Which angle has a measure equal to the sum of the m∠sqr and the m∠qrs?

180° - m∠RSQ

Proof:
Refer to the diagram shown below.
Let
x = m∠SQR
y = m∠QRS
z = m∠RSQ

w =  the external angle (supplement) of m∠RSQ

Because the sum of angles in a triangle is 180°, therefore
x + y + z = 180°
x + y = 180° - z
That is,
m∠SQR + m∠QRS = 180° - z        (1)

Because the sum of angles on one side of straight line is 180°, therefore
w + z = 180°
or
w = 180° - m∠RSQ        (2)

Equate equations (1) and (2) to obtain
m∠SQR + m∠QRS = completes∠RSQ
This completes the proof.