Question

find the measure of $\angle tsw$.
image of a circle with center s, points t, u, v, w on the circumference. arc tv is 91°, arc uw is 88°.
write your answer as a whole number or a decimal.
$m\angle tsw = \square ^\circ$

Explanation:

Step1: Recall circle angle sum

A full circle is \(360^\circ\), and vertical/straight angles sum to \(180^\circ\) for straight lines. But here, \(S\) is the center, so \( \angle TSW\) is related to the arcs. Wait, actually, \(TU\) and \(VW\) are diameters? Wait, no, \(TSU\) and \(VSW\) are straight lines (diameters). Wait, the arc \(TV\) is \(91^\circ\), arc \(UW\) is \(88^\circ\). Let's find the measure of arc \(TW\)? Wait, no, \( \angle TSW\) is a central angle? Wait, no, \(TS\) and \(SW\) are radii, and the angle between them. Wait, the sum of arcs around a circle is \(360^\circ\). The arcs: \(TV = 91^\circ\), \(UW = 88^\circ\), and the other arcs? Wait, since \(TU\) and \(VW\) are diameters (they pass through center \(S\)), so \( \angle TSW\) is a central angle, and the arc \(TW\) would be... Wait, maybe better: the sum of arcs \(TV + VW + WU + UT = 360^\circ\)? No, wait, the circle is divided into four arcs: \(TV\) (91°), \(VU\) (let's say), \(UW\) (88°), and \(WT\) (let's say). But since \(TSU\) and \(VSW\) are straight lines (diameters), the angle \( \angle TSW\) is a central angle, and the arc \(TW\) would correspond to it. Wait, alternatively, the sum of arcs on one side of a diameter is \(180^\circ\). Wait, arc \(TV + arc VW = 180^\circ\)? No, \(TSU\) is a diameter, so arc \(TU\) is \(180^\circ\), and \(VSW\) is a diameter, so arc \(VW\) + arc \(WU\) = \(180^\circ\)? Wait, maybe I made a mistake. Wait, the key is that \( \angle TSW\) is a central angle, and the measure of a central angle is equal to its arc. Wait, but let's calculate the remaining arc. The total circle is \(360^\circ\). The arcs given are \(91^\circ\) (TV) and \(88^\circ\) (UW). The other two arcs: let's call arc \(VW = x\) and arc \(UT = y\). Since \(TSU\) is a diameter, arc \(UT + arc TV = 180^\circ\)? Wait, no, \(TSU\) is a straight line, so arc \(TU\) (from T to U through S) is \(180^\circ\). So arc \(TV + arc VU = 180^\circ\)? Wait, no, T to V to U: arc TV is 91°, arc VU is... Wait, maybe the angle \( \angle TSW\) is equal to \(180^\circ - (91^\circ + 88^\circ)\)? Wait, no, let's think again. The sum of the arcs: arc TV (91°) + arc VW (let's say \(a\)) + arc WU (88°) + arc UT (let's say \(b\)) = 360°. But since \(TSU\) and \(VSW\) are diameters, arc TU (T to U through S) is 180°, so arc TV + arc VU = 180°? No, T to U through V: arc TV + arc VU = arc TU? No, T to U is a straight line (diameter), so arc TU is 180°, so the arc from T to U through the top (V) is arc TV + arc VU = 180°, and through the bottom (W) is arc TW + arc WU = 180°. Ah! There we go. So arc TW + arc WU = 180°, because TSWU is a semicircle (since TSU is a diameter, wait no, VSW is a diameter. Wait, \(VSW\) is a diameter, so \(V\) to \(W\) through \(S\) is 180°. Wait, maybe I confused the diameters. Let's label the points: T and U are on a horizontal diameter (TSU), V and W are on a vertical diameter (VSW). So the circle is divided into four arcs: TV (91°), VU (let's calculate), UW (88°), and WT (let's calculate). Since TSU is a horizontal diameter, the arc from T to U (through the top, V) is arc TV + arc VU = 180°, and through the bottom, W, is arc TW + arc WU = 180°. Similarly, VSW is a vertical diameter, so arc VU + arc UW = 180°, and arc TV + arc TW = 180°. Wait, let's use arc TW + arc WU = 180°, since T to W to U is a semicircle (because TSU is a diameter? No, TSU is horizontal, VSW is vertical. So T to W to U: T to W is arc TW, W to U is arc WU, and since TSU is a diameter, the angle at S for TSWU is a semicircle? Wait, no, \( \angle TSW\) is the angle between TS and SW. TS is horizontal left, SW is vertical down. So the arc between T and W is the arc that \( \angle TSW\) subtends. Wait, the measure of a central angle is equal to its arc. So to find \( \angle TSW\), we can find the measure of arc TW. The total circle is 360°, so arc TV (91°) + arc VU + arc UW (88°) + arc WT = 360°. But arc VU + arc UW = 180° (since VSW is a diameter, so V to U to W is a semicircle? No, V to W is a diameter, so arc VU + arc UW = arc VW? No, V to W is a diameter, so arc VW is 180°, so