Final answer:
To find m∠PQR in ΔOPQ, use the external angle theorem. After setting up and solving the equation 4x - 10 = 2x + 4, we find that x = 7°. Hence, m∠PQR = 18°.
Step-by-step explanation:
To find the measure of m∠PQR in ΔOPQ, we will use the fact that the sum of the angles in any triangle is 180°. Since ∠OPQ and ∠QOP are angles inside the triangle and m∠PQR is an external angle, m∠PQR is equal to the sum of the two non-adjacent internal angles.
Firstly, we can express the measures of the angles as:
m∠PQR = (4x - 10)°
m∠OPQ = (x + 9)°
m∠QOP = (x - 5)°
Since the sum of angles in a triangle is 180°, we can write the equation:
(x + 9) + (x - 5) + m∠POQ = 180°
However, since m∠POQ is not given, we use the external angle theorem which states that m∠PQR = m∠OPQ + m∠QOP
(4x - 10) = (x + 9) + (x - 5)
Solving the equation:
4x - 10 = 2x + 4
2x = 14
x = 7°
Now, plug x back into the expression for m∠PQR:
m∠PQR = (4(7) - 10)° = (28 - 10)° = 18°
Therefore, the measure of ∠PQR is 18°.