In a regular pentagon PQRST, PR intercept QS at O. Calculate the angle ROS.

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You need to remember that the measures of interior angles of a regular pentagon are equal. You may find its measures using the following formula:

`alpha = (180^o(n - 2))/n`

Replacing the number of sides, `n = 5` , yields:

`alpha = (180^o(5 - 2))/5 = 108^o`

Hence, each interior angle of a regular pentagon measures `108^o` .

The problem provides the information that the diagonals PR and QS meet at the point O.

Since the triangle QOR is an isosceles triangle because the angles `hat(OQR) = hat(ORQ) = 36^o` , you may evaluate the angle `hat(ROQ)` such that:

`hat(ROQ) = 180^o - (36 + 36^o) => hat(ROQ) = 180^o - 72^o = 108^o`

You may find the value of adjacent supplementary angle `hat(ROS) ` using the following formula, such that:

`hat(ROS) = 180^o - 108^o = 72^o`

**Hence, evaluating the measure of the angle `hat(ROS)` yields **`hat(ROS) = 72^o.`

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