The document outlines the essential learning competencies for Grade 10 mathematics focused on understanding circles, including definitions of key terms, the relationships among chords, arcs, central angles, and inscribed angles. It provides guidance for students on how to identify and measure these components and includes illustrative examples and activities for practice. The lesson aims to develop a comprehensive understanding of circles by exploring their properties and relationships.

1. Grade 10 Mathematics
Quarter II
Week 3

2. CLASSPOINT CODE:
Checking of Attendance

3. PRAYER LEADER
MARINEL KYLA GALUT

4. Most Essential Learning Competency
Derives inductively the
relations among chords, arcs,
central angles, and inscribed
angles.
(M10GE – IIc-1)

5. 1. Define a circle and describe other terms related to
circle and derive their relationships.
Objectives:
2. Identify the central angles and intercepted arcs of a
given circle.
3. Find the measure of the central angles and
intercepted arcs of a given circle.

6. GUIDE QUESTIONS
1. What is a circle?
2. What are the auxiliary parts of a circle?
3. How are these parts related?
4. What are central angles and intercepted arcs?
5. How are central angles and their intercepted
arcs related?

7. A circle is a set of all
points with the same
distance from a fixed
point called the center.
The center is used
to name a circle

8. ● O
ʘ O

9. Radius
● O
●
A 𝑂𝐴
And all
radii on
the same
circle are
congruent.

10. OA = 5 units
OB = 5 units
OC = 5 units
● O
● A
B●
●
C

11. ● O
● A
B●
Chord
𝐴𝐵
Chord
a line
segment
having
endpoints
on the circle

12. ●
● F
E●
D●
● C
● A
B●
𝑬𝑭
𝑪𝑫
Chords
O

13. ●
● F
E●
𝑬𝑭
Chord
Diamete
r
• Longest chord
• Twice the radius
Diameter

14. ●
●
´
𝐶𝐷
O
Secant
A line, segment or
ray that intersects
a circle at exactly
two points
D
●
C

15. ● Secants
●
H ●
G
´
𝑮𝑯
´
𝑿𝒀
●
O
Y
X
●

16. ● A
● I
● J
● N
R .
M .
W .
● O
Tangent
Tangent
a line,
segment or
a ray that
intersects a
circle at
exactly one
point

18. ● A
● E
● X
● C
● L
● Y
1.Which of the following is NOT a radii?
A.
B.
C.
D.

19. ● E
● X
● C
● L
● Y
● A
2. Which of the following is NOT a chord?
A.
B.
C.
D.

20. ● E
● X
● C
● L
● Y
● A
3. Which of the following is a diameter?
A.
B.
C.
D.

21. ● E
● X
● C
● L
● Y
4. Which is a secant line
● A
A.
B.
C.
D.

22. ● E
● X
● C
●
L
● Y
● A
5. Name a tangent line.
A.
B.
C.
D.

23. ● A
X
● C
Arc
is a part of a circle between any
two points. It is measured in
terms of degrees.
• minor
• semicircle
• major
Symbol
●

24. A semicircle is an
arc which is one-half of a
circle. It measures
exactly 1800
.
1800

25. A minor arc is less
than a semicircle and
its measure is
between 00
and 1800.

26. A major arc is greater
than a semicircle which
measures between 1800
and 3600.

27. ● A
●
X
●
Y
Semicircle
XEY
●
C
●
E
Semicircle
XCY
X
Y

28. ● A
●
X
●
Y
EX
●
C
●
E
X
Y
CY
EY
Minor Arcs
EX, EY, CY

29. ● A
●
●
CEY
C
●
E
Y
Major Arc

30. A central angle of a circle
is an angle formed by two radii
and its vertex is the center of
the circle.

31. ●
Y
● A
Referring to ʘ A
X
EAX E
●
●
EAY
Central angles.

32. An intercepted arc is created
when lines or subsets of a line
cuts across a section of the
circumference of a circle.

