 4x - 1 + 5 + 6 = 180 \]
\[ (5x + 8x + 4x) + (-1 + 5 + 6) = 180 \]
\[ 17x + 10 = 180 \]
Now, we will isolate \( x \):
\[ 17x = 180 - 10 \]
\[ 17x = 170 \]
\[ x = \frac{170}{17} = 10 \]
Now we will substitute \( x = 10 \) back into each angle:
-
First angle: \[ 5x - 1 = 5(10) - 1 = 50 - 1 = 49^\circ \]
-
Second angle: \[ 8x + 5 = 8(10) + 5 = 80 + 5 = 85^\circ \]
-
Third angle: \[ 4x + 6 = 4(10) + 6 = 40 + 6 = 46^\circ \]
Hence, the measures of each angle are \( 49^\circ \), \( 85^\circ \), and \( 46^\circ \).
Therefore, the correct response is:
46°, 49°, and 85°