find arc measures

MM2G3 Students will understand properties of circles.

MM2G3 a Understand and use properties of chords, tangents, and secants as an application of triangle similarity.

Find Arc MeasuresEssential Question:How do we use angle measures to find arc measures?

M2 Unit 3: Day 2

Monday, April 24, 2023

Lesson 6.2



ANSWER x = 60; y = 60ANSWER x = 35; y = 35

1.

Find x and y.

2.

Warm Ups



Daily Homework Quiz 1. Give the name that best describes the figure .

a. CD b. AB

c. FD d. EP



2. Tell how many common tangents the circles have .

ANSWER

One tangent; it is a vertical line through the point of tangency.

Daily Homework Quiz



3. Is AB tangent to C? Explain. .

ANSWER

Yes; 16 + 30 = 1156 = 34 so AB AC, and a line to a radius at its endpoint is tangent to the circle.

2 22

Daily Homework Quiz



4. Find x.

ANSWER 12

Daily Homework Quiz



http://free-stainedglasspatterns.com/2curvesround.html

Arcs in a stained glass window

http://free-stainedglasspatterns.com/2curvesround.html



8

ARCSThe part or portion on the circle from some point B to C is called an arc.

¼ABC

A

B

C»BC

Arcs :

Semicircle: An arc that is equal to 180°.

Example:

OA

B

CExample:



9

Minor Arc & Major Arc

»AB

¼ABC

Minor Arc : A minor arc is an arc that is less than 180°A minor arc is named using its endpoints with an “arc” above.

A

B

Example:

Major Arc: A major arc is an arc that is greater than 180°.

A major arc is named using its endpoints along with another point on the arc (in order).A

B

CExample:

O



Lesson 8-1: Circle Terminology 10

Example: ARCSIdentify a minor arc, a major arc, and a semicircle, given that is a diameter.

» » » », , ,DE EC CF DF

CD

AC

D

E

F

Minor Arc:

Major Arc: ¼ ¼ ¼ ¼, , ,CEF EDC DFE FCD

Semicircle: ¼ ¼ ¼ ¼, , ,CED CFD EDF ECF



Central Angles A central angle is an angle whose vertex is

the center of the circle and whose sides intersect the circle.

A

P

B

APB is a central angle

Central Angle(of a circle)

Central Angle(of a circle)

NOT A Central Angle

(of a circle)



Measuring Arcs

The measure of an arc is the same as the measure of its associated central angle.

A

P

B

100100

100

m APB

mAB

= °= °

R)



GUIDED PRACTICE

Identify the given arc as a major arc, minor arc, or

semicircle, and find the measure of the arc.

Identifying arcs

TQ is a minor arc, so m TQ = 120o.

1. TQ

SOLUTION

. QRT2

SOLUTION

QRT is a major arc, so m QRT= 240o.



GUIDED PRACTICE

. TQR3

SOLUTION

TQR is a semicircle, so m TQR = 180o.

. QS4

SOLUTION

QS is a minor arc, so m QS = 160o.



Identifying arcs



. TS5

SOLUTION

TS is a minor arc, so m TS = 80o.

. RST6

SOLUTION

RST is a semicircle, so m RST = 180o.



Identifying arcs



Find measures of arcs

RS7. RTS8. RST9.

SOLUTION

RS is a minor arc, so mRS = m RPS = 110o.7.

RTS is a major arc, so mRTS = 360o 110o = 250o.8. –

Find the measure of each arc of P, where RT is a diameter.

9. RT is a diameter, so RST is a semicircle, and mRST = 180o.



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

Arc Addition Postulate

¼ » »= +ABmABC m mBC




A recent survey asked teenagers if they would rather meet a famous musician, athlete, actor, inventor, or other person. The results are shown in the circle graph. Find the indicated arc measures.

Survey

10. mAC

SOLUTION

10. mAC mAB= + mBC= 29o + 108o

= 137o



11. mACD = mAC + mCD= 137o + 83o

= 220o


Survey

SOLUTION

11. mACD




EXAMPLE 2

mADC mAC= 360o – 12.

= 360o – 137o

= 223o


Survey

SOLUTION

12. mADC




EXAMPLE 2

13. mEBD = 360o – mED

= 360o – 61o

= 299o

13. mEBD


Survey

SOLUTION




Congruent arcs are two arcs with the same measure that are arcs of the same circle or of congruent circles.

Congruent Arc Definition

Congruent Circles DefinitionCongruent Circles DefinitionCongruent circles are two circles with the same radius.



EXAMPLE 3Identify congruent arcs

Tell whether the red arcs are congruent. Explain why or why not.

14. 15.

SOLUTION

14. CD EF because they are in the same circle and mCD = mEF

15. RS and TU have the same measure, but are not congruent because they are arcs of circles that are not congruent.



EXAMPLE 3Identify congruent arcs

16. VX YZ because they are in congruent circles and mVX = mYZ .


16.

SOLUTION



GUIDED PRACTICE


17.

AB CD because they are in congruent circles and mAB = mCD .

SOLUTION



GUIDED PRACTICE


18.

SOLUTION

MN and PQ have the same measure, but are not congruent because they are arcs of circles that are not congruent.



An angle whose vertex is the center of the circle.

An arc with endpoints that are the endpoints of a diameter.

An unbroken part of a circle.

Part of a circle measuring less than 180°

Part of a circle measuring between 180° and 360°

Definition ReviewCentral angle

Semicircle

Arc

Minor arc

Major



The measure of its central angle.

The difference between 360° and the measure of the related minor arc.

Two circles with the same radius

Two arcs with the same measure and in the same circle or congruent circles.

Definition ReviewMeasure of a minor arc

Measure of a major arc

Congruent Circles

Congruent Arcs



HomeworkPage 195-196# 1 -12 all, 15, # 19 – 24 all, 32 – 34 all.

find arc measures

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