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In this video, we will learn how to calculate angle measures that are created by intersecting lines in a circle.
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To do that, we’ll review some key terms we need to know when working with circles, consider three different circle theorems, and then use that information to solve some examples.
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If we have a circle, a chord is a line segment where both endpoints lie on the circle.
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A secant is a line that intersects the circle at two distinct points.
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The diameter is a line segment, where both endpoints lie on the circle and the midpoint is the center of the circle.
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The diameter is the longest chord of any circle.
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The radius is a line segment connecting the center of the circle to a point on the circle.
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A tangent is a line that intersects the circle at only one point, and the tangent makes a right angle with the radius at that point.
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Now that we’ve looked at lines and line segments in a circle, let’s consider angles and arcs.
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If you have two radii, the angle created between the two of them is called a central angle.
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The part of the circumference created by this central angle is known as an arc.
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An arc of the circle is a portion of the circumference.
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When we talk about the arc measure, we’re saying the degree measure of an arc, and it’s equal to the measure of the central angle that intercepts the arc.
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For example, if our central angle, pictured here, is 120 degrees, then we would say that the arc measure is 120 degrees.
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We can also say that the arc highlighted in yellow measures 120 degrees.
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When we do that, we’re saying that the arc is 120 degrees out of the full 360 degrees of the whole circle.
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And while we’ll be primarily focusing on degrees in this video, all of this information is equally true if you’re operating in radians.
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Let’s now consider some of the theorems, starting with the angles of intersecting chords theorem.
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Inside the circle, the line segment 𝐵𝐷 or chord 𝐵𝐷 intersects chord 𝐴𝐶 at this point.
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This intersection creates four different angles, labeled here one, two, three, and four.
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Since in this circle, the two chords 𝐴𝐶 and 𝐵𝐷 intersect inside the circle, the measure of angle one is equal to one-half times the measure of arc 𝐴𝐵 plus the measure of arc 𝐶𝐷.
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And likewise, the measure of angle two, one-half times the measure of arc 𝐵𝐶 plus the measure of arc 𝐴𝐷.
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Something else that we can say since vertical angles are congruent is that the measure of angle one is equal to the measure of angle three and the measure of angle two is equal to the measure of angle four.
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We’ve just looked at angles that are created when chords intersect.
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Now, let’s consider what angles are created when secants intersect.
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When two secants intersect outside of the circle, in this case, the intersection is here, they create this angle.
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In addition to that, it creates two intercepted arcs.
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Here, we have the first one in green and the second one in blue.
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If we call the angle created by the two secants angle one, and we’ll call the intercepted arc one 𝑥 and the intercepted arc two 𝑦, then the measure of angle one will be equal to the measure of the arc 𝑥 minus the measure of the arc 𝑦 divided by two.
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If we write this out, we’re saying that if two lines intersect outside the circle, then the measure of an angle formed by the two lines is one-half the positive difference of the measures of the intercepted arcs.
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In this case, we’re dealing with two secants that are intersecting outside the circle.
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But this is also true when two tangents intersect outside the circle.
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Let’s see what two tangents intersecting would look like.
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Now, we have two tangents intersecting.
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They form this angle that we can call angle one, which creates intercepted arc one and intercepted arc two.
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If we let the larger arc be 𝑥 and the smaller one be 𝑦, we would find the measure of angle one by subtracting the measure of arc 𝑥 minus the measure of arc 𝑦 and then dividing by two.
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Using this information, let’s try and solve for some missing values.
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Find the value of 𝑥.
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In the given figure, line segment 𝐶𝐸 and line segment 𝐴𝐸 intersect at point 𝐸.
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The angle created at their intersection is the one labeled 𝑥.
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And as these lines have intersected the circle, they have created two intercepted arcs.
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If we sketch the center of the circle, we can show arc 𝐷𝐵 that measures 71 degrees and arc 𝐶𝐴 that measures 144 degrees.
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Since the line segment 𝐶𝐸 and the line segment 𝐴𝐸 can both be described as secants of the circle, we can use the angles of intersecting secants theorem.
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Which tells us the angle created by the intersection of two secants, in this case, the measure of angle 𝐶𝐸𝐴, will be equal to one-half the difference of the two intercepted arcs.
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And that means we can say that 𝑥 is equal to 144 minus 71 divided by two.
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𝑥 is equal to 36.5 degrees.
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Here’s another example.
