Question

1. find m∠str

Explanation:

Step1: Use the inscribed - angle theorem

The measure of an inscribed angle is half the measure of its intercepted arc. Angles subtended by the same arc are equal.

Step2: Calculate \(m\angle STR\)

We know that the measure of an inscribed angle \(\angle SRQ\) and \(\angle STQ\) (i.e., \(\angle STR\)) subtend arcs.
Since the inscribed - angle theorem states that if two inscribed angles subtend the same arc, they are equal.
We use the property that the measure of an inscribed angle \(=\frac{1}{2}\) measure of its intercepted arc.
The arc \(SQ\) is given. The inscribed angle \(\angle STR\) subtends arc \(SR\).
We know that \(m\angle STR=\frac{1}{2}(46^{\circ})\) (because the inscribed angle \(\angle STR\) and the given \(46^{\circ}\) is related as per the circle's angle - arc relationship. Another way: In a circle, if we consider the fact that the angle \(\angle STR\) and the \(46^{\circ}\) angle (assuming the standard circle - angle properties where the angle we want is half of the given arc - related value. Wait, no, actually, using the property that \(\angle STR\) is an inscribed angle. If we assume the following:
Let's use the property that in a circle, if we have two points \(R\) and \(S\), and \(T\) on the circle. The measure of \(\angle STR\) is half of the measure of arc \(SR\). But wait, no, actually, if we consider the fact that the angle \(\angle STR\) and the \(46^{\circ}\) is related as follows:
We know that \(m\angle STR = 23^{\circ}\) (using the inscribed - angle formula \(m\angle=\frac{1}{2}\text{arc measure}\), assume the arc measure corresponding to \(\angle STR\) is \(46^{\circ}\) (by the property of circle angles. Wait, no, wait, actually, if we use the property that the angle \(\angle STR\) is an inscribed angle. If we assume that the arc \(SR\) is \(46^{\circ}\), then \(m\angle STR=\frac{1}{2}\times46^{\circ}\)

Answer:

\(23^{\circ}\)