Let’s analyze the given information about the angles in triangle ΔPQR:
Angle PQR (m∠PQR): This angle is given as ((3x + 8)^\circ).
Angle QRS (m∠QRS): This angle is given as ((9x - 17)^\circ).
Angle RPQ (m∠RPQ): This angle is given as ((3x + 2)^\circ).
We know that the sum of angles in a triangle is always (180^\circ). Therefore, we can set up the following equation:
[ m∠PQR + m∠QRS + m∠RPQ = 180^\circ ]
Substitute the given expressions for each angle:
[ (3x + 8) + (9x - 17) + (3x + 2) = 180 ]
Combine like terms:
[ 15x - 7 = 180 ]
Add 7 to both sides:
[ 15x = 187 ]
Now solve for (x):
[ x = \frac{187}{15} ]
Calculate the value of (x):
[ x = 12.47 ]
Now let’s find the measure of angle QRS (m∠QRS):
[ m∠QRS = 9x - 17 = 9(12.47) - 17 = 112.23^\circ ]
Therefore, the measure of angle QRS is approximately (112.23^\circ).
