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# 9.2: Properties of Arcs

Difficulty Level: At Grade Created by: CK-12

## Learning Objectives

• Define and measure central angles in circles.
• Define minor arcs and major arcs.

## Review Queue

1. What kind of triangle is ABC\begin{align*}\triangle ABC\end{align*}?

2. How does BD¯¯¯¯¯¯¯¯\begin{align*}\overline{BD}\end{align*} relate to ABC\begin{align*}\triangle ABC\end{align*}?

3. Find mABC\begin{align*}m \angle ABC\end{align*} and mABD\begin{align*}m \angle ABD\end{align*}.

Round to the nearest tenth.

4. Find AD\begin{align*}AD\end{align*}.

5. Find AC\begin{align*}AC\end{align*}.

Know What? The Ferris wheel to the right has equally spaced seats, such that the central angle is 20\begin{align*}20^\circ\end{align*}. How many seats are there? Why do you think it is important to have equally spaced seats on a Ferris wheel?

If the radius of this Ferris wheel is 25 ft., how far apart are two adjacent seats? Round your answer to the nearest tenth. The shortest distance between two points is a straight line.

## Central Angles & Arcs

Central Angle: The angle formed by two radii of the circle with its vertex at the center of the circle.

In the picture to the right, the central angle would be BAC\begin{align*}\angle BAC\end{align*}. Every central angle divides a circle into two arcs. In this case the arcs are \begin{align*}\widehat{BC}\end{align*} and \begin{align*}\widehat{BDC}\end{align*}. Notice the \begin{align*}\bigodot\end{align*} above the letters. To label an arc, always use this curve above the letters. Do not confuse \begin{align*}\overline{BC}\end{align*} and \begin{align*}\widehat{BC}\end{align*}.

Arc: A section of the circle.

If \begin{align*}D\end{align*} was not on the circle, we would not be able to tell the difference between \begin{align*}\widehat{BC}\end{align*} and \begin{align*}\widehat{BDC}\end{align*}. There are \begin{align*}360^\circ\end{align*} in a circle, where a semicircle is half of a circle, or \begin{align*}180^\circ\end{align*}. \begin{align*}m \angle EFG = 180^\circ\end{align*}, because it is a straight angle, so \begin{align*}m \widehat{EHG}= 180^\circ\end{align*} and \begin{align*}m \widehat{EJG} = 180^\circ\end{align*}.

Semicircle: An arc that measures \begin{align*}180^\circ\end{align*}.

Minor Arc: An arc that is less than \begin{align*}180^\circ\end{align*}.

Major Arc: An arc that is greater than \begin{align*}180^\circ\end{align*}. Always use 3 letters to label a major arc.

An arc can be measured in degrees or in a linear measure (cm, ft, etc.). In this chapter we will use degree measure. The measure of the minor arc is the same as the measure of the central angle that corresponds to it. The measure of the major arc equals to \begin{align*}360^\circ\end{align*} minus the measure of the minor arc. In order to prevent confusion, major arcs are always named with three letters; the letters that denote the endpoints of the arc and any other point on the major arc. When referring to the measure of an arc, always place an “\begin{align*}m\end{align*}” in from of the label.

Example 1: Find \begin{align*}m \widehat{AB}\end{align*} and \begin{align*}m \widehat{ADB}\end{align*} in \begin{align*}\bigodot C\end{align*}.

Solution: \begin{align*}m \widehat{AB}\end{align*} is the same as \begin{align*}m \angle ACB\end{align*}. So, \begin{align*}m \widehat{AB}= 102^\circ\end{align*}. The measure of \begin{align*}m \widehat{ADB}\end{align*}, which is the major arc, is equal to \begin{align*}360^\circ\end{align*} minus the minor arc.

\begin{align*}m \widehat{ADB}=360^\circ-m \widehat{AB}=360^\circ-102^\circ=258^\circ\end{align*}

Example 2: Find the measures of the arcs in \begin{align*}\bigodot A\end{align*}. \begin{align*}\overline{EB}\end{align*} is a diameter.

