Question

in the diagram of circle c, what is the measure of ∠1? 106° c 36° 1 17° 35° 70° 71°

Explanation:

Step1: Recall the formula for the measure of an angle formed by two secants outside a circle.

The measure of an angle formed by two secants intersecting outside a circle is half the difference of the measures of the intercepted arcs. The formula is \( m\angle1=\frac{1}{2}(m\mathrm{arc\;outside}-m\mathrm{arc\;inside}) \). First, we need to find the measure of the larger intercepted arc. The total circumference of a circle is \( 360^\circ \), but we can also find the larger arc by subtracting the given arc from \( 360^\circ \)? Wait, no, actually, the two intercepted arcs here: one is \( 106^\circ \), and the other arc? Wait, no, the angle inside the circle? Wait, no, the formula for an angle outside the circle: the measure of the angle is half the difference of the measures of the intercepted arcs. So the two arcs are the major arc and the minor arc. Wait, the given arc is \( 106^\circ \), and the other arc: wait, the smaller arc? Wait, no, the angle between the two secants: the intercepted arcs are the ones that are "cut off" by the secants. So the larger arc is \( 360^\circ - 106^\circ - 36^\circ \)? No, wait, no. Wait, the diagram: one arc is \( 106^\circ \), and the other arc (the one adjacent to the \( 36^\circ \))? Wait, no, let's re-examine. The angle \( \angle1 \) is formed by two secants. The intercepted arcs: the major arc and the minor arc. The major arc would be \( 360^\circ - 36^\circ - 106^\circ \)? No, that's not right. Wait, actually, the two arcs: one is \( 106^\circ \), and the other arc is \( 360^\circ - 106^\circ - 36^\circ \)? No, that's confusing. Wait, no, the correct approach: the measure of an angle formed outside the circle by two secants is \( \frac{1}{2}( \text{measure of the larger arc} - \text{measure of the smaller arc} ) \). So first, find the measure of the larger arc. The smaller arc is \( 36^\circ \)? No, wait, the given arc is \( 106^\circ \), and the other arc: wait, the total around the circle is \( 360^\circ \), but the two arcs intercepted by the secants: one is \( 106^\circ \), and the other arc is \( 360^\circ - 106^\circ - 36^\circ \)? No, that's not. Wait, no, the angle inside the circle? Wait, no, the formula is \( m\angle = \frac{1}{2}( \text{major arc} - \text{minor arc} ) \). So here, the major arc would be \( 360^\circ - 36^\circ = 324^\circ \)? No, that's not. Wait, maybe I made a mistake. Wait, the two arcs: one is \( 106^\circ \), and the other arc is \( 360^\circ - 106^\circ = 254^\circ \)? No, that's not. Wait, no, the angle is formed by two secants, so the intercepted arcs are the ones between the two secants. So the first secant cuts the circle at two points, creating an arc, and the second secant cuts the circle at two points, creating another arc. So the angle outside is half the difference of the measures of the intercepted arcs. So the larger arc minus the smaller arc, divided by 2. So here, the larger arc: let's see, the given arc is \( 106^\circ \), and the other arc (the one that's not \( 106^\circ \) or \( 36^\circ \))? Wait, no, the \( 36^\circ \) is an arc? Wait, the diagram shows a \( 36^\circ \) arc and a \( 106^\circ \) arc. Wait, maybe the two intercepted arcs are \( 106^\circ \) and \( (360^\circ - 106^\circ - 36^\circ) \)? No, that's not. Wait, let's think again. The formula for the measure of an angle formed by two secants outside the circle is \( m\angle = \frac{1}{2}( \text{measure of the far arc} - \text{measure of the near arc} ) \). So the far arc is the larger arc, the near arc is the smaller arc. So