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This time, instead of the intersection of two secants outside the circle, we have the intersection of a secant and a tangent outside the circle.
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Given that, in the shown figure, 𝑦 equals 𝑥 minus two and 𝑧 equals two 𝑥 plus two, determine the value of 𝑥.
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First, let’s start with what we know.
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Line segment 𝐷𝐴 and line segment 𝐵𝐴 intersect outside the circle at point 𝐴.
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Because that is true, we can say the measure of the angle formed by the two lines is one-half the positive difference of the measures of the intercepted arc.
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In this case, we already know that the measure of the angle formed by these two lines is 50 degrees, but that 50 degrees is equal to 𝑧 degrees minus 𝑦 degrees over two.
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Then, what we can do is substitute in two 𝑥 plus two in for 𝑧 and 𝑥 minus two in for 𝑦.
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Using this equation, we will be able to find the value of 𝑥.
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If we distribute the negative to the 𝑥 and the negative two, we will have 50 equals two 𝑥 plus two minus 𝑥 plus two divided by two.
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If we combine like terms, two 𝑥 minus 𝑥 equals positive 𝑥 and two plus two equals four.
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And so, we can say that 50 equals 𝑥 plus four over two.
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From there, we can get the two out of the denominator by multiplying both sides by two.
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We’ll have 100 equals 𝑥 plus four.
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And if 100 equals 𝑥 plus four and we subtract four from both sides, we see that 𝑥 equals 96.
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We know that 𝑦 was equal to 𝑥 minus two.
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And so, 𝑦 would be 94 degrees.
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𝑧 was equal to two 𝑥 plus two.
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If we multiply 96 by two and then add two, we get 194.
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𝑧 was then equal to 194 degrees.
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If we wanted to check, we can plug these values back in for 𝑧 and 𝑦 in the original equation we wrote, which says 50 degrees equals 𝑧 degrees minus 𝑦 degrees over two.
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194 minus 94 divided by two.
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194 minus 94 is 100 and 100 divided by two is 50.
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And so, we can say that, in the given figure, 𝑥 must be equal to 96.
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Here’s another example.
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Determine the measure of arc 𝐶𝐵.
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First, let’s see what we know.
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We know that the line segment 𝐶𝐴 and the line segment 𝐸𝐴 intersect outside the circle at point 𝐴.
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Both line segment 𝐶𝐴 and line segment 𝐸𝐴 are secants of the circle.
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The figure has also told us that the measure of arc 𝐶𝐵 is equal to the measure of arc 𝐸𝐷.
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We can also say that the sum of all arc measures in the circle must be 360 degrees.
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We’re interested in finding the measure of arc 𝐶𝐵.
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But in order to do that, we’ll need to find the measure of arc 𝐵𝐷.
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Based on what we said initially about the intersection outside the circle, we can say that the angle made by two intersecting lines outside the circle is half the positive difference of the intercepted arcs, which is the intersecting secant angles theorem.
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The angle created by these intersecting lines in our case is 34 degrees.
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34 degrees is equal to the measure of the intercepted arc 𝐶𝐸, which is 151 degrees, minus the measure of the intercepted arc 𝐵𝐷 and then divided it by two.
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We need to solve for the measure of the arc 𝐵𝐷.
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So, we multiply both sides of the equation by two, which gives us 68 equals 151 minus the measure of arc 𝐵𝐷.
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So, we subtract 151 from both sides, and we get negative 83 degrees equals the negative measure of arc 𝐵𝐷.
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We multiply both sides of the equation by negative one and flip the sides, and we get the measure of arc 𝐵𝐷 equals positive 83 degrees.
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But this is not our final answer.
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We’re still trying to find the measure of arc 𝐶𝐵.
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But remember, we know that all of these arcs must sum to 360 degrees.
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And we know that arc 𝐶𝐵 and arc 𝐸𝐷 must be equal.
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We could say that they’re equal to 𝑥 degrees.
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If we do that, we could then create the equation 151 plus 𝑥 plus 𝑥 plus 83 equals 360.
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If we combine like terms, 151 plus 83 equals 234.
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𝑥 plus 𝑥 equals two 𝑥.
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Therefore, 234 plus two 𝑥 equals 360 degrees.
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We subtract 234 from both sides of the equation, which tells us two 𝑥 equals 126.
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Divide both sides by two, and we see that 𝑥 equals 63.
