Geometry (B): READ: Arc Measures Reading

Arc Measures

Learning Objectives

Measure central angles and arcs of circles.
Find relationships between adjacent arcs.
Find relationships between arcs and chords.

Arc, Central Angle

In a circle, the central angle is formed by two radii of the circle with its vertex at the center of the circle. An arc is a section of the circle.

Minor and Major Arcs, Semicircle

A semicircle is half a circle. A major arc is longer than a semicircle and a minor arc is shorter than a semicircle.

An arc can be measured in degrees or in a linear measure (cm, ft, etc.). In this lesson we will concentrate on 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 $360^\circ$ minus the measure of the minor arc.

Minor arcs are named with two letters—the letters that denote the endpoints of the arc. In the figure above, the minor arc corresponding to the central angle $\angle{AOB}$ is called $\widehat{AB}$ . In order to prevent confusion, major arcs are named with three letters—the letters that denote the endpoints of the arc and any other point on the major arc. In the figure, the major arc corresponding to the central angle $\angle{AOB}$ is called $\widehat{ACB}$ .

Congruent Arcs

Two arcs that correspond to congruent central angles will also be congruent. In the figure below, $\angle{AOC} \cong \angle{BOD}$ because they are vertical angles. This also means that $\widehat{AC} \cong \widehat{DB}$ .

Arc Addition Postulate

The measure of the arc formed by two adjacent arcs is the sum of the measures of the two arcs.

In other words, $m\widehat{RQ}= m\widehat{RP} + m\widehat{PQ}$ .

Congruent Chords Have Congruent Minor Arcs

In the same circle or congruent circles, congruent chords have congruent minor arcs.

Proof. Draw the following diagram, in which the chords $\overline{DB}$ and $\overline{AC}$ are congruent.

Construct $\triangle{DOB}$ and $\triangle{AOC}$ by drawing the radii for the center $O$ to points $A, B, C$ and $D$ respectively.

Then, $\triangle AOC \cong \triangle BOD$ by the $SSS$ postulate.

This means that central angles, $\angle{AOC} \cong \angle{BOD}$ , which leads to the conclusion that $\widehat{AC} \cong \widehat{DB}$ .

Congruent Minor Arcs Have Congruent Chords and Congruent Central Angles

In the same circle or congruent circles, congruent chords have congruent minor arcs.

Proof. Draw the following diagram, in which the $\widehat{AC} \cong \widehat{DB}$ . In the diagram $\overline{DO}, \overline{OB}, \overline{AO}$ , and $\overline{OC}$ are radii of the circle.

Since $\widehat{AC} \cong \widehat{DB}$ , this means that the corresponding central angles are also congruent: $\angle{AOC} \cong \angle{BOD}$ .

Therefore, $\triangle AOC \cong \triangle BOD$ by the $SAS$ postulate.

We conclude that $\overline{DB} \cong\overline{AC}$ .

Here are some examples in which we apply the concepts and theorems we discussed in this section.

Example 1

Find the measure of each arc.

A. $m\widehat{ML}$

B. $m\widehat{PM}$

C. $m\widehat{LMQ}$

A. $m\widehat{ML} = m\angle{LOM} = 60^\circ$

B. $m\widehat{PM} = m\angle{POM} = 180^\circ - \angle{LOM} = 120^\circ$

C. $m\widehat{LMQ} = m\widehat{ML} + m\widehat{PM} + m\widehat{PQ} =60^\circ + 120^\circ + 60^\circ = 240^\circ$

Example 2

Find $m \widehat{AB}$ in circle $O$ . The measures of all three arcs must add to $360^\circ$ .

$x^\circ + 20^\circ + (4x)^\circ + 5^\circ + (3x)^\circ + 15^\circ &= 360^\circ\\ (8x)^\circ &= 320^\circ\\ x &= 40\\ m\widehat{AB} &= 60^\circ$

Example 3

The circle $x^2 + y^2 = 25$ goes through $A = (5, 0)$ and $B = (4, 3)$ . Find $m\widehat{AB}$ .

Draw the radii to points $A$ and $B$ .

We know that the measure of the minor arc $AB$ is equal to the measure of the central angle.

$\tan O = \frac{3} {4} \Rightarrow m\angle{AOB} = 36.90^\circ$

$m\widehat{AB} \approx 36.9^\circ$

Lesson Summary

In this section we learned about arcs and chords, and some relationships between them. We found out that there are major and minor arcs. We also learned that if two chords are congruent, so are the arcs they intersect, and vice versa.

The following questions are for your own review. The answers are listed below for you to check your work and understanding.

Review Questions

In the circle identify the following:
1. four radii
2. a diameter
3. two semicircles
4. three minor arcs
5. two major arcs
Find the measure of each angle in :
1. $m\angle{AOF}$
2. $m\angle{AOB}$
3. $m\angle{AOC}$
4. $m\angle{EOF}$
5. $m\angle{AOD}$
6. $m\angle{FOD}$
Find the measure of each angle in :
1. $m\widehat{MN}$
2. $m\widehat{LK}$
3. $m\widehat{MP}$
4. $m\widehat{MK}$
5. $m\widehat{NPL}$
6. $m\widehat{LKM}$
The students in a geometry class were asked what their favorite pie is. The table below shows the result of the survey. Make a pie chart of this information, showing the measure of the central angle for each slice of the pie.

Kind of pie	Number of students
apple	$6$
pumpkin	$4$
cherry	$2$
lemon	$3$
chicken	$7$
banana	$3$
total	$25$

Three identical pipes of diameter $14\;\mathrm{inches}$ are tied together by a metal band as shown. Find the length of the band surrounding the three pipes.
Four identical pipes of diameter $14\;\mathrm{inches}$ are tied together by a metal band as shown. Find the length of the band surrounding the four pipes.

Review Answers

1. $\overline{OQ}, \overline{OK}, \overline{OL}, \overline{OM}, \overline{ON}, \overline{OP}$
2. $\overline{KN}, \overline{QM}, \overline{PL}$
3. $\widehat{LKP}, \widehat{KPN}, \widehat{QPM}, \widehat{PML}, \widehat{NMK}, \widehat{MLQ}$
4. Some possibilities: $\widehat{NM}, \widehat{PK}, \widehat{QL}$ .
5. Some possibilities: $\widehat{MPK}, \widehat{NQL}$ .
1. $70^\circ$
2. $20^\circ$
3. $110^\circ$
4. $90^\circ$
5. $180^\circ$
6. $110^\circ$
1. $62^\circ$
2. $77^\circ$
3. $139^\circ$
4. $118^\circ$
5. $257^\circ$
6. $319^\circ$
$42 + 14\pi \approx\ 86\;\mathrm{in}$ .
$56 + 14\pi \approx\ 100\;\mathrm{in}$ .

Last modified: Tuesday, June 29, 2010, 11:25 AM