Question

angle rsu is complementary to angle ust. angle qsr is congruent to angle rsu. which statement is true about angles ust and qsr? they are complementary. they are congruent. they are supplementary. they are obtuse.

Explanation:

Step1: Analyze given angle relationships

We know that \(\angle RSU\) and \(\angle UST\) are complementary (sum to \(90^\circ\)), so \(\angle RSU+\angle UST = 90^\circ\). Also, \(\angle QSR\cong\angle RSU\) (given congruent angles).

Step2: Substitute congruent angle

Since \(\angle QSR=\angle RSU\), substitute \(\angle RSU\) with \(\angle QSR\) in the complementary angle equation: \(\angle QSR+\angle UST = 90^\circ\).

Step3: Determine relationship of \(\angle UST\) and \(\angle QSR\)

If the sum of two angles is \(90^\circ\), they are complementary. So \(\angle UST\) and \(\angle QSR\) are complementary. Wait, no, wait the options: Wait, let's re - check. Wait, the problem is about \(\angle UST\) and \(\angle QSR\). Wait, from the diagram, \(\angle RST\) is a right angle (\(90^\circ\)) because of the right - angle symbol at \(S\) between \(SR\) and \(ST\). So \(\angle RSU+\angle UST = 90^\circ\) (since \(\angle RST = 90^\circ\)). And \(\angle QSR\cong\angle RSU\) (given). So \(\angle QSR+\angle UST=\angle RSU+\angle UST = 90^\circ\), so they are complementary? Wait, but let's check the options. Wait, maybe I misread. Wait, the options are: "They are complementary", "They are congruent", "They are supplementary", "They are obtuse". Wait, no, wait the original problem: Wait, the text says "Angle RSU is complementary to angle UST. Angle QSR is congruent to angle RSU". So \(\angle RSU+\angle UST = 90^\circ\) (complementary), and \(\angle QSR=\angle RSU\) (congruent). So substitute \(\angle RSU\) with \(\angle QSR\) in \(\angle RSU+\angle UST = 90^\circ\), we get \(\angle QSR+\angle UST = 90^\circ\), so they are complementary? But wait, maybe the options are mis - ordered. Wait, no, let's re - evaluate. Wait, maybe the correct relationship is that they are congruent? No, wait \(\angle QSR=\angle RSU\), and \(\angle RSU\) and \(\angle UST\) are complementary, not congruent. Wait, I think I made a mistake. Wait, the right angle is between \(SR\) and \(ST\), so \(\angle RST = 90^\circ\), so \(\angle RSU+\angle UST=90^\circ\) (complementary). And \(\angle QSR\cong\angle RSU\) (given). So \(\angle QSR\) and \(\angle UST\): \(\angle QSR=\angle RSU\), so \(\angle QSR+\angle UST=\angle RSU+\angle UST = 90^\circ\), so they are complementary. But wait, the first option is "They are complementary". But wait, maybe I messed up. Wait, no, let's check again. Wait, the problem is "Which statement is true about angles UST and QSR?". So \(\angle QSR=\angle RSU\) (given), and \(\angle RSU+\angle UST = 90^\circ\) (complementary), so \(\angle QSR+\angle UST = 90^\circ\), so they are complementary. But wait, the options: the first option is "They are complementary", the second is "They are congruent", third "supplementary", fourth "obtuse". Wait, but if \(\angle QSR=\angle RSU\) and \(\angle RSU\) and \(\angle UST\) are complementary, then \(\angle QSR\) and \(\angle UST\) are complementary. But maybe the answer is "They are complementary". But wait, maybe I made a mistake. Wait, no, let's do it again.

1. Given \(\angle RSU\) and \(\angle UST\) are complementary: \(\angle RSU+\angle UST = 90^\circ\).
2. Given \(\angle QSR\cong\angle RSU\): \(\angle QSR=\angle RSU\).
3. Substitute \(\angle RSU\) with \(\angle QSR\) in the first equation: \(\angle QSR+\angle UST = 90^\circ\).

Since the sum of \(\angle UST\) and \(\angle QSR\) is \(90^\circ\), they are complementary.

Answer:

They are complementary