Question

$\overleftrightarrow{al}$ is tangent to circle $o$ at point $a$, and $\overleftrightarrow{ai}$ is a secant line. if $\text{m}\overarc{ai} = 134^\circ$, find $\text{m}\angle ial$.

answer attempt 1 out of 3

$\text{m}\angle ial = \square^\circ$

Explanation:

Step1: Recall Tangent-Secant Angle Theorem

The measure of an angle formed by a tangent and a secant drawn from a point outside the circle is half the difference of the measures of the intercepted arcs. The formula is \( m\angle IAL=\frac{1}{2}(m\overset{\frown}{AI'} - m\overset{\frown}{AI}) \), but here, the major arc and minor arc: the total circle is \( 360^\circ \), so the arc opposite to \( \overset{\frown}{AI} \) (the major arc) is \( 360^\circ - 134^\circ=226^\circ \)? Wait, no, actually, the angle formed by tangent and secant is half the measure of the intercepted arc that is not adjacent. Wait, correction: the angle between tangent and secant is half the measure of the intercepted arc. Wait, no, the correct formula is: if a tangent and a secant intersect at a point on the circle (wait, no, \( AL \) is tangent at \( A \), and \( AI \) is a secant, so the angle \( \angle IAL \) is formed by tangent \( AL \) and secant \( AI \) at point \( A \) (on the circle). Wait, no, when the tangent and secant meet at the point of tangency, the angle is half the measure of the intercepted arc. Wait, the arc \( \overset{\frown}{AI} \) is \( 134^\circ \), so the angle between tangent and chord (since \( AI \) is a chord here, as \( AL \) is tangent at \( A \)): the measure of the angle between a tangent and a chord is half the measure of the intercepted arc. Wait, yes! The theorem is: the measure of an angle formed by a tangent and a chord is equal to half the measure of the intercepted arc. Wait, but in this case, the chord is \( AI \), and the tangent is \( AL \), so the angle \( \angle IAL \) intercepts arc \( \overset{\frown}{AI} \)? No, wait, no. Wait, the angle between tangent and chord is half the measure of the intercepted arc. Wait, if the tangent is at \( A \), and the chord is \( AI \), then the intercepted arc is the arc that is opposite to the angle, i.e., the arc that is not containing the angle. Wait, no, let's re - state the theorem: The measure of an angle formed by a tangent and a chord is half the measure of the intercepted arc. The intercepted arc is the arc that lies in the interior of the angle. Wait, no, actually, the angle between tangent and chord is half the measure of the intercepted arc. Wait, let's check the formula. If the tangent is \( AL \) (tangent at \( A \)) and chord is \( AI \), then \( \angle IAL \) is equal to half the measure of the arc \( \overset{\frown}{AI} \)? No, that can't be, because if the arc \( \overset{\frown}{AI} \) is \( 134^\circ \), then half of that is \( 67^\circ \), but wait, maybe I got the theorem wrong. Wait, no, the correct theorem: the measure of an angle formed by a tangent and a secant (when the secant passes through the circle) at the point of tangency is half the measure of the intercepted arc. Wait, no, when the tangent and secant meet at the point of tangency, the angle is half the measure of the intercepted arc. Wait, let's calculate the other arc. The total circumference is \( 360^\circ \), so the arc that is not \( \overset{\frown}{AI} \) is \( 360 - 134 = 226^\circ \)? No, that's not right. Wait, no, the angle between tangent and chord is half the measure of the intercepted arc. Wait, I think I made a mistake. Let's recall: the angle between a tangent and a chord is equal to half the measure of the intercepted arc. So if the chord is \( AI \), and the tangent is \( AL \), then the intercepted arc is \( \overset{\frown}{AI} \)? No, that would mean \( \angle IAL=\frac{1}{2}m\overset{\frown}{AI} \), but that would be \( 67^\circ \), but wait, let's think again. Wait, the tangent is perpendicular to the radius at the point of tangency, but here we have a secant. Wait, no, the formula for the angle formed by tangent and secant: when the tangent and secant are drawn from a point outside the circle, the angle is half the difference of the intercepted arcs. But in this case, the point is on the circle (since \( AL \) is tangent at \( A \), and \( AI \) is a secant passing through \( A \)), so the angle is formed at the point of tangency. So the correct theorem is: the measure of an angle formed by a tangent and a chord is half the measure of the intercepted arc. Wait, but if the arc \( \overset{\frown}{AI} \) is \( 134^\circ \), then the angle \( \angle IAL \) is half of the arc that is supplementary? Wait, no, let's draw this mentally. The tangent \( AL \) at \( A \), chord \( AI \). The angle between tangent and chord is equal to half the measure of the intercepted arc. The intercepted arc is the arc that is "cut off" by the chord \( AI \) and lies on the opposite side of the angle. Wait, actually, the angle between tangent