here, the near arc is \( 36^\circ \)? No, wait, the \( 106^\circ \) is one arc, and the other arc: let's calculate the other arc. The total circle is \( 360^\circ \), so the other arc (the one not \( 106^\circ \)) is \( 360^\circ - 106^\circ = 254^\circ \)? No, that can't be. Wait, no, maybe the two arcs are \( 106^\circ \) and \( (360^\circ - 106^\circ - 36^\circ) \)? Wait, no, the \( 36^\circ \) is an arc? Wait, the diagram: one arc is \( 106^\circ \), another arc is \( 36^\circ \)? No, that doesn't add up. Wait, maybe the \( 36^\circ \) is not an arc but an angle? No, the diagram shows \( 36^\circ \) as an arc. Wait, no, the \( 36^\circ \) is an arc measure. Wait, let's check the formula again. The measure of an angle formed by two secants outside the circle is \( \frac{1}{2}( \text{measure of the intercepted major arc} - \text{measure of the intercepted minor arc} ) \). So here, the major arc would be \( 360^\circ - 36^\circ = 324^\circ \)? No, that's not. Wait, I think I messed up. Wait, the two secants: one secant cuts the circle at two points, creating an arc of \( 106^\circ \), and the other secant cuts the circle at two points, creating an arc of \( 36^\circ \)? No, that can't be. Wait, no, the angle inside the circle? Wait, no, the angle outside. Wait, let's look at the answer choices. The options are \( 17^\circ \), \( 35^\circ \), \( 70^\circ \), \( 71^\circ \). Let's try to calculate. The formula is \( m\angle1 = \frac{1}{2}( \text{larger arc} - \text{smaller arc} ) \). So the larger arc: let's see, the \( 106^\circ \) is one arc, and the other arc (the one that's opposite) is \( 360^\circ - 106^\circ - 36^\circ \)? No, that's \( 360 - 106 - 36 = 218^\circ \)? No, that's not. Wait, maybe the two arcs are \( 106^\circ \) and \( (360^\circ - 106^\circ) \)? No, that's \( 254^\circ \), then \( \frac{1}{2}(254 - 36) = \frac{1}{2}(218) = 109^\circ \), which is not an option. So I must have misidentified the arcs. Wait, maybe the \( 36^\circ \) is not an arc but a central angle? No, the diagram shows \( 36^\circ \) as an arc. Wait, another approach: the angle formed by two secants outside the circle is half the difference of the intercepted arcs. So the two arcs are the ones that are "between" the secants. So one arc is \( 106^\circ \), and the other arc is \( 360^\circ - 106^\circ - x \), but no. Wait, maybe the \( 36^\circ \) is a typo? No, the diagram shows \( 36^\circ \). Wait, wait, maybe the two arcs are \( 106^\circ \) and \( (180^\circ - 36^\circ) \)? No, that doesn't make sense. Wait, let's check the answer options. The options include \( 35^\circ \), \( 70^\circ \), \( 71^\circ \), \( 17^\circ \). Wait, maybe the formula is \( m\angle1 = \frac{1}{2}(106^\circ - 36^\circ) \)? No, that would be \( \frac{1}{2}(70) = 35^\circ \). Wait, that's one of the options (35°). Wait, is that possible? Wait, if the angle is half the difference of the two arcs, but maybe the arcs are \( 106^\circ \) and \( 36^\circ \), but that would be \( \frac{1}{2}(106 - 36) = \frac{1}{2}(70) = 35^\circ \). Oh! Maybe I had the formula reversed. Wait, no, the formula is \( \frac{1}{2}( \text{major arc} - \text{minor arc} ) \), but if the major arc is \( 106^\circ \) and the minor arc is \( 36^\circ \), then \( \frac{1}{2}(106 - 36) = 35^\circ \), which is option B. So that must be it. So the measure of \( \angle1 \) is \( \frac{1}{2}(106^\circ - 36^\circ) \).

Step2: Calculate the measure of \( \angle1 \).

First, find the difference of the arcs: \( 106^\circ - 36^\circ = 70^\circ \). Then, take half of that: \( \frac{1}{2} \times 70^\circ = 35^\circ \).

Answer:

\( 35^\circ \) (Option B: \( 35^\circ \))