Question

in the given circle, tangent ab and secant cd intersect at point e as shown. what is the value represented by x?
a. 220
b. 140
c. 70
d. 110

Explanation:

Step1: Recall the tangent-secant angle theorem

The measure of an angle formed by a tangent and a secant intersecting outside a circle is half the difference of the measures of the intercepted arcs. Also, a straight line is \(180^\circ\), so the angle adjacent to \(110^\circ\) (formed by the tangent and the secant inside the circle - related to the intercepted arc) can be found first. The angle between the tangent \(AB\) and the secant \(CD\) at point \(E\) (the one supplementary to \(110^\circ\)) is \(180^\circ - 110^\circ=70^\circ\). But wait, actually, the formula for the angle formed by a tangent and a secant intersecting at a point outside the circle is \(\theta=\frac{1}{2}( \text{major arc}-\text{minor arc})\), but here if we consider the angle inside (the \(110^\circ\)) and the arc \(x\), we know that the measure of the angle formed by a tangent and a chord (or secant) at the point of tangency is half the measure of the intercepted arc. Wait, no, when the tangent and secant intersect at the point of tangency? No, here \(E\) is the point of tangency? Wait, \(AB\) is tangent at \(E\)? Wait, the diagram shows \(AB\) is tangent, \(CD\) is secant, intersecting at \(E\), which is the point of tangency (since \(AB\) is tangent at \(E\)). So the angle between tangent \(AB\) and chord \(EC\) (part of secant \(CD\)) is equal to half the measure of the intercepted arc \(EC\)? Wait, no, the correct theorem: the measure of an angle formed by a tangent and a chord is equal to half the measure of the intercepted arc. But here, the angle given is \(110^\circ\), which is adjacent to the angle between tangent and chord. Wait, let's correct: the straight line \(AB\) is a tangent at \(E\), so the angle between tangent \(AB\) and secant \(ED\) (or \(EC\)): the angle between tangent and secant at the point of tangency is equal to half the measure of the intercepted arc. But the angle shown as \(110^\circ\) is supplementary to the angle between tangent and chord (or secant) at \(E\). Wait, actually, the sum of the angle between tangent and secant (at \(E\)) and the \(110^\circ\) angle is \(180^\circ\) (since they are adjacent and form a linear pair). Then, the angle between tangent \(AB\) and secant \(EC\) (let's call it \(\alpha\)) is \(180^\circ - 110^\circ = 70^\circ\). Now, by the tangent - chord angle theorem, \(\alpha=\frac{1}{2}x\)? Wait, no, the tangent - chord angle theorem states that the measure of an angle formed by a tangent and a chord is equal to half the measure of the intercepted arc. Wait, if the angle between tangent \(AB\) and chord \(EC\) is \(\alpha\), then \(\alpha=\frac{1}{2}\times\) measure of arc \(EC\)? Wait, no, maybe I got it reversed. Wait, the angle formed by a tangent and a secant outside the circle is \(\frac{1}{2}(\text{major arc}-\text{minor arc})\), but here the intersection is at the point of tangency (inside the circle? No, \(E\) is on the circle because \(AB\) is tangent at \(E\), so \(E\) is on the circle. So when a tangent and a secant intersect at a point on the circle (the point of tangency), the measure of the angle is half the measure of the intercepted arc. Wait, the angle between tangent \(AB\) and secant \(CD\) at \(E\) (on the circle) is equal to half the measure of the intercepted arc \(EC\). But the angle given is \(110^\circ\), which is adjacent to that angle. Wait, let's think differently. The total circumference is \(360^\circ\), but maybe the arc \(x\) and the arc opposite to it (the one intercepted by the \(110^\circ\) angle) have a relationship. Wait, no, the angle between the tangent and the secant at \(E\) (the angle inside the circle, \(110^\circ\)) and the arc \(x\): actually, the measure of an angle formed by a tangent and a secant at the point of tangency is equal to half the measure of the intercepted arc. But if the angle between tangent \(AB\) and secant \(ED\) is \(\theta\), then \(\theta=\frac{1}{2}x\), and \(\theta + 110^\circ=180^\circ\) (linear pair). So \(\theta = 70^\circ\), then \(70^\circ=\frac{1}{2}x\)? No, that would make \(x = 140^\circ\). Wait, let's recall the correct theorem: The measure of an angle formed by a tangent and a secant drawn from a point on the circle (the point of tangency) is equal to half the measure of the intercepted arc. Wait, no, when the angle is formed by a tangent and a chord (secant is a chord extended) at the point of tangency, the angle is equal to half the measure of the intercepted arc. So if the tangent is \(AB\) at \(E\), and the secant is \(ECD\), then the angle between \(AB\) and \(EC\) is equal to half the measure of arc \(EC\) (which is \(x\)). But the angle between \(AB\) and \(ED\) is supplementary to the angle between \(AB\) and \(EC\). Wait, the angle given is \(110^\circ\), which is the angle between \(AB\) and \(ED\). So the angle between \(AB\) and \(EC\) is \(180^\circ - 110^\circ = 70^\circ\), and this angle is equal to half the measure of arc \(EC\) (which is \(x\))? No, that would be wrong. Wait, no, the correct formula is: the measure of an angle formed by a tangent and a secant intersecting at a point outside the circle is \(\frac{1}{2}(\text{major arc}-\text{minor arc})\), but here the intersection is at the point of tangency (on the circle), so the angle is equal to half the measure of the intercepted arc. Wait, I think I mixed up. Let's use the linear pair and the inscribed angle. The angle between tangent \(AB\) and secant \(CD\) at \(E\) (on the circle) and the arc \(x\): the angle \(110^\circ\) and the arc \(x\) - the measure of the angle formed by a tangent and a secant at the point of tangency is equal to half the measure of the intercepted arc. Wait, no, the angle between tangent and secant at \(E\) (the one that is not \(110^\circ\)) is equal to half the arc \(x\). And since \(110^\circ\) and that angle are supplementary (linear pair), so \(110^\circ+\frac{1}{2}x = 180^\circ\)? No, that would be if the angle is outside. Wait, no, let's draw this mentally: \(AB\) is a tangent at \(E\), \(CD\) is a secant passing through \(E\) and \(C\) (and \(D\) outside). The angle between \(AB\) (tangent) and \(ED\) (part of secant) is \(110^\circ\), so the angle between \(AB\) and \(EC\) is \(180 - 110=70^\circ\). Now, by the tangent - chord angle theorem, the measure of the angle between a tangent and a chord is equal to half the measure of the intercepted arc. So the angle between \(AB\) (tangent) and \(EC\) (chord) is equal to half the measure of arc \(EC\) (which is \(x\)). Wait, that would mean \(70^\circ=\frac{1}{2}x\), so \(x = 140^\circ\). Yes, that makes sense. So the steps are:

Step1: Find the angle between tangent and chord

The angle between tangent \(AB\) and chord \(EC\) is supplementary to \(110^\circ\) (linear pair), so \(180^\circ - 110^\circ = 70^\circ\).

Step2: Apply tangent - chord angle theorem

The measure of the angle between a tangent and a chord is half the measure of the intercepted arc. Let the intercepted arc be \(x\) (arc \(EC\)). So \(70^\circ=\frac{1}{2}x\). Solving for \(x\), we multiply both sides by 2: \(x = 2\times70^\circ=140^\circ\).

Answer:

B. 140