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Measures of arcs and central angles

We can use a few more theorems to find the measures of arcs and central angles of circles.  Let’s begin by stating a few theorems:

THEOREM:  The measure of a central angle is equal to the measure of the arc it intersects.

THEOREM:  The measure of a major arc (an arc greater than a semicircle) is equal to \(360^\circ \) minus the measure of the corresponding minor arc.

THEOREM:  Vertical angles are equal.

EXAMPLE:  Find the measure of the arc \(\widehat {IKH}\)

Circles-Measures-of-Arcs-and-Central-Angles-1

SOLUTION: \(\widehat {IKH}\) is a major arc, so, by a theorem above, its measure is \(360^\circ  - m\widehat {IH}\). Then we must find the measure of \(\widehat {IH}\).

Again, by a theorem above, we know that the measure of the central angle corresponding to \(\widehat {JK}\) must be \(70^\circ \). Now we can conclude, by the fact that straight angles measure \(180^\circ \), that the central angle corresponding to \(\widehat {IH}\) equals \(180 - 70 - 70 = 40\). That is, \(m\widehat {IH} = 40^\circ \).

So we can conclude that \(m\widehat {IKH} = 360^\circ  - 40^\circ  = 320^\circ \).

 

EXAMPLE:  Find \(m\angle JKI\)

Circles-Measures-of-Arcs-and-Central-Angles-2

SOLUTION:  Since \(\widehat {FKG} = 50^\circ \), we can conclude, by a theorem above, that \(\angle FKG = 50^\circ \). Then, by the vertical angle theorem, we know that \(\angle JKL = \angle FKG\). That is, \(\angle JKL = 50^\circ \).

Below you can download some free math worksheets and practice.


Downloads:
4116 x

Find the measure of the arc or central angle indicated. Assume that lines which appear to be diameters are actual diameters.

This free worksheet contains 10 assignments each with 24 questions with answers.

Example of one question:

Circles-Measures-of-arcs-and-central-angles-Easy

Watch bellow how to solve this example:

 

Downloads:
2832 x

Find the measure of the arc or central angle indicated. Assume that lines which appear to be diameters are actual diameters.

This free worksheet contains 10 assignments each with 24 questions with answers.

Example of one question:

Circles-Measures-of-arcs-and-central-angles-Medium

Watch bellow how to solve this example:

 

Downloads:
2811 x

Find the measure of the arc or central angle indicated. Assume that lines which appear to be diameters are actual diameters.

This free worksheet contains 10 assignments each with 24 questions with answers.

Example of one question:

Circles-Measures-of-arcs-and-central-angles-Hard

Watch bellow how to solve this example:

 
 
 

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