Lesson Video: Inscribed Angles in a Circle | Nagwa Lesson Video: Inscribed Angles in a Circle | Nagwa

# Lesson Video: Inscribed Angles in a Circle Mathematics • Third Year of Preparatory School

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In this video, we will learn how to identify theorems of finding the measure of an inscribed angle with respect to its subtended arc or central angle subtended by the same arc and the measures of inscribed angles in a semicircle.

15:44

### Video Transcript

In this video, we will learn how to find the measures of inscribed angles using the relationship between angles and arcs.

Before we talk about these angle relationships, letβs remember what an inscribed angle is. Itβs an angle where the vertex and two endpoints all lie on the circumference of the circle, on the outside of the circle. We could measure this inscribed angle in degrees. If this inscribed angle measures π degrees, then the arc created between these two endpoints will be two π degrees. Another way to say that is an inscribed angle measures half of the subtended arc thatβs created by that angle. 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 thatβs created by these endpoints, which here is two π.

Itβs also worth noting a special case. The special case is the case where the angle inscribed has endpoints that are at either end of a circleβs diameter. In this case, the subtended arc is 180 degrees, which makes the inscribed angle a right angle. And again, we can move that vertex and still create a right angle as long as the end points donβt move.

Before we move on, we should also remind ourselves of central angles. In a central angle, the vertex is the center of the circle. And when weβre dealing with a central angle, its angle measure will be equal to the subtended arcβs angle measure. The way weβve drawn it here, it will be equal to two π. And that means when an inscribed angle shares the endpoints with a central angle, the inscribed angle will be half the central angle.

Another thing we should know about arcs and circles is what happens when we have parallel chords. If we have chords like these that run parallel, arc π΄π· will be equal to arc π΅πΆ. That is to say, in a circle, arcs between parallel chords are congruent. In the circle weβve drawn here, the measure of arc π΄π· will be equal to the measure of arc π΅πΆ. Weβre now ready to use these circle theorems to calculate unknown angles.

In the figure, π is the center and the measure of angle ππ΄π΅ equals 59.5 degrees. What is the measure of angle π΄ππ΅? What is the measure of angle π΄πΆπ΅?

Weβre told that π is the center of this circle and that the measure of angle ππ΄π΅ is 59.5 degrees. We want to find the measure of angle π΄ππ΅ and the measure of π΄πΆπ΅. We see that the points π΄, π, and π΅ form a triangle. Both line segment ππ΅ and line segment ππ΄ 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. This means we can say that line segment ππ΄ is equal to line segment ππ΅. And it will mean that triangle π΄ππ΅ is an isosceles triangle.

In an isosceles triangle, the two angles opposite the radii are equal to each other. And that means we could say that angle π΄π΅π is also equal to 59.5 degrees. Since these three angles form a triangle, they must sum to 180 degrees. And so, we substitute the values we do know for angle ππ΄π΅ and angle π΄π΅π. We add the two angles we know, and we get 119 degrees. And then to solve for angle π΄ππ΅, we subtract 119 degrees from both sides, and we find that angle π΄ππ΅ is equal to 61 degrees. Thatβs the answer to part one.

Part two is a little bit less straightforward. We notice that both of these angles share the endpoints π΄, π΅, which means theyβre both subtended by the arc π΄π΅. But we need to make a clarification here. Angle π΄ππ΅ is a central angle that is subtended by arc π΄π΅, while angle π΄πΆπ΅ is an inscribed angle subtended by arc π΄π΅. And we remember that the central angle subtended by two points on a circle is twice the inscribed angle subtended by those two points. 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.

Based on that, we can say that the measure of angle π΄πΆπ΅ will be equal to one-half the measure of angle π΄ππ΅. So, we plug in 61 degrees for angle π΄ππ΅. Half of 61 degrees is 30.5 degrees. And so, the measure of angle π΄πΆπ΅ equals 30.5 degrees.

Hereβs another example.

From the figure, what is π₯?

Letβs start with what we know. We have angle π΄πΆπ΅, which measures 101 degrees. And we have angle π΄ππ΅. In this case, weβre talking about the reflex of angle π΄ππ΅. Thatβs the one thatβs greater than 180 degrees, which measures two π₯ plus eight degrees. Angle π΄πΆπ΅ and angle π΄ππ΅ share the endpoints π΄ and π΅. But because the vertex of angle π΄ππ΅ is the center of the circle, we say that angle π΄ππ΅ is a central angle for this circle. While the vertex for angle π΄πΆπ΅ is on the outside of the circle, making angle π΄πΆπ΅ an inscribed angle of the circle. And these three facts point us to the central angle theorem.

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. In this diagram, the inscribed angle is π degrees, and that would make the central angle two π degrees. By this, we can say that the measure of angle π΄ππ΅ is going to be equal to two times the measure of angle π΄πΆπ΅. The measure of the central angle will be equal to two times the measure of the inscribed angle. And so, we can say that two π₯ plus eight will be equal to two times 101. When we multiply two times 101, we get 202. And now, weβre ready to solve for π₯. Subtract eight from both sides. Two π₯ equals 194. Then, divide both sides by two, and we find that π₯ equals 97.

In our next example, we have some intersecting chords in a circle.

Given that the measure of angle π΄π΅π· equals 44 degrees and the measure of angle πΆπΈπ΄ equals 72 degrees, find π₯, π¦, and π§.

