find the measure of each lettered angle
DESCRIPTION
Warm up. Find the measure of each lettered angle. UNIT OF STUDY Lesson 6.3 ARCS AND ANGLES. TOPIC VII - CIRCLES. Warm up = (FRI Starting Point). You will learn …. (FRI Ending Point ). To discover relationships between an inscribed angle of a circle and its intercepted arc. - PowerPoint PPT PresentationTRANSCRIPT
Find the measure of each lettered angle.
Warm up
UNIT OF STUDYLesson 6.3
ARCS AND ANGLES
TOPIC VII - CIRCLES
Warm up = (FRI Starting Point)
You will learn ….(FRI Ending Point)
To discover relationships
between an inscribed angle of a
circle and its intercepted arc
Content….(FRE - Research)
Complete your investigations, and as
you see the power point presentation,
on your notebook and/or your packet,
write down the key concepts and
complete the conjectures.
Central AnglesAngle whose vertex is at the center of a circle.
AOB is a central angle of circle O
A
D
B
Inscribed Angle
Angle that has its vertex on the circle and its sides are chords.
ABC is an inscribed angles of circle O
A
B
O
D
O
Two types of angles in a circleARCS AND ANGLES
ARCS AND ANGLES
It is two points on the circle and the continuous (unbroken) part of the circle between the two points.
A
o
Example: AB
Minor arc is an arc that is smaller than a semicircle and are named by their end points
Major arc is an arc that is larger than a semicircle and are named by their end points and a point on the arc C
Example: ABC
B
Arc Definition
ARCS AND ANGLES
C
o
m CAR = ½ m COR
The measure of an angle inscribed in a circle is one-half (1/2) the measure of the intercepted arc.
A
R
Inscribed Angle Properties
100o
50o
Incribed Angle Conjecture
ARCS AND ANGLES
A
AQB APB
Inscribed angles that intercept the same arc are congruent
P
B
Inscribed Angle Intercepting the same arc
80o
Incribed Angle intercepting Conjecture
Q
80o
ARCS AND ANGLES
A
Angles inscribed in a semicircle are right angles
B
Angles Inscribed in a Semicircle
Angle Inscribed in a semicircle Conjecture90o
90o
90o
ARCS AND ANGLES
A
The opposite angles of a cyclic quadrilateral are supplementary
B
Cyclic Quadrilaterals
Cycle quadrilaterals Conjecture
101o
132o
79o
48o
ARCS AND ANGLES
By the Cyclic Quadrilateral Conjecture,
w +100° = 180°, so w = 80°.
PSR is an inscribed angle for PR.
m PR = 47°+73° = 120°, so by the Inscribed Angle Conjecture,
x = ½(120°) =60°.
By the Cyclic Quadrilateral Conjecture,
x + y = 180°.
Substituting 60° for x and solving the equation gives y =120°.
By the Inscribed Angle Conjecture, w = ½ (47° + z).
Substituting 80° for w and solving the equation gives z = 113°.
Cyclic Quadrilaterals
Find each lettered measure
Find each lettered measure.
Cyclic Quadrilaterals
ARCS AND ANGLES
A
Parallel lines intercept congruent arcs on a circle.
B
Arcs by Parallel lines
Parallel lines intercepted Arcs Conjecture
secant
D
C
AD BC
Practice (FRI Skill development)
Arcs Length
TOPIC VII - CIRCLES
ARCS LENGTH
It is two points on the circle and the continuous (unbroken) part of the circle between the two points.
Minor arc is an arc that is smaller than a semicircle and are named by their end points
Arc Definition
C
o
m COR = 100o
The measure of the minor arc is the measure of the central angle.
A
R
100o
CR = 100o
The measure of the arc from 12:00 to 4:00 is equal to the measure of the angle formed by the hour and minute hands
A circular clock is divided into 12 equal arcs, so the measure of each hour is
360 or 30°. 12
ARCS LENGTH
Because the minute hand is longer, the tip of theminute hand must travel farther than the tip of the hour hand even though they both move 120° from 12:00 to 4:00.
So the arc length is different even though the arc measure is the same!
ARCS LENGTH
The arc measure is 90°, a full circle measures 360°, and 90° = 1. 360° 4
The arc measure is half of the circlebecause 180° = 1 360° 2
The arc measure is one-third of the circle because 120° = 1 360° 3
The arc length is some fraction of the circumference of its circle.
ARCS LENGTH
ARCS LENGTH
The arc measure is 90°, a full circle measures 360°, and 90° = 1. 360° 4
The arc measure is half of the circle because 180° = 1 360° 2
To find the arcs length we have to follow this stepsStep 1: find what fraction of the circle each arc
For AB and CED find what fraction of the circle each arc is
ARCS LENGTH
Circle TC = 2(12 m)C= 24 m
Circle O C= 2 (4 in.)C= 8 in
Step 2: Find the circumference of each circle
Step 3: Combine the circumferences to find the length of the arcs
Circle T Length of AB= 90° 2 (12m) 360°Or AB= 90° 24 m 360°
AB = 18.84 m
Circle O Length of CD= 180° 2 (4 in) 360°Or CD= 180° 8 in 360°
CD = 12.56 in
Step 1: find what fraction of the circle the arc is
The length of an arc equals the measure of the arc divided by 360° times the circumference
ARCS LENGTHArc Length conjecture
l = x . 2 r3600
ARCS LENGTH
Remember:
The arc is part of a circle and its length is a part of the circumference of a circle.
The measure of an arc is calculated in units of degrees, but arc length is calculated in units of distance (foot, meters, inches, centimeter.
ARCS LENGTHExample: