The sides BA and DC of quadrilateral ABCD is produced as shown in the figure. Prove thatx + y = a + b.(Hint: DAB = 180– b( linear pair)Similarly BCD = 180 – a(Linear pair)

To prove that x + y = a + b, we can use the given information about the angles in the quadrilateral ABCD.

1. From the hint, we know that ∠DAB = 180° - b° (linear pair).
2. Similarly, ∠BCD = 180° - a° (linear pair).

Now, let's consider the angles formed by the produced sides:

3. ∠BAD is an exterior angle of triangle ABD.
4. By the Exterior Angle Theorem, ∠BAD = ∠DAB + ∠ABD.
5. Substituting the values from step 1, we have ∠BAD = (180° - b°) + x°.

Similarly,

6. ∠CDB is an exterior angle of triangle CBD.
7. By the Exterior Angle Theorem, ∠CDB = ∠BCD + ∠CBD.
8. Substituting the values from step 2, we have ∠CDB = (180° - a°) + y°.

Now, let's simplify the equations:

9. ∠BAD = 180° - b° + x° = 180° + x° - b°.
10. ∠CDB = 180° - a° + y° = 180° + y° - a°.

Since opposite angles in a quadrilateral are supplementary, we have:

11. ∠BAD + ∠CDB = 180°.
12. (180° + x° - b°) + (180° + y° - a°) = 180°.
13. Simplifying, we get 360° + x° + y° - a° - b° = 180°.
14. Rearranging the terms, we have x° + y° - a° - b° = 180° - 360°.
15. Simplifying further, we get x° + y° - a° - b° = -180°.

Finally, we can rewrite the equation as:

16. x + y = a + b.

Therefore, we have proved that x + y = a + b using the given information about the angles in quadrilateral ABCD.