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In this video, weβll learn how to find the measures of inscribed angles subtended by the same arc or by congruent arcs.
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To begin, weβll recap the meaning of some of those key terms before looking at a theorem that will help us to solve problems involving missing angles.
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An inscribed angle is the angle thatβs formed by the intersection of a pair of chords on the circumference of a circle.
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In the diagram, angle π΄π΅πΆ is the inscribed angle.
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Now, this angle is also said to be subtended by the arc π΄πΆ.
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There are a number of properties that apply to such angles.
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And in this video, weβre mostly going to investigate one such property.
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That is, angles subtended by the same arc are equal.
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In this diagram, that means the measure of angle π΄π·πΆ is equal to the measure of angle π΄π΅πΆ, since both angles are subtended from arc π΄πΆ.
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In a similar way, the measure of angle π·π΄π΅ is equal to the measure of angle π·πΆπ΅.
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This time, theyβre both subtended from arc π΅π·.
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This property is sometimes equivalently referred to as angles in the same segment are equal.
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But itβs also sometimes informally referred to as the bow tie property, since the pair of inscribed angles form the shape of a bow tie.
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Itβs important to note that thatβs an informal definition and should not be referred to in a mathematical proof or otherwise.
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And an incredibly powerful aspect of this property is that we can construct any number of angles subtended from arc π΄πΆ and theyβll all be equal as shown.
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Similarly, any number of angles can be subtended from arc π΅π· or even π΅πΈ.
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And those will also all be equal.
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So before we demonstrate the application of this property, letβs have a look at a very short geometric proof.
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To complete this proof, weβll add the center of the circle.
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Letβs define that to be π.
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And then weβre going to construct a pair of radii, that is, the radii π΄π and ππΆ.
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Then we apply a known property.
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That is, the inscribed angle is half the central angle that subtends the same arc.
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Or more simply, the angle at the center is double the angle at the circumference.
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Then weβre going to define this angle at the center, angle π΄ππΆ, as being equal to two π₯.
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Now we could have alternatively chosen π₯ degrees or π or π¦.
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But it does make sense to choose a multiple of two because it will make the further calculations a little bit more simple.
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Then angle π΄π·πΆ is half the size of this.
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So itβs a half times two π₯, which is simply π₯ degrees.
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But then we can apply the same rule to calculate the measure of angle π΄π΅πΆ.
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Once again, itβs half the measure of the angle at the center, so itβs half times two π₯, which is π₯ degrees.
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And so we can conclude that these angles are congruent.
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The measure of angle π΄π΅πΆ is equal to the measure of angle π΄π·πΆ, as we required.
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So now we have the property and a proof; weβre going to demonstrate a simple application.
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Given that the measure of angle π΅π΄π· is equal to 36 degrees and the measure of angle πΆπ΅π΄ is equal to 37 degrees, find the measure of angle π΅πΆπ· and the measure of angle πΆπ·π΄.
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Letβs begin by adding the angles that we know onto the diagram.
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The measure of angle π΅π΄π· is equal to 36 degrees, and the measure of angle πΆπ΅π΄ is equal to 37.
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We are looking to calculate the measure of angle π΅πΆπ·, which is this one, and the measure of angle πΆπ·π΄, which is this one.
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We now observe that the first unknown angle, thatβs π΅πΆπ·, is subtended by the same arc π΅π· as angle π΅π΄π·.
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And we know that inscribed angles subtended by the same arc are equal.
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So the measure of angle π΅π΄π· must be equal to the measure of angle π΅πΆπ·.
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But of course, weβve now seen that thatβs 36 degrees.
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In a similar way, we observed that angle π΄π·πΆ is subtended from the same arc as angle π΄π΅πΆ.
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And so, these two angles are congruent.
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The measure of angle π΄π΅πΆ must be equal to the measure of angle π΄π·πΆ.
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And thatβs 37.
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So, using the property of inscribed angles subtended from the same arc, we find the measure of angle π΅πΆπ· is equal to 36 degrees and the measure of angle πΆπ·π΄ is equal to 37 degrees.
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Now, in this example, we looked at how to solve problems involving numerical values for the angles.
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In our next example, letβs have a look at how to apply the same property, but to solve problems involving algebraic expressions.
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If the measure of angle π΅π΄π· is equal to two π₯ plus two degrees and the measure of angle π΅πΆπ· is equal to π₯ plus 18 degrees, determine the value of π₯.
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Letβs begin by adding the measure of angle π΅π΄π· and the measure of angle π΅πΆπ· to the diagram.
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In doing so, we see that each of these inscribed angles is subtended from arc π΅π·.
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And so we quote one of the theorems that we use when working with inscribed angles.
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That is, angles subtended by the same arc are equal, or alternatively angles in the same segment are equal.
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This must mean that angle π΅π΄π· is equal to angle π΅πΆπ·.
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This allows us to form and solve an equation in π₯.
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The measure of angle π΅π΄π· is two π₯ plus two degrees, and the measure of angle π΅πΆπ· is π₯ plus 18 degrees.
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So two π₯ plus two must be equal to π₯ plus 18.
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To solve this equation for π₯, letβs begin by subtracting π₯ from both sides, giving us π₯ plus two equals 18.
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Finally, we can isolate the π₯ by subtracting two from both sides.
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18 minus two is equal to 16.
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And so weβve determined the value of π₯; itβs 16.
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Now, it probably comes as no surprise that we can extend the properties of inscribed angles to work with distinct circles or even congruent arcs.