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This tells us that both arc 𝐶𝐵 and arc 𝐸𝐷 is equal to a measure of 63 degrees.
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We were primarily interested in the measure of arc 𝐶𝐵, and we found that that measure is 63 degrees.
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In our next example, we have two chords that intersect inside a circle.
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In the given figure, find the measure of arc 𝐴𝐶 plus the measure of arc 𝐵𝐷.
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Let’s see what we know.
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By looking at the figure, we can see that line segment 𝐵𝐴 and line segment 𝐶𝐷 are chords that intersect inside a circle, which means we can think about the angles of intersecting chords theorem.
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Which tells us that the measure of angle one is equal to one-half the measure of arc 𝑃𝑆 plus the measure of arc 𝑄𝑅.
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In our figure, we’re interested in the measure of arc 𝐴𝐶 plus the measure of arc 𝐵𝐷.
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Based on what we know about intersecting chords in a circle, we can say that 112 degrees is equal to one-half times the measure of arc 𝐴𝐶 plus the measure of arc 𝐵𝐷.
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And if we multiply both sides of this equation by two, we would have 224 degrees is equal to the measure of arc 𝐴𝐶 plus the measure of arc 𝐵𝐷.
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And that’s what we’re looking for, the measure of arc 𝐴𝐶 plus the measure of arc 𝐵𝐷, which is 224 degrees.
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In this example, we haven’t been given an image.
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And so, we’ll have to sketch one on our own.
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The point 𝐴 is outside a circle with center 𝑀.
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The line between 𝐴 and 𝐶 intersects the circle at 𝐵 and 𝐶, and the line between 𝐴 and 𝐸 meets the circle at points 𝐷 and 𝐸.
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Given that the measure of angle 𝐶𝑀𝐸 equals 130 degrees and the measure of angle 𝐵𝑀𝐷 is equal to 56 degrees, find the measure of angle 𝐴.
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We’ve been given a description of our circle and lines, but we haven’t been given an image.
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The best place to start here is with sketching.
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We know we have a circle and that point 𝐴 is outside the circle.
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We also know that there is a line between 𝐴 and 𝐶; there’s a line with endpoints 𝐴 and 𝐶.
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And this line intersects the circle at 𝐵 and 𝐶.
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If we draw a line from 𝐴 to the circle, we know that the endpoints of this line were 𝐴 and 𝐶, and the other intersection along the circle was point 𝐵.
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Similarly, we have a line between 𝐴 and 𝐸 that meets the circle at 𝐷 and 𝐸.
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We’ll draw another line from point 𝐴.
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The endpoint is 𝐸 and its other intersection point is 𝐷.
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The circle has a center 𝑀.
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We’ve been told the measure of angle 𝐶𝑀𝐸, which would be this angle, is 130 degrees.
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And we’ve been told the measure of angle 𝐵𝑀𝐷, which would be this angle, and that measures 56 degrees.
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And we want to know the measure of angle 𝐴.
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Now, of course, when we look at our sketch, we know that these angles are a little bit off.
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But this sketch gives us enough information to figure out how we’re going to try and solve for the measure of angle 𝐴.
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Because we have two lines intersecting outside of a circle, then the angle created by the two lines intersecting outside the circle is half the positive difference between the intercepted arcs.
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And that means the measure of angle 𝐴 is the angle created outside the circle.
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It’s going to be one-half the measure of arc 𝐶𝐸 minus the measure of arc 𝐷𝐵.
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Arc 𝐶𝐸 measures 130 degrees; arc 𝐵𝐷 measures 56 degrees.
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130 minus 56 equal 74, and half of 74 is 37.
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And so, a circle under these conditions will have the angle measure 𝐴 equal to 37 degrees.
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And now, we can review the key points.
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The intersecting secant angle theorem tells us that the angle made by two secants intersecting outside a circle is half the positive difference between the intercepted arc measures.
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This is true for two secants, two tangents, and when one secant and one tangent intersect on the outside of a circle.
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And then, we have our intersecting chords theorem, which tells us that the measure of angle one equals one-half the measure of arc 𝐴𝐵 plus the measure of arc 𝐶𝐷.
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And by extension, the measure of angle two equals one-half times the measure of arc 𝐵𝐶 plus the measure of arc 𝐴𝐷.
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And because we know the definitions of vertical angles, we know that when chords intersect, the measure of angle one is equal to the measure of angle three, and the measure of angle two is equal to the measure of angle four.