arc VU + arc UW = 180°. Therefore, arc VU = 180° - 88° = 92°. Then arc TV + arc VU = 91° + 92° = 181°, which can't be, since T to U through V should be 180°. Wait, I must have messed up the arcs. Let's start over. The circle has center S, with horizontal diameter TU (T left, U right) and vertical diameter VW (V top, W bottom). So the four arcs are: top-left (T to V): 91°, top-right (V to U): let's call it \(x\), bottom-right (U to W): 88°, bottom-left (W to T): let's call it \(y\). Now, since TU is a diameter, the sum of top arcs (T to V to U) is 180°: 91° + \(x\) = 180° ⇒ \(x\) = 89°. Since VW is a diameter, the sum of right arcs (V to U to W) is 180°: \(x\) + 88° = 180° ⇒ \(x\) = 92°. Wait, contradiction. So my initial assumption about the arcs is wrong. Wait, the problem says "Find the measure of \( \angle TSW\)". \( \angle TSW\) is the angle at S between T (left on horizontal diameter) and W (bottom on vertical diameter). So TS is horizontal (left), SW is vertical (down). So the angle between horizontal left and vertical down is 90°? But no, because the arcs are not 90° each. Wait, no, the arcs given are 91° (T to V) and 88° (U to W). So T to V is 91°, U to W is 88°. Let's find the measure of \( \angle TSW\). The angle \( \angle TSW\) is a central angle, and the arc it subtends is T to W. The sum of arcs T to V (91°), V to U (let's say \(a\)), U to W (88°), and W to T (let's say \(b\)) is 360°. Also, since TU is a diameter, the sum of arcs T to V to U (91° + \(a\)) is 180°, so \(a\) = 180° - 91° = 89°. Since VW is a diameter, the sum of arcs V to U to W (\(a\) + 88°) is 180°, so \(a\) + 88° = 180° ⇒ \(a\) = 92°. Wait, now \(a\) is both 89° and 92°, which is impossible. Therefore, my mistake is in assuming which arcs are which. Wait, the arc labeled 91° is T to V (top-left), and 88° is U to W (bottom-right). So the arc from T to W (bottom-left) and V to U (top-right). Let's use the fact that the sum of arcs around the center is 360°, and the angle \( \angle TSW\) is equal to the measure of arc TW. The arc TW is opposite to arc VU? No, let's calculate the sum of the known arcs: 91° + 88° = 179°. Then the remaining two arcs (VU and TW) sum to 360° - 179° = 181°. But since TU and VW are diameters, the sum of arc TV + arc TW = 180° (because T to V to W to T is a semicircle? No, T to W is vertical? Wait, no, T is left, W is bottom, so the arc from T to W is a quarter-circle? No, not necessarily. Wait, maybe the angle \( \angle TSW\) is 90° plus or minus? No, the arcs are not 90° each. Wait, another approach: the measure of a central angle is equal to its arc. The angle \( \angle TSW\) is formed by two radii, TS and SW. The arc between T and W is the arc that is not 91° or 88°. Wait, the total circle is 360°, so the arc from T to W is (360° - 91° - 88° - arc VU)/2? No, this is confusing. Wait, maybe the key is that the sum of the arcs adjacent to \( \angle TSW\) and the given arcs. Wait, the angle \( \angle TSW\) is a central angle, and the sum of the arcs around the center: arc TV (91°) + arc VU + arc UW (88°) + arc WT = 360°. But since TU and VW are diameters, arc TV + arc VU = 180° (because T to U through V is a semicircle) and arc UW + arc WT = 180° (because U to W to T is a semicircle). Wait, that makes sense! So arc TV + arc VU = 180° ⇒ 91° + arc VU = 180° ⇒ arc VU = 89°. Then arc UW + arc WT = 180° ⇒ 88° + arc WT = 180° ⇒ arc WT = 92°. But arc WT is the arc subtended by \( \angle TSW\), so \( \angle TSW\) = arc WT = 92°? Wait, no, arc WT is from W to T, but \( \angle TSW\) is from T to S to W, so the arc is from T to W, which is the same as arc WT. Wait, but 88° + 92° = 180°, which matches the semicircle. And 91° + 89° = 180°, which also matches. So then \( \angle TSW\) is equal to the measure of arc TW, which is 92°? Wait, but let's check: if arc TV is 91°, arc VU is 89°, arc UW is 88°, arc WT is 92°, then total is 91+89+88+92=360, which works. So the measure of \( \angle TSW\) is 92°, because it's a central angle subtended by arc TW, which is 92°.

Step2: Confirm the calculation

Arc TV = 91°, arc UW = 88°. Since TU and VW are diameters, arc TV + arc VU = 180° ⇒ arc VU = 180 - 91 = 89°. Then arc UW + arc WT = 180° ⇒ arc WT = 180 - 88 = 92°. Since \( \angle TSW\) is a central angle, its measure equals arc WT, so \( m\angle TSW = 92^\circ \).

Answer:

92