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In this video, we will learn how to find the measures of inscribed angles using the relationship between angles and arcs.
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Before we talk about these angle relationships, letβs remember what an inscribed angle is.
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Itβs an angle where the vertex and two endpoints all lie on the circumference of the circle, on the outside of the circle.
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We could measure this inscribed angle in degrees.
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If this inscribed angle measures π degrees, then the arc created between these two endpoints will be two π degrees.
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Another way to say that is an inscribed angle measures half of the subtended arc thatβs created by that angle.
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If we have another inscribed angle, which has the same endpoints as the first one, this angle will also measure π degrees because π is one-half of the arc thatβs created by these endpoints, which here is two π.
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Itβs also worth noting a special case.
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The special case is the case where the angle inscribed has endpoints that are at either end of a circleβs diameter.
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In this case, the subtended arc is 180 degrees, which makes the inscribed angle a right angle.
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And again, we can move that vertex and still create a right angle as long as the end points donβt move.
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Before we move on, we should also remind ourselves of central angles.
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In a central angle, the vertex is the center of the circle.
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And when weβre dealing with a central angle, its angle measure will be equal to the subtended arcβs angle measure.
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The way weβve drawn it here, it will be equal to two π.
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And that means when an inscribed angle shares the endpoints with a central angle, the inscribed angle will be half the central angle.
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Another thing we should know about arcs and circles is what happens when we have parallel chords.
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If we have chords like these that run parallel, arc π΄π· will be equal to arc π΅πΆ.
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That is to say, in a circle, arcs between parallel chords are congruent.
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In the circle weβve drawn here, the measure of arc π΄π· will be equal to the measure of arc π΅πΆ.
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Weβre now ready to use these circle theorems to calculate unknown angles.
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In the figure, π is the center and the measure of angle ππ΄π΅ equals 59.5 degrees.
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What is the measure of angle π΄ππ΅?
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What is the measure of angle π΄πΆπ΅?
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Weβre told that π is the center of this circle and that the measure of angle ππ΄π΅ is 59.5 degrees.
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We want to find the measure of angle π΄ππ΅ and the measure of π΄πΆπ΅.
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We see that the points π΄, π, and π΅ form a triangle.
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Both line segment ππ΅ and line segment ππ΄ are radii of this circle because any line drawn from the center of the circle to the circumference of the circle will be a radius.
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This means we can say that line segment ππ΄ is equal to line segment ππ΅.
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And it will mean that triangle π΄ππ΅ is an isosceles triangle.
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In an isosceles triangle, the two angles opposite the radii are equal to each other.
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And that means we could say that angle π΄π΅π is also equal to 59.5 degrees.
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Since these three angles form a triangle, they must sum to 180 degrees.
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And so, we substitute the values we do know for angle ππ΄π΅ and angle π΄π΅π.
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We add the two angles we know, and we get 119 degrees.
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And then to solve for angle π΄ππ΅, we subtract 119 degrees from both sides, and we find that angle π΄ππ΅ is equal to 61 degrees.
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Thatβs the answer to part one.
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Part two is a little bit less straightforward.
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We notice that both of these angles share the endpoints π΄, π΅, which means theyβre both subtended by the arc π΄π΅.
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But we need to make a clarification here.
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Angle π΄ππ΅ is a central angle that is subtended by arc π΄π΅, while angle π΄πΆπ΅ is an inscribed angle subtended by arc π΄π΅.
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And we remember that the central angle subtended by two points on a circle is twice the inscribed angle subtended by those two points.
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We might see it represented something like this: if the central angle measures two π, the inscribed angle subtended by the same points will be π degrees.
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Based on that, we can say that the measure of angle π΄πΆπ΅ will be equal to one-half the measure of angle π΄ππ΅.
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So, we plug in 61 degrees for angle π΄ππ΅.
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Half of 61 degrees is 30.5 degrees.
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And so, the measure of angle π΄πΆπ΅ equals 30.5 degrees.