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In particular, if a pair of circles are congruent, then inscribed angles subtended by congruent arcs will be equal.
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What this means for our diagram is that if arcs π΄πΆ and π·πΉ are congruent, then the measure of angle π΄π΅πΆ must be equal to the measure of angle π·πΈπΉ.
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And so we see that whilst it might be tempting to look for the typical bow tie shape to help us solve problems involving inscribed angles, this isnβt always the most sensible route.
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In our next example, weβll demonstrate that.
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Given that the measure of angle πΉπΈπ· is equal to 14 degrees and the measure of angle πΆπ΅π΄ is equal to two π₯ minus 96 degrees, calculate the value of π₯.
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So letβs look at the diagram.
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We quickly see that arc π΄πΆ is congruent to arc π·πΉ.
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And we know that inscribed angles subtended by congruent arcs are going to be equal in measure.
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So this means that angle π΄π΅πΆ is going to be equal in measure to angle π·πΈπΉ.
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Weβre in fact told that the measure of angle π΄π΅πΆ or πΆπ΅π΄ is two π₯ minus 96.
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And the measure of angle πΉπΈπ·, which is the same as π·πΈπΉ, is 14 degrees.
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Since these angles are equal, we can form and solve an equation in π₯.
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That is, two π₯ minus 96 equals 14.
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To solve for π₯, we add 96 to both sides, giving us two π₯ is equal to 110.
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And finally, we divide through by two, and that gives us π₯ is equal to 55.
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And so given the information about angles πΉπΈπ· and πΆπ΅π΄, we can deduce that π₯ is equal to 55.
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Now, in all our previous examples, weβve considered congruent, that is, identical, circles.
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What do we do if weβre given a pair of concentric circles?
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Remember, concentric circles are a pair of circles which share the same center.
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We also know that any two circles will always be similar to one another.
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And so, we can say that inscribed angles subtended by two arcs of equal measure in these circles with the same center must be equal to one another.
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In our next example, weβll demonstrate what that would look like.
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In the figure, line segment π΄πΈ and line segment π΅πΆ pass through the midpoint of the circles.
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Given that the measure of angle πΉπΈπ· is 50 degrees and the measure of angle πΆπ΅π΄ is equal to two π₯ minus 10 degrees, find π₯.
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We begin by adding the relevant measurements to our diagram.
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πΉπΈπ· is 50 degrees, and πΆπ΅π΄ is two π₯ minus 10 degrees.
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Now, we do know that angles subtended from the same arc are equal.
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But we also know that angles subtended from arcs with equal measure are equal.
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And this is really useful when weβre working with a pair of concentric circles, because weβre able to say that the measure of arc πΉπ· is equal to the measure of arc πΆπ΄.
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Theyβre both equal to this angle here.
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Since the measure of those two arcs are equal, then the measure of any angles subtended from the arcs must also be equal.
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In other words, the measure of angle πΉπΈπ· must be equal to the measure of angle πΆπ΅π΄.
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And so, we can say that 50 must be equal to two π₯ minus 10.
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Then, we simply have an equation that we can solve for π₯.
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Weβll begin by adding 10 to both sides, giving us 60 equals two π₯.
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And then we divide through by two, giving us 30 is equal to π₯, or π₯ is equal to 30.
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In our previous examples, weβve used the properties of inscribed angles in a circle to find missing values.
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Now, it in fact follows that we can apply a reverse property to prove statements about circles.
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In other words, if we have two congruent angles subtended from the same line segment and on the same side of that line segment, then their vertices and the endpoints of the line segment must lie on a circle.
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In the diagram, for instance, because the measure of angle π΄π΅πΆ is equal to the measure of angle π΄π·πΆ and these two angles are subtended from the line segment π΄πΆ and theyβre on the same side, then the points π΄, π΅, πΆ, π· must all lie on a circle.
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In our final example, weβll use this information to decide whether a circle passes through four given points.
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Given that the measure of angle π΅πΆπ΄ equals 61 degrees and the measure of angle π·π΄π΅ equals 98 degrees, can a circle pass through the points π΄, π΅, πΆ, and π·?
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Remember, if there are a pair of congruent angles subtended by the same line segment and on the same side of it, then their vertices and the segmentβs endpoints lie on a circle in which that segment is a chord.
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Well, we have a line segment π΅π΄, from which angle π΅πΆπ΄ and π΅π·π΄ are subtended.
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The angles lie on the same side of that line segment.
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So if angle π΅πΆπ΄ is equal to angle π΅π·π΄, then all four of our points must lie on the circumference of a circle.
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Now, weβre given that the measure of angle π΅πΆπ΄ is 61 degrees and the measure of angle π΅π΄π· is 98.
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Since triangle π΅π·π΄ is isosceles, we can calculate the measure of angle π΅π·π΄ by subtracting 98 from 180 and then dividing by two.
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And that gives us that the measure of angle π΅π·π΄ is 41 degrees.
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So we see that the measure of angle π΅πΆπ΄ is not equal to the measure of angle π΅π·π΄.
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Since these angles are not equal, we observe that a circle cannot pass through the points, and the answer is no.
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Letβs recap the key concepts from this lesson.
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In this lesson, we learned that inscribed angles subtended by the same arc are equal.
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We also saw that inscribed angles subtended by congruent arcs or arcs of equal measure are also equal.
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Finally, we learned that the reverse of these ideas is also true.
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If there are two congruent angles subtended by the same line segment and on the same side of that line segment, then their vertices and the endpoints of that line segment lie on a circle.
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And in that circle, the line segment is a chord.