1.

Take the full triangle.

LP +

LR +

LT = 180

LR +

LT = 90. since

LP = 90. -------(1)

In isosceles triangle RQS,

LRSQ =

LRQS

Hence Sum of angles in triangle RQS

LR +

LRQS +

LRSQ = 180

LR = 180 - 2(

LRSQ) --------------(2) since

LRSQ =

LRQS

In isosceles triangle TUS, similarly

LT = 180 - 2(

LUST) ------------(3)

From the figure we know that,

LUST +

Lx +

LQSR = 180

LQSR = 180 - x -

LUST -------------------(4)

Substitue (2) (3) and (4) in (1)

we get,

180 - 2*

LQSR + 180 - 2*(

LUST) = 90

360 - 2*(

LQSR) - 2*(180 - x -

LUST) = 90

On cancelling it out,

2x = 90 alone remains.

x = 45.

Phew... !!

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D- Day December 30 2011. Hoping for the happiest new year celebrations !

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