Two congruent triangles, ΔQRS and ΔWXY, with given angles and side lengths. Determine the measure of angle W.

We are told that triangle QRS is congruent to triangle WXY. That means all corresponding sides and angles are equal.

Let’s match the vertices based on the order:
△QRS ≅ △WXY
So:
Q ↔ W
R ↔ X
S ↔ Y

That means:
∠Q = ∠W
∠R = ∠X
S = ∠Y

From the diagram:
In △QRS:
- ∠Q = 63°
- ∠S = 37°
- Side QR = 50 mm

In △WXY:
- ∠X = 80° → which corresponds to ∠R (since R ↔ X)
- Side XY = 75 mm → but we don’t need this for angle calculation
- Side YW = 83 mm → also not needed for angle

Wait — let’s double-check the correspondence.

If △QRS ≅ △WXY, then:

Vertex Q corresponds to vertex W
Vertex R corresponds to vertex X
Vertex S corresponds to vertex Y

Therefore:

Angle at Q (∠Q) corresponds to angle at W (∠W)
Angle at R (∠R) corresponds to angle at X (∠X)
Angle at S (∠S) corresponds to angle at Y (∠Y)

We are given in △QRS:
∠Q = 63°
∠S = 37°
So we can find ∠R using triangle angle sum:

Sum of angles in a triangle = 180°
So ∠R = 180° - ∠Q - ∠S = 180° - 63° - 37° = 80°

Now, since ∠R corresponds to ∠X, and we’re told ∠X = 80°, that matches! Good.

Now, we want m∠W. Since ∠W corresponds to ∠Q, and ∠Q = 63°, then ∠W must also be 63°.

Alternatively, we could compute from △WXY directly if we knew two angles, but we only know one angle (∠X = 80°). But because of congruence, we don’t need to — we just map the known angles.

So:
m∠W = m∠Q = 63°

Final Answer:
63