Letβs start by listing what we know. Angle π΄π΅π· equals 44 degrees, angle πΆπΈπ΄ measures 72 degrees. These two chords intersect at point πΈ. And that means we can say that angle π΅πΈπ· and angle πΆπΈπ΄ are vertical angles, which means their measure will be equal to one another. They are congruent angles. And in this case, that means that angle π΅πΈπ· is also equal to 72 degrees. The points πΈ, π΅, and π· form a triangle, which means that their three angles must sum to 180 degrees. And we can substitute what we know for these three angles into this equation. 72 plus 44 equals 116. 116 plus π§ equals 180. So, we subtract 116 from both sides. And we find that π§ equals 64 degrees.

We wonβt be able to follow the same procedure to find π₯ and π¦. So, weβll need to think about some of the circle theorems. 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·. We could write them as arc π΄π· intercepts angle π΄π΅π·. But thereβs another angle in this circle that also intercepts the same arc, and that would be angle π΄πΆπ·. Because both of these angles are subtended by the same arc, we can say that the measure of angle π΄πΆπ· will be equal to the measure of angle π΄π΅π·. And that means π₯ will be equal to 44 degrees.

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. If we wanted to confirm this, we could see that angle πΆπ΄π΅ intercepts arc πΆπ΅ and angle πΆπ·π΅ intercepts arc πΆπ΅. And so, we found that π₯ equals 44 degrees, and both π¦ and π§ equals 64 degrees.

In our next example, weβll have a diameter to consider.

Given that line segment π΄π΅ is a diameter in circle π and the measure of angle π΅ππ· equals 59 degrees, find the measure of angle π΄πΆπ· in degrees.

Letβs put what we know into the diagram. Angle π΅ππ· measures 59 degrees, and weβre trying to find the measure of angle π΄πΆπ·. If we start with what we know about angle π΅ππ·, since π΅ππ· has a vertex at the center of the circle, π΅ππ· is a central angle. And because angle π΅ππ· is a central angle, its subtended arc, arc π΅π·, also measures 59 degrees. Weβre also interested in angle π΄πΆπ·. But angle π΄πΆπ· is not a central angle. Itβs an inscribed angle because its vertex is on the circumference of the circle, as are both endpoints.

The arc associated with angle π΄πΆπ· would be arc π΄π·. We have a partial measurement for this arc, but weβre missing the distance from π΄ to π΅. But because we know that π΄π΅ is a diameter, it cuts the circle in half. And that means the measure of arc π΄π΅ is 180 degrees. 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·.

If we plug in what we know, the measure of arc π΄π· is 239 degrees. Because angle π΄πΆπ· is an inscribed angle and it has a subtended arc measure of 239 degrees, we can find out the exact measure of angle π΄πΆπ·. The measure of the inscribed angle π΄πΆπ· will be one-half its subtended arc, arc π΄π·. Since that arc is 239 degrees, we take half of that and we get 119.5 degrees for the measure of angle π΄πΆπ·.

In our final example, weβll look at how parallel chords can give us information about arc measures.

Given that the line segment π΄π΅ is a diameter of the circle and line segment π·πΆ is parallel to line segment π΄π΅, find the measure of angle π΄πΈπ·.

Weβre interested in the measure of angle π΄πΈπ·; thatβs this measure. And weβve been given a few other pieces of information. We know line segment π·πΆ is parallel to line segment π΄π΅. We know line segment π΄π΅ is the diameter. And on the figure, angle πΆπ΅π΄ has been labeled as 68.5 degrees.

At first, it might not seem like thereβs a very clear direction for where to go here. But if we start with the measure of angle πΆπ΄π΅, using that information, we could find the measure of arc πΆπ΄. Since angle πΆπ΄π΅ is an inscribed angle, its arc will be two times the measure of that inscribed angle. Arc π΄πΆ will then be equal to two times 68.5, which is 137 degrees. And because we know that line segment π΄π΅ is a diameter, arc π΄π΅ must be equal to 180 degrees. We can also say that arc π΄π΅ will be equal to arc π΅πΆ plus arc πΆπ΄.

We know π΄π΅ needs to be 180 degrees and arc πΆπ΄ is 137 degrees. To solve for the measure of arc π΅πΆ, we can subtract 137 from both sides of the equation. And we get the measure of arc π΅πΆ is 43 degrees. And hereβs where our parallel chords come into play. When you have parallel chords, their intercepted arcs are going to be congruent. And that means because arc πΆπ΅ equals 43 degrees, arc π·π΄ also equals 43 degrees.

And at this point, we began to see that arc π·π΄ is subtended by the angle π΄πΈπ·. Since angle π΄πΈπ· is an inscribed angle, its angle measure, the measure of angle π΄πΈπ·, is going to be equal to one-half the measure of arc π΄π·. We know that the measure of arc π΄π· is 43 degrees, and one-half of 43 is 21.5. And so, we can say that the measure of angle π΄πΈπ· is 21.5 degrees.

Before we finish, letβs quickly review the key points. If you have a central angle that measures two π degrees, its intercepted arc will also measure two π degrees. While an inscribed angle that intercepts the same arc will have half the angle measure, only π degrees. 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. We also can say that the angles subtended by the same arc on a circle will be equal. And finally, the arcs between parallel chords will always be congruent.

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