PQRST is a regular pentagon. U is the reflection of R in QS.how big is angle UST
To find the measure of angle UST, we can use the property of reflection. When a point is reflected across a line, the angle of incidence (the angle between the incident ray and the reflected ray) is equal to the angle of reflection (the angle between the reflected ray and the line of reflection).
In this case, U is the reflection of R in QS. So, the measure of angle USR is equal to the measure of angle UQS (angle of incidence), and the measure of angle UST is equal to the measure of angle UQS (angle of reflection).
Since PQRST is a regular pentagon, all of its angles are equal. Let's assume that the measure of each angle is x.
Since the angle USR is equal to the angle UQS, we can write:
x = x
Now, let's find the measure of angle UQS. Since PQRST is a regular pentagon, the sum of all its interior angles is (n - 2) * 180 degrees, where n is the number of sides.
In this case, n = 5 (pentagon), so the sum of the interior angles of the pentagon is (5 - 2) * 180 = 3 * 180 = 540 degrees.
Since all angles of a regular pentagon are equal, each angle measures 540 / 5 = 108 degrees.
Therefore, the measure of angle UQS (and angle USR) is 108 degrees.
Since the measure of angle UST is equal to the measure of angle UQS, the measure of angle UST is also 108 degrees.