Question

no additional details were added for this assignment.
what is the measure of ∠qts?
use the keypad to enter your answer in the box.
the measure of ∠qts is degrees

Explanation:

Step1: Recall the central angle theorem

The central angle theorem states that the central angle is twice the inscribed angle subtended by the same arc. Here, \( QT \) and \( ST \) are radii, so \( \triangle QTS \) is isosceles? Wait, no, actually, the arc \( QS \) subtends the inscribed angle at the circumference (the \( 73.6^\circ \) angle) and the central angle \( \angle QTS \). Wait, no, correction: the inscribed angle theorem says that an angle subtended by an arc at the center is twice the angle subtended at the circumference. Wait, in the diagram, the \( 73.6^\circ \) angle is an inscribed angle? Wait, no, actually, \( QT \) and \( ST \) are radii, so \( T \) is the center. So the angle at the center \( \angle QTS \) and the angle at the circumference (the \( 73.6^\circ \)): wait, no, the angle given is between two chords from the circumference to a point on the circumference? Wait, no, looking at the diagram, \( T \) is the center (since \( QT \) and \( ST \) are radii, as they go from \( T \) to the circle). So the angle at the center \( \angle QTS \) and the inscribed angle over the same arc \( QS \). Wait, the inscribed angle theorem: the measure of a central angle is twice the measure of the inscribed angle subtended by the same arc. Wait, but here, the angle given is \( 73.6^\circ \), which is an inscribed angle? Wait, no, maybe the triangle formed by the two radii and the chord \( QS \), and the other angle is at the circumference. Wait, actually, the angle at the circumference (the \( 73.6^\circ \)) and the central angle \( \angle QTS \): the central angle should be twice the inscribed angle? Wait, no, wait: if the inscribed angle is \( \theta \), the central angle is \( 2\theta \). But wait, in this case, maybe the angle given is the inscribed angle, so the central angle \( \angle QTS \) would be \( 180 - 2\times73.6 \)? Wait, no, that doesn't make sense. Wait, no, let's think again. \( QT = ST \) (radii), so \( \triangle QTS \) is isosceles? Wait, no, the angle at the top (the \( 73.6^\circ \)) is an inscribed angle, and \( \angle QTS \) is the central angle? Wait, no, maybe the angle given is the angle between two chords from a point on the circumference, and \( T \) is the center. Wait, the key is that \( QT \) and \( ST \) are radii, so \( T \) is the center. The arc \( QS \) subtends the angle at the center \( \angle QTS \) and the angle at the circumference (the \( 73.6^\circ \))? No, the angle at the circumference subtended by arc \( QS \) would be half the central angle. Wait, maybe the angle given is the angle between two chords from a point on the circumference, and we need to find the central angle. Wait, no, actually, the triangle \( QTS \): \( QT \) and \( ST \) are radii, so \( QT = ST \), so \( \triangle QTS \) is isosceles? Wait, no, the angle at the top (the \( 73.6^\circ \)) is an angle in the triangle? Wait, no, the diagram shows a circle with center \( T \), points \( Q \) and \( S \) on the circle, so \( QT \) and \( ST \) are radii. Then there's another point on the circle (the top) connected to \( Q \) and \( S \), forming an angle of \( 73.6^\circ \). So that angle is an inscribed angle subtended by arc \( QS \), so the central angle \( \angle QTS \) subtended by arc \( QS \) is twice that, so \( 2 \times 73.6 = 147.2^\circ \)? Wait, no, that can't be, because the sum of angles in a triangle is \( 180^\circ \). Wait, maybe I got it wrong. Wait, \( QT \) and \( ST \) are radii, so \( QT = ST \), so \( \triangle QTS \) is isosceles with \( QT = ST \). The angle at the top (the \( 73.6^\circ \)) is an angle outside? No, wait, the angle given is between the two chords from the top point to \( Q \) and \( S \), so that's an inscribed angle. The central angle \( \angle QTS \) is related to that. Wait, the inscribed angle theorem: the measure of an inscribed angle is half the measure of its subtended central angle. So if the inscribed angle is \( 73.6^\circ \), then the central angle \( \angle QTS \) is \( 2 \times 73.6 = 147.2^\circ \)? But wait, let's check the triangle. \( QT = ST \), so \( \triangle QTS \) has \( QT = ST \), so the base angles are equal? Wait, no, \( \angle QTS \) is the vertex angle. Wait, maybe the angle given is the angle between the two radii? No, the angle given is \( 73.6^\circ \) between the two chords from the circumference. Wait, I think I made a mistake. Let's start over.

In a circle, the central angle is twice the inscribed angle subtended by the same arc. So if we have an inscribed angle \( \angle QPS = 73.6^\circ \) (where \( P \) is the top point), then the central angle \( \angle QTS \) (where \( T \) is the center) subtended by arc \( QS \) is \( 2 \times 73.6^\circ = 147.2^\circ \)? But wait, \( QT \) and \( ST \) are radii, so \( \triangle QTS \) is isosceles with \( QT = ST \), so the base angles \( \angle TQS \) and \( \angle TSQ \) are equal. But the sum of angles in a triangle is \( 180^\circ \), so \( \angle QTS + \angle TQS + \angle TSQ = 180^\circ \). But if \( \angle QTS \) is the central angle, then the inscribed angle over arc \( QS \) is half of it, so \( \angle QPS = \frac{1}{2} \angle QTS \), so \( \angle QTS = 2 \times 73.6^\circ = 147.2^\circ \). Wait, but that would mean the other two angles in \( \triangle QTS \) are \( (180 - 147.2)/2 = 16.4^\circ \) each, which is possible. So the measure of \( \angle QTS \) is \( 147.2^\circ \)? Wait, no, wait, maybe the angle given is the angle between the two radii? No, the diagram shows the angle at the top (circumference) as \( 73.6^\circ \), and \( T \) is the center. So the central angle is twice the inscribed angle. So \( \angle QTS = 2 \times 73.6^\circ = 147.2^\circ \). Wait, but let's confirm. The inscribed angle theorem: an angle formed by two chords in a circle with the vertex on the circle (inscribed angle) is half the measure of the central angle formed by the same two chords. So yes, if the inscribed angle is \( 73.6^\circ \), the central angle \( \angle QTS \) is \( 2 \times 73.6 = 147.2 \) degrees.

Step2: Calculate the central angle

Given the inscribed angle is \( 73.6^\circ \), the central angle \( \angle QTS \) is \( 2 \times 73.6^\circ \).
\[ 2 \times 73.6 = 147.2 \]

Answer:

147.2