Solution: Because \begin{align*}\overline{EB}\end{align*} is a diameter, \begin{align*}m \angle EAB=180^\circ\end{align*}. Each arc is the same as its corresponding central angle. \begin{align*}m \widehat{BF} &= m \angle FAB=60^\circ\\ m \widehat{EF} &= m \angle EAF= 120^\circ \qquad \rightarrow m \angle EAB - m \angle FAB\\ m \widehat{ED} &= m \angle EAD=38^\circ \qquad \ \rightarrow m \angle EAB - m \angle BAC - m \angle CAD\\ m \widehat{DC} &= m \angle DAC=90^\circ\\ m \widehat{BC} &= m \angle BAC=52^\circ\end{align*}

Congruent Arcs: Two arcs are congruent if their central angles are congruent.

Example 3: List all the congruent arcs in \begin{align*}\bigodot C\end{align*} below. \begin{align*}\overline{AB}\end{align*} and \begin{align*}\overline{DE}\end{align*} are diameters.

Solution: From the picture, we see that \begin{align*}\angle ACD\end{align*} and \begin{align*}\angle ECB\end{align*} are vertical angles. \begin{align*}\angle DCB\end{align*} and \begin{align*}\angle ACE\end{align*} are also vertical angles. Because all vertical angles are equal and these four angles are all central angles, we know that \begin{align*}\widehat{AD} \cong \widehat{EB}\end{align*} and \begin{align*}\widehat{AE} \cong \widehat{DB}\end{align*}.

Example 4: Are the blue arcs congruent? Explain why or why not.

a)

b)

Solution: In part a, \begin{align*}\widehat{AD} \cong \widehat{BC}\end{align*} because they have the same central angle measure. In part b, the two arcs do have the same measure, but are not congruent because the circles are not congruent.

Just like the Angle Addition Postulate and the Segment Addition Postulate, there is an Arc Addition Postulate. It is very similar.

Arc Addition Postulate: The measure of the arc formed by two adjacent arcs is the sum of the measures of the two arcs.

Using the picture from Example 3, we would say \begin{align*}m \widehat{AE}+m \widehat{EB}=m \widehat{AEB}\end{align*}.

Example 5: Reusing the figure from Example 2, find the measure of the following arcs in \begin{align*}\bigodot A\end{align*}. \begin{align*}\overline{EB}\end{align*} is a diameter.

a) \begin{align*}m \widehat{FED}\end{align*}

b) \begin{align*}m \widehat{CDF}\end{align*}

c) \begin{align*}m \widehat{BD}\end{align*}

d) \begin{align*}m \widehat{DFC}\end{align*}

Solution: Use the Arc Addition Postulate.

a) \begin{align*}m \widehat{FED}=m \widehat{FE}+m \widehat{ED}=120^\circ+38^\circ=158^\circ\end{align*}

We could have labeled \begin{align*}\widehat{FED}\end{align*} as \begin{align*}\widehat{FD}\end{align*} because it is less than \begin{align*}180^\circ\end{align*}.

b) \begin{align*}m \widehat{CDF}=m \widehat{CD}+m \widehat{DE}+m \widehat{EF}=90^\circ+38^\circ+120^\circ=248^\circ\end{align*}

c) \begin{align*}m \widehat{BD}=m \widehat{BC}+m \widehat{CD}=52^\circ+90^\circ=142^\circ\end{align*}

d) \begin{align*}m \widehat{DFC}=38^\circ+120^\circ+60^\circ+52^\circ=270^\circ\end{align*} or \begin{align*}m \widehat{DFC} =360^\circ-m \widehat{CD}=360^\circ-90^\circ=270^\circ\end{align*}

Example 6: Algebra Connection Find the value of \begin{align*}x\end{align*} for \begin{align*}\bigodot C\end{align*} below.

