Logical way to think about it without guess and check :

①∠x is a multiple of 3.

Since the ratio given is 1 : 3 : 2,

1 + 3 + 2 = 6

Sum of ∠x,y,z is equal to 6 times of ∠x

This tells us that the sum must be a multiple of 18 since 3 × 6 = 18

②We need to add ∠w to sum of ∠x,y,z (a multiple of 5 to a multiple of 18) to get a total of 360° (angles at a point).

Now 180° is obviously a multiple of 18 but since the sum of ∠x,y,z is greater than ∠w, and looking at the figure, we know that the sum must be greater than 180° .

This means ∠w must be less than 180°

③Now a multiple of 5 ends with either 5 or 0.
(5,10,15,20,25,30,...)

Multiples of 18 are as such:
18,36,54,72,90
108,126,144,162,180
And so on.

It always ends with a 8,6,4,2 or 0.

Since 360° ends with a 0, we know that ∠w cannot end with a 5.

Because in the ones place, 5 added to any of those five digits cannot give a ones place that ends with 0.

In this case, we can only get 0 by adding 0 to 0.


④So for the sum of ∠x,y,z, we need to only consider multiples of 18 that end with a 0 at the back.

Looking back at ②, we realise this occurs every multiple of 90

90,180,270,360,450...

From this it is clear that the sum must be 270°.

For ∠w we only need to consider those ending with a 0 and is less than 180°

Since we already know the sum of ∠x,y,z is 270°,

Then ∠w = 360° - 270° = 90°

Alternatively,

①Sum of ∠x,y,z is a multiple of 18 as explained previously. ∠w is a multiple of 5 (given)

② Since angles at a point add to 360° and sum of ∠x,y,z is greater than ∠w,

∠w is less than half of 360°, which means it is less than 180° (or a straight angle, which is basically angles on a line)

Sum of ∠x,y,z is more than half of 360°, which means more than 180°.

③ Realise that 360° is a multiple of BOTH 5 and 18.

(Multiples of 5 end with 0 or 5. 360 ends with 0 and = 5 × 72)

(36 is a multiple of 18 since 18 × 2 = 36, and 360 is just 36 × 10 so clearly 360 is also a multiple. 360 = 18 × 20)

④ If we subtract sum of ∠x,y,z from 360°, we get ∠w.

But we are subtracting a multiple of 18 from another multiple of 18. So the result is also a multiple of 18.

Likewise, if we subtract ∠w from 360°, we get sum of ∠x,y,z.

But we are subtracting a multiple of 5 from another multiple of 5 So the result is also a multiple of 5.

This tells that BOTH ∠w and the sum of ∠x,y,z are multiples of BOTH 5 and 18.

⑤ If a number is both a multiple of 5 and 18, it has to be a multiple of 90 as they have no common factors.

The lowest common multiple of 5 and 18
= 5 × 18 = 90

So this tells us that ∠w and sum∠x,y,z have to be multiples of 90.

Only possibility is : ∠w = 90° and sum∠x,y,z = 270°