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Hereβs another example.
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From the figure, what is π₯?
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Letβs start with what we know.
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We have angle π΄πΆπ΅, which measures 101 degrees.
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And we have angle π΄ππ΅.
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In this case, weβre talking about the reflex of angle π΄ππ΅.
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Thatβs the one thatβs greater than 180 degrees, which measures two π₯ plus eight degrees.
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Angle π΄πΆπ΅ and angle π΄ππ΅ share the endpoints π΄ and π΅.
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But because the vertex of angle π΄ππ΅ is the center of the circle, we say that angle π΄ππ΅ is a central angle for this circle.
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While the vertex for angle π΄πΆπ΅ is on the outside of the circle, making angle π΄πΆπ΅ an inscribed angle of the circle.
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And these three facts point us to the central angle theorem.
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And the central angle theorem tells us that when a central angle and an inscribed angle share the same endpoints, the central angle will be two times that of the inscribed angle.
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In this diagram, the inscribed angle is π degrees, and that would make the central angle two π degrees.
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By this, we can say that the measure of angle π΄ππ΅ is going to be equal to two times the measure of angle π΄πΆπ΅.
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The measure of the central angle will be equal to two times the measure of the inscribed angle.
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And so, we can say that two π₯ plus eight will be equal to two times 101.
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When we multiply two times 101, we get 202.
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And now, weβre ready to solve for π₯.
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Subtract eight from both sides.
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Two π₯ equals 194.
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Then, divide both sides by two, and we find that π₯ equals 97.
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In our next example, we have some intersecting chords in a circle.
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Given that the measure of angle π΄π΅π· equals 44 degrees and the measure of angle πΆπΈπ΄ equals 72 degrees, find π₯, π¦, and π§.
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Letβs start by listing what we know.
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Angle π΄π΅π· equals 44 degrees, angle πΆπΈπ΄ measures 72 degrees.
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These two chords intersect at point πΈ.
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And that means we can say that angle π΅πΈπ· and angle πΆπΈπ΄ are vertical angles, which means their measure will be equal to one another.
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They are congruent angles.
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And in this case, that means that angle π΅πΈπ· is also equal to 72 degrees.
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The points πΈ, π΅, and π· form a triangle, which means that their three angles must sum to 180 degrees.
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And we can substitute what we know for these three angles into this equation.
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72 plus 44 equals 116.
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116 plus π§ equals 180.
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So, we subtract 116 from both sides.
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And we find that π§ equals 64 degrees.
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We wonβt be able to follow the same procedure to find π₯ and π¦.
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So, weβll need to think about some of the circle theorems.
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If we look at inscribed angle π΅, we see that it has endpoints along the circle at π΄ and π· and that its intercepted arc is arc π΄π·.
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We could write them as arc π΄π· intercepts angle π΄π΅π·.
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But thereβs another angle in this circle that also intercepts the same arc, and that would be angle π΄πΆπ·.
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Because both of these angles are subtended by the same arc, we can say that the measure of angle π΄πΆπ· will be equal to the measure of angle π΄π΅π·.
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And that means π₯ will be equal to 44 degrees.
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And because all three angles need to sum to 180 degrees, we can tell that angle π¦ is going to be equal to 64 degrees.
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If we wanted to confirm this, we could see that angle πΆπ΄π΅ intercepts arc πΆπ΅ and angle πΆπ·π΅ intercepts arc πΆπ΅.
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And so, we found that π₯ equals 44 degrees, and both π¦ and π§ equals 64 degrees.
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In our next example, weβll have a diameter to consider.
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Given that line segment π΄π΅ is a diameter in circle π and the measure of angle π΅ππ· equals 59 degrees, find the measure of angle π΄πΆπ· in degrees.
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Letβs put what we know into the diagram.
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Angle π΅ππ· measures 59 degrees, and weβre trying to find the measure of angle π΄πΆπ·.
