1. Warm up
Find the measure of each lettered angle.

2. UNIT OF STUDY Lesson 6.3 ARCS AND ANGLES
TOPIC VII - CIRCLES
UNIT OF STUDY Lesson 6.3 ARCS AND ANGLES

3. Warm up = (FRI Starting Point)

4. You will learn ….
(FRI Ending Point)
To discover relationships between an inscribed angle of a circle and its intercepted arc

5. 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.

6. Arcs and Angles Two types of angles in a circle Central Angles
Angle whose vertex is at the center of a circle. O B
 AOB is a central angle of circle O D
Inscribed Angle B
Angle that has its vertex on the circle and its sides are chords. O A
 ABC is an inscribed angles of circle O D

7. Arcs and Angles Arc Definition
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 A
Example: AB
Major arc is an arc that is
larger than a semicircle and are named by their end points and a point on the arc o B C
Example: ABC

8. Arcs and Angles Inscribed Angle Properties
Incribed Angle Conjecture
The measure of an angle inscribed in a circle is one-half (1/2) the measure of the intercepted arc.
100o o R
50o A
m CAR = ½ m COR

9. Arcs and Angles Inscribed Angle Intercepting the same arc
Incribed Angle intercepting Conjecture
Inscribed angles that intercept the same arc are congruent
80o
80o P B Q
 AQB   APB

10. Arcs and Angles Angles Inscribed in a Semicircle
Angle Inscribed in a semicircle Conjecture A
90o
90o
Angles inscribed in a semicircle are right angles
90o B

11. Cyclic Quadrilaterals
Arcs and Angles
Cyclic Quadrilaterals
Cycle quadrilaterals Conjecture A
101o
The opposite angles of a cyclic quadrilateral are supplementary
132o
48o
79o B

12. Cyclic Quadrilaterals
Arcs and Angles
Cyclic Quadrilaterals
Find each lettered measure
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°.
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°.

13. Cyclic Quadrilaterals
Find each lettered measure.

14. Arcs and Angles Arcs by Parallel lines
secant
Parallel lines intercepted Arcs Conjecture A D
Parallel lines intercept congruent arcs on a circle. B C
AD  BC

15. Practice
(FRI Skill development)

20. TOPIC VII - CIRCLES
Arcs Length

21. Arcs LenGTH Arc Definition m COR = 100o CR = 100o
It is two points on the circle and the continuous (unbroken) part of the circle between the two points. C
Minor arc is an arc that is
smaller than a semicircle and are named by their end points
100o
The measure of the minor arc is the measure of the central angle. o R
m COR = 100o A
CR = 100o

22. Arcs LenGTH
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

23. Arcs LenGTH Because the minute hand is longer, the tip of the
minute 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!

24. The arc length is some fraction of the circumference of its circle.
Arcs LenGTH
The arc measure is 90°, a full circle measures 360°, and 90° = 1.
360°
The arc measure is half of the circle
because 180° = 1
360°
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.

25. Arcs LenGTH To find the arcs length we have to follow this steps
Step 1: find what fraction of the circle each arc
For AB and CED find what fraction of the circle each arc is
The arc measure is 90°, a full circle measures 360°, and 90° = 1.
360°
The arc measure is half of the circle because
180° = 1
360°

26. Arcs LenGTH Step 2: Find the circumference of each circle
Circle T
C = 2(12 m)
C= 24 m
Circle O
C= 2 (4 in.)
C= 8  in
Step 1: find what fraction of the circle the arc is
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
AB = m
Circle O
Length of CD= 180° 2 (4 in)
360°
Or CD= 180° 8 in
CD = in

27. l = x . 2 r 3600 Arcs LenGTH Arc Length conjecture
The length of an arc equals the measure of the arc divided by 360° times the circumference
l = x . 2 r
3600

28. 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.

29. Arcs LenGTH
Example: