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# 9.2: Properties of Arcs

Difficulty Level: At Grade Created by: CK-12

## Learning Objectives

• Define and measure central angles, 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. Use the trig ratios.

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 on this ride? Why do you think it is important to have equally spaced seats on a Ferris wheel?

## Central Angles & Arcs

Recall that a straight angle is 180\begin{align*}180^\circ\end{align*}. If take two straight angles and put one on top of the other, we would have a circle. This means that a circle has 360, 180+180\begin{align*}360^\circ, \ 180^\circ + 180^\circ\end{align*}. This also means that a semicircle, or half circle, is 180\begin{align*}180^\circ\end{align*}.

Arc: A section of the circle.

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

To label an arc, place a curve above the endpoints. You may want to use 3 points to clarify.

\begin{align*}\widehat{EHG} \ \text{and} \ \widehat{EJG} \ \text{are semicircles} \qquad \quad m \widehat{EHG} =180^\circ\end{align*}

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

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.

The central angle is \begin{align*}\angle BAC\end{align*}.

The minor arc is \begin{align*}\widehat{BC}\end{align*}.

The major arc is \begin{align*}\widehat{BDC}\end{align*}.

Every central angle divides a circle into two arcs.

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 is \begin{align*}360^\circ\end{align*} minus the measure of the minor arc.

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}= m\widehat{ACB}\end{align*}. So, \begin{align*}m\widehat{AB}= 102^\circ\end{align*}.

\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 \rightarrow 180^\circ - 60^\circ\\ m\widehat{ED} & = m \angle EAD = 38^\circ \ \rightarrow 180^\circ - 90^\circ - 52^\circ\\ 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 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: \begin{align*}\angle ACD = \angle ECB\end{align*} because they are vertical angles. \begin{align*}\angle DCB = \angle ACE\end{align*} because they are also vertical angles.

\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:

a) \begin{align*}\widehat{AD} \cong \widehat{BC}\end{align*} because they have the same central angle measure and in the same circle.

b) The two arcs have the same measure, but are not congruent because the circles have different radii.

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

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

\begin{align*}m\widehat{AD} + m\widehat{DB} = m\widehat{ADB}\end{align*}

Example 5: Find the measure of the 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{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*}

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{DFC} = 38^\circ + 120^\circ + 60^\circ + 52^\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:

\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 the balance of the Ferris wheel.

## Review Questions

• Questions 1-6 use the definition of minor arc, major arc, and semicircle.
• Question 7 is similar to Example 3.
• Questions 8 and 9 are similar to Example 5.
• Questions 10-15 are similar to Example 1.
• Questions 16-18 are similar to Example 4.
• Questions 19-26 are similar to Example 2 and 5.
• Questions 27-29 are similar to Example 6.
• Question 30 is a challenge.

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*}.

Find the measure of the minor arc and the major arc in each circle below.

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*}\widehat{GAB}\end{align*}
4. \begin{align*}\widehat{FG}\end{align*}
5. \begin{align*}\widehat{EDB}\end{align*}
6. \begin{align*}\widehat{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. Challenge What can you conclude about \begin{align*}\bigodot A\end{align*} and \begin{align*}\bigodot B\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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