Big Ideas Math Geometry, 2014
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Big Ideas Math Geometry, 2014 View details
2. Finding Arc Measures
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Exercise 4 Page 539

A minor arc is an arc that measures less than 180^(∘). A major arc is an arc that measures greater than 180^(∘). A semicircle is an arc with endpoints that lie on a diameter.

Type: Minor arc
Measure: 160^(∘)

Practice makes perfect

An arc is a portion of a circle defined by two endpoints. A central angle separates the circle into two arcs — the major arc and the minor arc. The measures of these arcs are related to the measure of the central angle.

Arc Measure
A minor arc is the shortest arc connecting two endpoints on a circle. The measure is less than 180^(∘) and is equal to the measure of its related central angle.
A major arc is the longest arc connecting two endpoints on a circle. The measure is greater than 180^(∘) and is equal to 360^(∘) minus the measure of the minor arc with the same endpoints.
A semicircle is an arc with endpoints that lie on a diameter. The measure of a semicircle is 180^(∘).
Now, let's consider the given diagram.
The arc that we are looking at is QS. It is the shortest arc that connects the endpoints Q and S. Therefore, QS is a minor arc. To find its measure, we will need to use the Arc Addition Postulate. With this postulate and knowing that mRST= 180^(∘), we can find mRS.
mRST=mRS+mST
180^(∘)=mRS+ 80^(∘)
Solve for mRS
100^(∘)=mRS
mRS=100^(∘)
Let's add the obtained measure to our diagram.
Finally, using the Arc Addition Postulate once more, we can find mQS.
mQS=mQR+mRS
mQS= 60^(∘)+ 100^(∘)
mQS=160^(∘)