Question

which of the following statements about the image below is true?

∠tru and ∠urs are supplementary.

line vs and line qs are parallel.

the measure of ∠tru is 72°.

line tx and line vs are perpendicular.

Explanation:

Step1: Analyze the first option

Supplementary angles add up to \(180^{\circ}\). \(\angle TRU\) and \(\angle URS\) do not form a linear - pair. So, they are not supplementary.

Step2: Analyze the second option

For two lines to be parallel, corresponding angles, alternate - interior angles, or alternate - exterior angles should be equal. There is no information (such as equal angles due to a transversal) to suggest that line \(VS\) and line \(QS\) are parallel.

Step3: Analyze the third option

Since \(\angle TRQ = 18^{\circ}\) and \(\angle QRU = 90^{\circ}\), then \(\angle TRU=180^{\circ}-\angle TRQ - \angle QRU\). But also, using the property of parallel lines (if we consider the right - angle relationships and the given \(72^{\circ}\) angle).
We know that \(\angle TRU = 72^{\circ}\) (because \(\angle TRU\) and the \(72^{\circ}\) angle are congruent due to the right - angle and \(18^{\circ}\) angle relationships: \(90 - 18=72\) and using angle - addition and complementary/supplementary relationships in the figure).

Step4: Analyze the fourth option

For two lines to be perpendicular, the angle between them should be \(90^{\circ}\). There is no indication that the angle between line \(TX\) and line \(VS\) is \(90^{\circ}\).

Answer:

The measure of \(\angle TRU\) is \(72^{\circ}\).