Question

what is the measure of the angle in question? (image of a circle with points u, t, f, s and intersection v, with arcs 53° and 175° marked)

Explanation:

Step1: Recall the theorem for intersecting chords

When two chords intersect inside a circle, the measure of the angle formed is equal to half the sum of the measures of the intercepted arcs. The formula is \( \text{Angle} = \frac{1}{2}(\text{arc}_1 + \text{arc}_2) \). First, we need to find the measure of the arc opposite to the \( 53^\circ \) arc. The total circumference of a circle is \( 360^\circ \), so the arc opposite to \( 53^\circ \) (let's call it arc \( UT \)) can be found by subtracting the known arcs from \( 360^\circ \). Wait, actually, the two arcs intercepted by the angle at \( V \) are the \( 53^\circ \) arc (arc \( FS \)) and the arc \( UT \). Wait, no, the given arc is \( 175^\circ \) (arc \( UT \)? Wait, looking at the diagram, the circle has an arc labeled \( 175^\circ \) and another arc \( 53^\circ \). Wait, when two chords intersect at \( V \), the angle at \( V \) intercepts arc \( FS \) ( \( 53^\circ \)) and arc \( UT \). Wait, actually, the measure of the angle formed by two intersecting chords is half the sum of the measures of the intercepted arcs. So first, we need to find the measure of the other intercepted arc. The total circle is \( 360^\circ \), but wait, maybe the \( 175^\circ \) is one arc, and the arc opposite to \( 53^\circ \) is \( 360 - 175 - 53 - \text{other arc} \)? Wait, no, maybe I misread. Wait, the two chords are \( FS \) and \( UT \), intersecting at \( V \). So the angle at \( V \) (the "?") intercepts arc \( FS \) ( \( 53^\circ \)) and arc \( UT \). Wait, the arc labeled \( 175^\circ \) is arc \( UT \)? Wait, no, maybe the arc \( UT \) is \( 175^\circ \), and the arc \( FS \) is \( 53^\circ \). Wait, no, let's correct. The formula for the angle formed by two intersecting chords: \( \angle FVT = \frac{1}{2}(\text{arc } FT + \text{arc } SU) \)? Wait, no, maybe the diagram has arc \( FS = 53^\circ \) and arc \( UT = 175^\circ \), and the other two arcs (arc \( FU \) and arc \( ST \))? Wait, no, when two chords intersect, say \( FS \) and \( UT \) intersect at \( V \), then the angle at \( V \) is \( \frac{1}{2}(\text{arc } FT + \text{arc } SU) \)? Wait, I think I made a mistake. Let's recall the correct theorem: The measure of an angle formed by two intersecting chords is equal to half the sum of the measures of the intercepted arcs. So if chords \( AB \) and \( CD \) intersect at \( E \), then \( \angle AEC = \frac{1}{2}(\text{arc } AC + \text{arc } BD) \). So in this case, chords \( FS \) and \( UT \) intersect at \( V \), so the angle at \( V \) (let's call it \( \angle FVT \)) is \( \frac{1}{2}(\text{arc } FT + \text{arc } SU) \)? Wait, no, arc \( FS \) is \( 53^\circ \), and arc \( UT \) is \( 175^\circ \). Wait, maybe the arc opposite to \( 53^\circ \) is \( 175^\circ \), and the other two arcs: the total circle is \( 360^\circ \), so the sum of all arcs is \( 360^\circ \). So arc \( FS = 53^\circ \), arc \( UT = 175^\circ \), then the other two arcs (arc \( FU \) and arc \( ST \)) would sum to \( 360 - 53 - 175 = 132^\circ \)? No, that doesn't seem right. Wait, maybe the \( 175^\circ \) is the arc \( UT \), and the arc \( FS \) is \( 53^\circ \), so the angle at \( V \) is \( \frac{1}{2}(53^\circ + 175^\circ) \)? Wait, let's calculate that: \( \frac{1}{2}(53 + 175) = \frac{1}{2}(228) = 114^\circ \). Wait, is that correct? Wait, no, maybe the other arc. Wait, no, the theorem is that the angle is half the sum of the intercepted arcs. So if the two chords intersect, the angle is half the sum of the arcs that are "across" from the angle. So in the diagram, the angle at \( V \) is between the two chords, so it intercepts arc \( FS \) ( \( 53^\circ \)) and arc \( UT \) ( \( 175^\circ \))? Wait, no, that can't be, because \( 53 + 175 = 228 \), half of that is \( 114 \). Wait, but let's check again. Wait, maybe the arc \( UT \) is not \( 175^\circ \), but the arc opposite to \( 53^\circ \) is \( 360 - 175 - 53 = 132^\circ \)? No, that's not. Wait, maybe I misread the diagram. Let's re-express: the circle has four arcs: arc \( FS = 53^\circ \), arc \( UT = 175^\circ \), and the other two arcs (arc \( FU \) and arc \( ST \)). When two chords intersect at \( V \), the angle at \( V \) is formed by chords \( FV \) and \( UV \), so the intercepted arcs are arc \( FS \) (between \( F \) and \( S \)) and arc \( UT \) (between \( U \) and \( T \))? Wait, no, the intercepted arcs are the arcs that are cut off by the two pairs of vertical angles. So the vertical angles at \( V \) will have intercepted arcs: one angle intercepts arc \( FS \) and arc \( UT \), and the other angle intercepts the other two arcs. Wait, the formula is: if two chords intersect at a point inside the circle, then the measure of the angle is half the sum of the measures of the intercepted arcs. So let's denote the angle at \( V \) as \( \theta \), then \( \theta = \frac{1}{2}(\text{arc } FS + \text{arc } UT) \). We know arc \( FS = 53^\circ \), and we need to find arc \( UT \). Wait, the total circle is \( 360^\circ \), so arc \( UT = 360 - 53 - \text{arc } FU - \text{arc } ST \). But that's not helpful. Wait, maybe the \( 175^\circ \) is arc \( UT \), so arc \( UT = 175^\circ \), arc \( FS = 53^\circ \), then \( \theta = \frac{1}{2}(53 + 175) = \frac{228}{2} = 114^\circ \). Yes, that makes sense. So the measure of the angle is half the sum of the two intercepted arcs.

Step2: Apply the formula

Using the formula for the angle formed by two intersecting chords: \( \text{Angle} = \frac{1}{2}(\text{arc}_1 + \text{arc}_2) \). Here, \( \text{arc}_1 = 53^\circ \) (arc \( FS \)) and \( \text{arc}_2 = 175^\circ \) (arc \( UT \)). Plugging in the values: \( \text{Angle} = \frac{1}{2}(53 + 175) \). Calculate the sum inside the parentheses: \( 53 + 175 = 228 \). Then take half of that: \( \frac{228}{2} = 114 \).

Answer:

\( 114^\circ \)