Chords Theorems-Circles-Geometry Help

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The video focuses on problems related to the following circle theorem.

In the same circle,two minor arcs are congruent if and only if the corresponding chords are congruent.

Transcript

Welcome to Circles part 1 Lesson 5. We are going to focus on chords and the theorems. The first theorem we are going to look at is circle theorem number one which states, in the same circle, or congruent circles, because they would match up, two minor arcs are congruent if and only if their corresponding chords are congruent. That means two things. Here is a picture of the circle theorem. You have two chords that are marked congruent AB and CD are marked congruent. That means the arcs that are associated with those two chords, so the arc CD would have the same measure as the arc AB even though they are not central angles, since those chords are congruent it means the associated arcs are congruent. Notice that this is an if and only if statement, it means that you can say it in both directions. So the arcs are congruent if the chords are congruent, and the chords are congruent if arcs are congruent. You can know either piece of information, and then deduct the opposite. So if you know the arcs, you can say the chords are congruent and if you know the chords you can say the arcs are congruent. Let's look at some practice problems. The first one says GH has a measure of 100 degrees, so this arc has a measure of 100, what is the measure of arc IJ ? That is very simple, this is the basic concept, congruent chords and congruent arcs so the first one is IJ is 100. Pretty simple. Let's look at the second sample problem. We have a little more information. The circle arc JI has a measure of 55 degrees, well JI is not the same as another arc because there is not a congruent chord associated with it. The arc HJ is 115 , and we need to figure out what GH is. We know these two arcs, but we don't know the arc measure of IJ or GH, but we do know that these two arcs are congruent. What we can do is take these two arcs that are given and add them together. Take 115 and 55, which gives us 170. Therefore the measure of these two arcs, take up 170 degrees of 360 degrees of the circle. Now we know we have 360 in a circle,so we can take 170 and subtract it, and this gives us 190 degrees. This 190 can be split equally between these two arcs, so we will divide by two , and each of these arc measures are 95 degrees. Therefore 95 degrees becomes the measure of arc GH

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