The Triangle Angle Sum Theorem states that the sum of the angles in a triangle is always \(180^\circ\). Let's denote the three angles of the triangle as follows:
- Angle at the top: \( (8x + 5)^\circ \)
- Angle at the left corner: \( (5x - 1)^\circ \)
- Angle at the right corner: \( (4x + 6)^\circ \)
According to the Triangle Angle Sum Theorem, we can set up the following equation:
\[ (8x + 5) + (5x - 1) + (4x + 6) = 180 \]
Now, let's simplify the equation step by step:
- Combine like terms:
- \( 8x + 5x + 4x = 17x \)
- \( 5 - 1 + 6 = 10 \)
So the equation becomes:
\[ 17x + 10 = 180 \]
- Subtract 10 from both sides:
\[ 17x = 170 \]
- Divide by 17:
\[ x = 10 \]
Now that we know \(x\), we can find the measures of each angle by substituting \(x = 10\) back into the angle expressions:
-
Angle at the top: \[ 8x + 5 = 8(10) + 5 = 80 + 5 = 85^\circ \]
-
Angle at the left corner: \[ 5x - 1 = 5(10) - 1 = 50 - 1 = 49^\circ \]
-
Angle at the right corner: \[ 4x + 6 = 4(10) + 6 = 40 + 6 = 46^\circ \]
Now we have the measures of the angles: \(85^\circ\), \(49^\circ\), and \(46^\circ\).
Thus, the angles of the triangle are:
- \(46^\circ\)
- \(49^\circ\)
- \(85^\circ\)
The correct response from the options provided is: 46°, 49°, and 85°.
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