By applying the Triangle Sum Theorem and the Exterior Angle Theorem to triangle PQR, we can determine that the measure of angle QRS is 48°.
The student is working on a problem involving triangle angles and algebraic expressions in geometry.
To find the measure of angle QRS in triangle PQR, we must apply the Triangle Sum Theorem, which states that the sum of the interior angles in any triangle is 180°.
Given the expressions for angles PQR and RPQ, we can find the third interior angle (angle PRQ) by subtracting the sum of angles PQR and RPQ from 180°.
First, we need to calculate the measure of angle PRQ using the Triangle Sum Theorem:
m∠PQR + m∠RPQ + m∠PRQ = 180°
(3x + 20)° + (3x - 8)° + m∠PRQ = 180°
6x + 12 + m∠PRQ = 180°
m∠PRQ = 180° - (6x + 12)
Next, since line PR is extended through R to S, angle QRS is an exterior angle to triangle PQR, and by the Exterior Angle Theorem, it is equal to the sum of the two opposite interior angles (PQR and RPQ). Thus:
m∠QRS = m∠PQR + m∠RPQ
m∠QRS = (3x + 20)° + (3x - 8)°
m∠QRS = 6x + 12°
Finally, we substitute this value into the expression given for angle QRS to find the value of 'x', and then calculate the measure of angle QRS.
(10x - 12)° = 6x + 12°
4x = 24°
x = 6°
Now we can plug this 'x' value into the expression for angle QRS:
m∠QRS = (10(6) - 12)°
m∠QRS = (60 - 12)°
m∠QRS = 48°
So, the measure of angle QRS is 48°.