m∠QSW < m∠WQS
<h3>Explanation:</h3>The question is asking which angle is larger, the one at vertex S or the one at vertex Q? You are expected to understand how angles are named and how angle measure works.
Angles are named by listing points on the rays that make them up, with the vertex point in the middle of the list. ∠QSW is the angle whose vertex is at S. (Since there is only one angle with its vertex at S, it could also be called ∠S.) The letters Q and W help you identify that rays SQ and SW form the sides of the angle.
Likewise, ∠WQS is the angle whose vertex is at Q. It could also be called ∠Q without any confusion. The other letters in the name tell you that rays QW and QS are the sides of the angle.
When the rays that make the sides of an angle are closer together, the angle has a smaller measure.
In a triangle, there are a several different ways to determine which of the angles is larger.
- the larger angle will be opposite the longer side
- the base point of an altitude line will be closer to the larger (acute) angle
- the rays that make up the larger angle will be farther apart (at some given radius from the vertex)
- relative to a perpendicular bisector between the vertices, the vertex of the third angle will be on the side closer to the larger angle (another way to say the relationship of the second point above)
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<em>Your triangle</em>
Line segment QW (opposite ∠S) is shorter than line segment SW (opposite ∠Q) so m∠S < m∠Q.
A line segment drawn from W perpendicular to QS is closer to Q, so ∠Q is the larger angle.
Thus we must conclude the measure of ∠QSW is less than the measure of ∠WQS:
... m∠QSW < m∠WQS
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<em>Comment on appearances</em>
Here, you're specifically told to go by the appearance of the angles. In most cases, you cannot, as the figures are rarely drawn to scale, and are often drawn to be intentionally misleading.
