Back To CourseGeometry: High School
15 chapters | 160 lessons
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Jeff teaches high school English, math and other subjects. He has a master's degree in writing and literature.
Here's a clock. This particular time, 3 o'clock, is a memorable one. When I was in high school, it was what I always hoped to see when I looked up, waiting for the final bell to ring.
But this clock is more than just a way of telling time. It's also a circle. And the minute hand? It's a radius, or a line extending from the circle's center to its edge, or circumference. Yep, we're talking geometry here. If we take the hour hand and extend it to the circumference, then it's also a radius.
These two lines show us three o'clock. And this angle here? It's called a central angle. A central angle is the angle formed by two radii in a circle.
How many total degrees are in a circle? In other words, in one hour, how many degrees does the minute hand travel? 360. So our central angle will be somewhere between 0, like this clock at 12 o'clock, and 360, like this clock also at 12 o'clock. If we put points on the circle like this, then we can name our angle. This is angle AOB.
Oh, and if we draw a line segment known as a chord right here, we make a triangle. A chord is just a line with endpoints on a circle. Now, what do we know about this triangle? First, if this chord is our second hand, then our clock is totally broken. But let's say it's just a chord.
The two radius lines are equal, since they're both traveling the same distance from the center of the circle. So this is an isosceles triangle. That means angle OAB and angle OBA are the same.
Let's practice a bit with central angles. First, let's talk time. 3 o'clock is exactly one quarter of the circle, so this central angle is one quarter of 360, or 90 degrees. What about 6 o'clock? That's half the circle, so it's 180 degrees.
Now let's look at a triangle involving a central angle. What's the measure of the central angle in this circle? We can see that angle OAB is 50 degrees. Since both OA and OB are radius lines, they are equal, and this is an isosceles triangle. That means angle OBA is also 50 degrees. Since the sum of the interior angles of a triangle is 180, then the central angle, angle AOB, must be 180 - 100, or 80 degrees.
Now here's a funny-looking clock. Did our minute and hour hands break? No, but this clock is completely non-functional as a timepiece. Instead of two radii, where the lines originate at the circle's center, we have two lines that start at the circumference.
This is an inscribed angle, or the angle formed by points on the circle's circumference. In this angle, which we call angle ACB, point C is the vertex and points A and B are the endpoints.
Inscribed angles have some special properties. First, with fixed endpoints, the measure of the inscribed angle remains the same regardless of the location of the vertex. So we can move C anywhere along the circumference and this angle doesn't change. That means that when we have two inscribed angles, as we see here, the angles are the same.
Second, when they share endpoints, the measure of an inscribed angle is half the measure of a central angle. So in this circle, angle AOB is twice angle ACB. This works as long as point C, the vertex of the inscribed angle, isn't on the arc formed by the central angle. If it is in that arc, well, it'll look kind of backwards. And then its measure would be the supplement of half the central angle.
Let's practice what we just learned. Here's a circle with two inscribed angles. We know that angle ACB is 28 degrees. What is angle ADB? Since both angles share endpoints, then angle ADB must also be 28 degrees.
Okay, what about this one? What's the measure of the central angle, angle AOB? We know the inscribed angle, angle ACB, is 22 degrees. So the central angle will be twice that, or 44 degrees.
In this example, we know the central angle, angle AOB, is 60 degrees. So what is the inscribed angle, angle ACB? This is just the reverse of the last one. It's half of the central angle, or 30 degrees.
Let's get a little trickier. Look at this one. What is the inscribed angle, angle ACB? We have an isosceles triangle here and we know this angle is 40 degrees. So this angle is also 40 degrees. That makes our central angle 180 - 80, or 100 degrees. And we know the inscribed angle is half the central angle, so angle ACB equals 50 degrees.
Here's another tricky one. What is the measure of angle CAD? Well, there are two inscribed angles that share endpoints. We know the angles are the same, but we don't know either one. Fortunately, we know a couple of the other angles, and we can utilize our geometry wizard skills.
We know this angle, angle BXD, is 100 degrees. So that means angle AXC is also 100 degrees. Why? They're vertical angles, or the angles opposite each other. They're always going to be the same. Well, look what we just did. We just found two of three angles inside a triangle. If angle ACB is 28 degrees. then angle CAD is just 180 minus 128, or 52 degrees.
In summary, the central angle in a circle is the angle formed by two radius lines. An inscribed angle is the angle formed by points on the circle's circumference. There are a few key things to know about central and inscribed angles. With central angles, a chord connecting the points on the circumference forms an isosceles triangle.
Central angles are also twice the measure of inscribed angles that share the same endpoints. Also, two inscribed angles that share endpoints will be equal, even if their vertices are not the same. And back to our clock, it's now the equivalent of 3 o'clock for this lesson. We made it!
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Back To CourseGeometry: High School
15 chapters | 160 lessons