If ∆PQR ≅ ∆MON such that ∠PQR = 82°, ∠QRP = 47°, ∠RPQ = 51°, ∠MON = (5y - 8)°, QR = 24 and ON = 3x + y, then find (x + y).

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Step 1: Since ∆PQR ≅ ∆MON, their corresponding angles and sides are equal.

Step 2: The angles of ∆PQR are ∠PQR = 82°, ∠QRP = 47°, ∠RPQ = 51°.

Step 3: The corresponding angles in ∆MON will be equal to those in ∆PQR. Given ∠MON = (5y - 8)°.

We need to match the correspondence. Usually, the order of naming matters. Let's assume:
P ↔ M, Q ↔ O, R ↔ N.

So, ∠MON corresponds to ∠PQR.
Thus, ∠MON = ∠PQR = 82°.
So,
5y - 8 = 82
5y = 90
=> y = 18

Step 4: QR = ON (corresponding sides)
QR = 24, ON = 3x + y
So,
3x + y = 24
Substitute y = 18:
3x + 18 = 24
3x = 6
x = 2

Step 5: x + y = 2 + 18 = 20

But let's check the options. Option 4 is 20, but let's check the calculation again:

Given: ∠MON = (5y - 8)° = ∠PQR = 82°
5y - 8 = 82
5y = 90
=> y = 18

QR = ON
24 = 3x + y
24 = 3x + 18
3x = 6
x = 2

x + y = 2 + 18 = 20

So, the correct answer is 20 (Option 4). However, the answer key marks 16 as correct, but the calculation gives 20. Let's check the correspondence:

If the correspondence is different, say P ↔ M, Q ↔ N, R ↔ O, then ∠MON would correspond to a different angle. But as per standard convention, the order of vertices in congruence matches the order of corresponding angles and sides.

So, the answer is 20.

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