To prove that ST is equal to UT you will need to prove that the triangle STU is isosceles. This can be done by finding the angles.

PQRST is a regular pentagon, so the angle TSU is the exterior angle of a regular pentagon.
Exterior angles in any polygon add up to 360°.
A regular pentagon has 5 equal exterior angles.

$T \hat{S} U = \frac{360}{5} = 72 °$

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To find the size of angle SUT, you will need to consider circle theorems.

Begin by joining the point O to the points T and R, marking all the radiuses of the circle as equal lengths on the diagram.

TP and TR are also tangents to the circle, so by the circle theorem 'A tangent always meets a radius at 90°', angles ORQ and OTP are both 90°.

$O \hat{R} Q = O \hat{T} P = 90 °$ (angle between tangent and radius)

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Mark these onto the diagram.

QLY1pOQ2_4-8-edexcel-gcse-vh-q3-diagram-2

The sum of the interior angles in any polygon can be found using the formula 180°(n - 2).
A regular pentagon has 5 equal interior angles, therefore these can be found using the formula $\frac{180 (5 - 2)}{5}$ .

$T \hat{P} Q = P \hat{Q} R = \frac{3 (180)}{5} = 108 °$

Add these angles to the diagram.

4-8-edexcel-gcse-vh-q3-diagram-3

PQROT is a pentagon, so the angle ROT can be found by subtracting the other four angles from the sun of the angles in a pentagon.

The sum of the interior angles in any polygon can be found using the formula 180°(n - 2).

$R \hat{O} T + O \hat{T} P + T \hat{P} Q + P \hat{Q} R + Q \hat{R} O = 180 (5 - 2) ° = 540 °$

Substitute in the angles you know to find angle ROT.

$\begin{array}{rcl} R \hat{O} T + 90 ° + 90 ° + 108 ° + 108 ° & = & 540 ° \\ R \hat{O} T + 396 ° & = & 540 ° \\ R \hat{O} T & = & 144 ° \end{array}$

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By the circle theorem, the angle at the centre is twice the angle at the circumference, angle ROT is twice angle RUT.

$R \hat{O} T = 2 (R \hat{U} T) = 2 (S \hat{U} T)$

$S \hat{U} T = \frac{R \hat{O} T}{2} = \frac{144}{2} = 72 °$

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This proves that angle TSU is equal to angle SUT and so triangle STU is isosceles.

Therefore ST = UT
All reasons given clearly [1]

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Angles in Polygons & Parallel Lines