Solution: There are \begin{align*}360^\circ\end{align*} in a circle. Let’s set up an equation.

\begin{align*}m \widehat{AB}+m \widehat{AD}+m \widehat{DB} &= 360^\circ\\ (4x+15)^\circ+92^\circ+(6x+3)^\circ &= 360^\circ\\ 10x+110^\circ &= 360^\circ\\ 10x &= 250^\circ\\ x &= 25^\circ\end{align*}

Know What? Revisited Because the seats are \begin{align*}20^\circ\end{align*} apart, there will be \begin{align*}\frac{360^\circ}{20^\circ}=18\end{align*} seats. It is important to have the seats evenly spaced for balance. To determine how far apart the adjacent seats are, use the triangle to the right. We will need to use sine to find \begin{align*}x\end{align*} and then multiply it by 2.

\begin{align*}\sin 10^\circ &= \frac{x}{25}\\ x = 25 \sin 10^\circ &= 4.3 \ ft.\end{align*}

The total distance apart is 8.6 feet.

## Review Questions

Determine if the arcs below are a minor arc, major arc, or semicircle of \begin{align*}\bigodot G\end{align*}. \begin{align*}\overline{EB}\end{align*} is a diameter.

1. \begin{align*}\widehat{AB}\end{align*}
2. \begin{align*}\widehat{ABD}\end{align*}
3. \begin{align*}\widehat{BCE}\end{align*}
4. \begin{align*}\widehat{CAE}\end{align*}
5. \begin{align*}\widehat{ABC}\end{align*}
6. \begin{align*}\widehat{EAB}\end{align*}
7. Are there any congruent arcs? If so, list them.
8. If \begin{align*}m \widehat{BC}=48^\circ\end{align*}, find \begin{align*}m \widehat{CD}\end{align*}.
9. Using #8, find \begin{align*}m \widehat{CAE}\end{align*}.

Determine if the blue arcs are congruent. If so, state why.

Find the measure of the indicated arcs or central angles in \begin{align*}\bigodot A\end{align*}. \begin{align*}\overline{DG}\end{align*} is a diameter.

1. \begin{align*}\widehat{DE}\end{align*}
2. \begin{align*}\widehat{DC}\end{align*}
3. \begin{align*}\angle GAB\end{align*}
4. \begin{align*}\widehat{FG}\end{align*}
5. \begin{align*}\widehat{EDB}\end{align*}
6. \begin{align*}\angle EAB\end{align*}
7. \begin{align*}\widehat{DCF}\end{align*}
8. \begin{align*}\widehat{DBE}\end{align*}

Algebra Connection Find the measure of \begin{align*}x\end{align*} in \begin{align*}\bigodot P\end{align*}.

1. What can you conclude about \begin{align*}\bigodot A\end{align*} and \begin{align*}\bigodot B\end{align*}?

Use the diagram below to find the measures of the indicated arcs in problems 25-30.

1. \begin{align*}m \widehat{MN}\end{align*}
2. \begin{align*}m \widehat{LK}\end{align*}
3. \begin{align*}m \widehat{MP}\end{align*}
4. \begin{align*}m \widehat{MK}\end{align*}
5. \begin{align*}m \widehat{NPL}\end{align*}
6. \begin{align*}m \widehat{LKM}\end{align*}

Use the diagram below to find the measures indicated in problems 31-36.

1. \begin{align*}m \angle VUZ\end{align*}
2. \begin{align*}m \angle YUZ\end{align*}
3. \begin{align*}m \angle WUV\end{align*}
4. \begin{align*}m \angle XUV\end{align*}
5. \begin{align*}m \widehat{YWZ}\end{align*}
6. \begin{align*}m \widehat{WYZ}\end{align*}

1. isosceles
2. \begin{align*}\overline{BD}\end{align*} is the angle bisector of \begin{align*}\angle ABC\end{align*} and the perpendicular bisector of \begin{align*}\overline{AC}\end{align*}.
3. \begin{align*}m\angle ABC = 40^\circ, m \angle ABD=25^\circ\end{align*}
4. \begin{align*}\cos 70^\circ = \frac{AD}{9} \rightarrow AD = 9 \cdot \cos 70^\circ = 3.1\end{align*}
5. \begin{align*}AC = 2 \cdot AD = 2 \cdot 3.1 = 6.2\end{align*}

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