and chord is half the measure of the intercepted arc. So if the arc \( \overset{\frown}{AI} \) is \( 134^\circ \), then the angle \( \angle IAL \) is half of \( (360 - 134)^\circ \)? No, that's not. Wait, no, I think I confused the two cases. Case 1: angle formed outside the circle (tangent and secant from outside): \( \frac{1}{2}(major arc - minor arc) \). Case 2: angle formed at the point of tangency (tangent and chord): \( \frac{1}{2}(measure of intercepted arc) \). Wait, in this problem, the tangent \( AL \) and secant \( AI \) meet at \( A \) (the point of tangency), so it's case 2. Wait, but the arc \( \overset{\frown}{AI} \) is \( 134^\circ \), so the angle \( \angle IAL \) should be half of the arc that is not \( \overset{\frown}{AI} \)? No, that doesn't make sense. Wait, let's calculate the other arc. The total circle is \( 360^\circ \), so the arc opposite to \( \overset{\frown}{AI} \) (the major arc) is \( 360 - 134 = 226^\circ \), but that's not right. Wait, no, the angle between tangent and chord is equal to half the measure of the intercepted arc. So if the chord is \( AI \), the intercepted arc is \( \overset{\frown}{AI} \), but that would be \( 67^\circ \), but let's check with the formula. Wait, no, the correct formula is: the measure of an angle formed by a tangent and a chord is half the measure of the intercepted arc. So if the arc \( \overset{\frown}{AI} \) is \( 134^\circ \), then \( m\angle IAL=\frac{1}{2}(360 - 134)^\circ \)? No, that would be \( \frac{1}{2}(226) = 113^\circ \), which is wrong. Wait, I think I messed up the theorem. Let's look up the correct theorem: The measure of an angle formed by a tangent and a secant drawn from a point outside the circle is half the difference of the measures of the intercepted arcs. But in this case, the point is on the circle (since \( AL \) is tangent at \( A \), and \( AI \) is a secant passing through \( A \)), so the angle is formed at the point of tangency. So the angle between tangent and chord is equal to half the measure of the intercepted arc. Wait, no, when the tangent and chord meet at the point of tangency, the angle is equal to half the measure of the intercepted arc. So if the arc \( \overset{\frown}{AI} \) is \( 134^\circ \), then the angle \( \angle IAL \) is half of the arc that is supplementary? Wait, no, let's take an example. If the arc is \( 180^\circ \) (a semicircle), the angle between tangent and chord (diameter) would be \( 90^\circ \), which is half of \( 180^\circ \). So that works. So if the arc is \( 134^\circ \), the angle should be half of \( 134^\circ \)? No, that would be \( 67^\circ \), but wait, in the example with semicircle, arc is \( 180^\circ \), angle is \( 90^\circ \), which is half. So that formula holds. Wait, but in the problem, the arc \( \overset{\frown}{AI} \) is \( 134^\circ \), so the angle \( \angle IAL \) is \( \frac{1}{2}m\overset{\frown}{AI} \)? But that would be \( 67^\circ \), but let's check the total. Wait, no, maybe the arc is the major arc. Wait, no, the angle between tangent and chord is half the measure of the intercepted arc. So if the chord is \( AI \), the intercepted arc is \( \overset{\frown}{AI} \), so \( m\angle IAL=\frac{1}{2}\times134^\circ = 67^\circ \)? Wait, no, that can't be, because when the arc is \( 180^\circ \), the angle is \( 90^\circ \), which is half. So that formula is correct. Wait, but I think I made a mistake earlier. Let's re - derive. The tangent is perpendicular to the radius, but here we have a chord. The angle between tangent and chord: let \( OA \) be the radius, so \( OA\perp AL \) (since tangent is perpendicular to radius at point of tangency). The central angle for arc \( \overset{\frown}{AI} \) is \( \angle AOI \) with measure \( 134^\circ \). The triangle \( OAI \) is isosceles ( \( OA = OI \) radii). The angle \( \angle OAI \) is \( \frac{180^\circ - 134^\circ}{2}=23^\circ \). Then, since \( OA\perp AL \), \( \angle OAL = 90^\circ \). So \( \angle IAL=\angle OAL-\angle OAI = 90^\circ - 23^\circ = 67^\circ \). Yes, that matches. So the measure of \( \angle IAL \) is \( 67^\circ \).

Step2: Calculate the angle

Using the formula for the angle between tangent and chord: \( m\angle IAL=\frac{1}{2}m\overset{\frown}{AI} \). Wait, no, in the derivation above, we got \( 67^\circ \), which is \( \frac{1}{2}(180 - (180 - 134)) \)? No, the correct way is from the radius - tangent perpendicularity and the isosceles triangle. But the formula that the angle between tangent and chord is half the measure of the intercepted arc gives \( m\angle IAL=\frac{1}{2}m\overset{\frown}{AI} \)? Wait, no, in the derivation, we saw that \( \angle IAL = 90^\circ-\angle OAI \), and \( \angle OAI=\frac{180 - m\overset{\frown}{AI}}{2} \), so \( \angle IAL = 90^\circ-\frac{180 - m\overset{\frown}{AI}}{2}=\frac{m\overset{\frown}{AI}}{2} \). Yes, so \( m\angle IAL=\frac{1}{2}\times134^\circ = 67^\circ \).

Answer:

\( 67 \)