the movement of the progress bar may be uneven because questions can be worth more or less (including zero) depending on your answer. which of the following statements about the image below is true? line vs and line qs are parallel. line tx and line vs are perpendicular. ∠tru and ∠urs are supplementary. the measure of ∠tru is 72°.

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Accepted Answer

# Brief Explanations:
1. Analyze "Line VS and Line QS are parallel": These lines share a common point $ S $ (or $ R $), so they can't be parallel. Eliminate this.
2. Analyze "Line TX and Line VS are perpendicular": The angle between them is $ 72^\circ $, not $ 90^\circ $, so not perpendicular. Eliminate this.
3. Analyze " $ \angle TRU $ and $ \angle URS $ are supplementary": $ \angle TRU + \angle URS = 180^\circ $ (they form a linear pair), so they are supplementary.
4. Analyze "The measure of $ \angle TRU $ is $ 72^\circ $": From the diagram, $ \angle TRU = 90^\circ - 18^\circ = 72^\circ $? Wait, no—wait, let's re - check. Wait, actually, the angle with $ 72^\circ $ is related to another angle. But for the supplementary angle: $ \angle TRU $ and $ \angle URS $ form a straight line, so their sum is $ 180^\circ $, so they are supplementary. But wait, let's check the last option again. Wait, maybe I made a mistake. Wait, the angle between $ TQ $ and $ TR $ is $ 18^\circ $, and $ RU $ is perpendicular to $ QS $ (since there's a right angle at $ R $ for $ RU $ and $ QS $). So $ \angle TRU = 90^\circ - 18^\circ = 72^\circ $? Wait, no, maybe the correct answer is the third option? Wait, no, let's re - evaluate:

Wait, the options:

- Line VS and QS: share a point, not parallel.
- TX and VS: the angle between them is $ 72^\circ $, not $ 90^\circ $, so not perpendicular.
- $ \angle TRU $ and $ \angle URS $: they are adjacent and form a linear pair, so supplementary (sum to $ 180^\circ $).
- Measure of $ \angle TRU $: Let's see, the angle with $ 18^\circ $: if $ RU \perp QS $ (right angle at $ R $ for $ RU $ and $ QS $), then $ \angle QRU = 90^\circ $, and $ \angle TRQ = 18^\circ $, so $ \angle TRU = 90^\circ - 18^\circ = 72^\circ $? Wait, but then the third option: $ \angle TRU $ and $ \angle URS $ are supplementary. Let's check the sum: if $ \angle TRU = 72^\circ $, then $ \angle URS = 180^\circ - 72^\circ = 108^\circ $, so they are supplementary. But also, the last option says $ \angle TRU = 72^\circ $. Wait, maybe I messed up. Wait, let's look at the diagram again (as per the given angles: $ 18^\circ $ and $ 72^\circ $).

Wait, the angle between $ TQ $ and $ TR $ is $ 18^\circ $, $ RU $ is perpendicular to $ QS $ (right angle), so $ \angle TRU = 90^\circ - 18^\circ = 72^\circ $. But also, $ \angle TRU $ and $ \angle URS $ are supplementary. Wait, but which one is correct? Wait, let's check the options again:

Option 3: $ \angle TRU $ and $ \angle URS $ are supplementary. Since they form a linear pair (they are adjacent and their non - common sides form a straight line), their sum is $ 180^\circ $, so they are supplementary.

Option 4: The measure of $ \angle TRU $ is $ 72^\circ $. Let's calculate: if $ \angle QRT = 18^\circ $ and $ \angle QRU = 90^\circ $ (right angle), then $ \angle TRU = \angle QRU-\angle QRT = 90^\circ - 18^\circ = 72^\circ $. Wait, so both option 3 and 4 seem correct? No, that can't be. Wait, maybe the diagram has $ RU \perp QS $ (right angle at $ R $ for $ RU $ and $ QS $), and $ \angle TQR = 18^\circ $, and the angle at $ W $ is $ 72^\circ $. Wait, maybe I made a mistake in the angle calculation.

Wait, let's start over:

1. Parallel lines: Lines that never meet. Line VS and QS meet at a point, so not parallel. Eliminate.

2. Perpendicular lines: Angle between them is $ 90^\circ $. The angle between TX and VS is $ 72^\circ $, so not perpendicular. Eliminate.

3. Supplementary angles: Two angles that sum to $ 180^\circ $. $ \angle TRU $ and $ \angle URS $ are adjacent and form a straight line (linear pair), so their sum is $ 180^\circ $. So they are supplementary.

4. Measure of $ \angle TRU $: Let's see, if $ \angle QRT = 18^\circ $ and $ RU $ is perpendicular to $ QS $ ( $ \angle QRU = 90^\circ $ ), then $ \angle TRU = 90^\circ - 18^\circ = 72^\circ $. But wait, if $ \angle TRU = 72^\circ $, then $ \angle URS = 180^\circ - 72^\circ = 108^\circ $, so they are supplementary. But which option is correct? Wait, maybe the diagram has $ RU $ perpendicular to $ QS $, and $ TX $ and $ QS $ with some angle. Wait, maybe the correct answer is the third option ( $ \angle TRU $ and $ \angle URS $ are supplementary) because a linear pair of angles is always supplementary. The last option: if $ \angle TRU = 72^\circ $, but let's check the angle at $ W $ is $ 72^\circ $, maybe $ \angle TRU $ is equal to that angle (corresponding angles if lines are parallel, but earlier we saw lines aren't parallel). Wait, maybe I made a mistake. Let's re - check the options:

- Option 1: Wrong (intersecting lines).
- Option 2: Wrong (angle is $ 72^\circ $, not $ 90^\circ $).
- Option 3: $ \angle TRU $ and $ \angle URS $ form a linear pair, so supplementary (sum to $ 180^\circ $): correct.
- Option 4: Let's see, the angle with $ 18^\circ $: if $ RU \perp QS $, then $ \angle TRU = 90 - 18 = 72^\circ $, but is that accurate? Maybe, but the third option is about supplementary angles, which is a fundamental property (linear pair is supplementary). So the correct answer is the third option.

Wait, but maybe I was wrong. Wait, let's check again. The angle $ \angle TRU $ and $ \angle URS $: they are adjacent and their outer sides form a straight line, so they are supplementary. So this statement is true.

# Answer:
$ \boldsymbol{\angle TRU} $ and $ \boldsymbol{\angle URS} $ are supplementary. (The option corresponding to " $ \angle TRU $ and $ \angle URS $ are supplementary" )

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