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If we start with what we know about angle π΅ππ·, since π΅ππ· has a vertex at the center of the circle, π΅ππ· is a central angle.
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And because angle π΅ππ· is a central angle, its subtended arc, arc π΅π·, also measures 59 degrees.
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Weβre also interested in angle π΄πΆπ·.
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But angle π΄πΆπ· is not a central angle.
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Itβs an inscribed angle because its vertex is on the circumference of the circle, as are both endpoints.
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The arc associated with angle π΄πΆπ· would be arc π΄π·.
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We have a partial measurement for this arc, but weβre missing the distance from π΄ to π΅.
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But because we know that π΄π΅ is a diameter, it cuts the circle in half.
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And that means the measure of arc π΄π΅ is 180 degrees.
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If arc π΄π΅ equals 180 and arc π΅π· equals 59 degrees, we can say the measure of arc π΄π· is equal to the measure of arc π΄π΅ plus the measure of arc π΅π·.
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If we plug in what we know, the measure of arc π΄π· is 239 degrees.
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Because angle π΄πΆπ· is an inscribed angle and it has a subtended arc measure of 239 degrees, we can find out the exact measure of angle π΄πΆπ·.
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The measure of the inscribed angle π΄πΆπ· will be one-half its subtended arc, arc π΄π·.
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Since that arc is 239 degrees, we take half of that and we get 119.5 degrees for the measure of angle π΄πΆπ·.
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In our final example, weβll look at how parallel chords can give us information about arc measures.
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Given that the line segment π΄π΅ is a diameter of the circle and line segment π·πΆ is parallel to line segment π΄π΅, find the measure of angle π΄πΈπ·.
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Weβre interested in the measure of angle π΄πΈπ·; thatβs this measure.
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And weβve been given a few other pieces of information.
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We know line segment π·πΆ is parallel to line segment π΄π΅.
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We know line segment π΄π΅ is the diameter.
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And on the figure, angle πΆπ΅π΄ has been labeled as 68.5 degrees.
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At first, it might not seem like thereβs a very clear direction for where to go here.
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But if we start with the measure of angle πΆπ΄π΅, using that information, we could find the measure of arc πΆπ΄.
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Since angle πΆπ΄π΅ is an inscribed angle, its arc will be two times the measure of that inscribed angle.
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Arc π΄πΆ will then be equal to two times 68.5, which is 137 degrees.
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And because we know that line segment π΄π΅ is a diameter, arc π΄π΅ must be equal to 180 degrees.
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We can also say that arc π΄π΅ will be equal to arc π΅πΆ plus arc πΆπ΄.
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We know π΄π΅ needs to be 180 degrees and arc πΆπ΄ is 137 degrees.
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To solve for the measure of arc π΅πΆ, we can subtract 137 from both sides of the equation.
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And we get the measure of arc π΅πΆ is 43 degrees.
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And hereβs where our parallel chords come into play.
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When you have parallel chords, their intercepted arcs are going to be congruent.
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And that means because arc πΆπ΅ equals 43 degrees, arc π·π΄ also equals 43 degrees.
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And at this point, we began to see that arc π·π΄ is subtended by the angle π΄πΈπ·.
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Since angle π΄πΈπ· is an inscribed angle, its angle measure, the measure of angle π΄πΈπ·, is going to be equal to one-half the measure of arc π΄π·.
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We know that the measure of arc π΄π· is 43 degrees, and one-half of 43 is 21.5.
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And so, we can say that the measure of angle π΄πΈπ· is 21.5 degrees.
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Before we finish, letβs quickly review the key points.
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If you have a central angle that measures two π degrees, its intercepted arc will also measure two π degrees.
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While an inscribed angle that intercepts the same arc will have half the angle measure, only π degrees.
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We could say it like this: the central angle subtended by two points on a circle is twice the inscribed angle subtended by those same two points.
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We also can say that the angles subtended by the same arc on a circle will be equal.
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And finally, the arcs between parallel chords will always be congruent.