33. ●
● C
Y
● A
EC is a minor
arc intercepted
by inscribed
angle EYC.
E
●

34. ●
Y
●
EX is a minor arc
intercepted by
central angle EAX
and also
intercepted by
inscribed angle
EYX. ●
E
X
● A

35. The sum of the measures of the central
angles of a circle with no common
interior points is 3600
.
3600

36. In the figure, angles 1, 2, 3, and 4
are examples of central angles
without common interior points.
1
3
2
4
Thus,
m + m + m + m = 3600

37. G
F
C
Central FCG
intercepts
minor arc FG

38. E
I
C
Central angle
ECI intercepts
minor arc EI

39. A
G
B
C
Central angle
ACG intercepts
major arc GBA

40. A
C
Minor arc AI
is intercepted
by central
angle ACI
I

41. C D
F
H
Major arc FDH
is intercepted
by central
angle FCH

42. Arc Addition Postulates
The measure of an arc
formed by two adjacent non-
overlapping arcs (arc that share
exactly one point) is equal to
the sum of the measures of
these two arcs.

43. Using circle D,
mAC = mAB + mBC
mAC = 600
+ 900
mAC = 1500
900
B
A
C
D
600
H

44. ACTIVITY
Determine the measures of the
given arcs and angles.

45. N
M
Y
L
X
Z
P
O
C
150
R
8
1
2
7
6
5
4
3
450
300
Q
1. mor m
Answer: The measure
of is 450
.
450
A. 300
B. 750
C. 450
D. 150

46. N
M
Y
G
X
Z
P
O
C
150
R
8
1
2
7
6
5
4
3
450
300
Q
450
2. Measure of arc
LM
Answer: The measure of arc LM is 450
because the measure of that
intercepts arc LM is 450
.
N
M
Y
L
X
Z
O
C
150
R
8
1
2
7
6
5
4
3
450
300
Q
450
N
M
Y
L
X
Z
P
O
C
150
R
8
1
2
7
6
5
4
3
450
300
Q
450
A. 750
B. 450
C. 300
D. 150

47. N
M
Y
G
X
Z
P
O
C
150
R
8
1
2
7
6
5
4
3
450
300
Q
450
3. Measure of arc XQ
Answer: Arc XQ is 300
. and are vertical
angles and a pair of vertical angles are
congruent. It follows that their intercepted
arcs also are congruent.
N
M
Y
L
X
Z
P
O
C
150
R
8
1
2
7
6
5
4
3
450
300
Q
450
N
M
Y
L
X
Z
P
O
C
150
R
8
1
2
7
6
5
4
3
450
300
Q
450
A. 750
B. 450
C. 300
D. 150

48. : • Defined Circle
• Described terms
related to circle
• Derived their
relationships
• Identified and
measured the
central angles and
intercepted arcs
of a circle

49. D
1. The angle intercepting
arc GA. A
E
G
F
B
H
I
C
Answer:

50. D
2. The angle intercepting
arc DH.
A
E
G
F
B
H
I
C
Answer:

51. D
3. The arc intercepted by
ACH
A
E
G
F
B
H
I
C
Answer: AH

52. N
M
Y
G
X
Z
P
O
C
150
R
8
1
2
7
6
5
4
3
450
300
Q
450
1. Measure of
Answer: 150 N
M
Y
G
X
Z
P
O
C
150
R
8
1
2
7
6
5
4
3
450
300
Q
450
N
M
Y
L
X
Z
P
O
C
150
R
8
1
2
7
6
5
4
3
450
300
Q
450

53. N
M
Y
G
X
Z
P
O
C
150
R
8
1
2
7
6
5
4
3
450
300
Q
450
2. Measure of
arc PO
Answer: 450
N
M
Y
G
X
Z
P
O
C
150
R
8
1
2
7
6
5
4
3
450
300
Q
450
N
M
Y
G
X
Z
P
O
C
150
R
8
1
2
7
6
5
4
3
450
300
Q
450
N
M
Y
L
X
Z
P
O
C
150
R
8
1
2
7
6
5
4
3
450
300
Q
450

54. N
M
Y
G
X
Z
P
O
C
150
R
8
1
2
7
6
5
4
3
450
300
Q
450
3. Measure of
Answer: 450
N
M
Y
G
X
Z
P
O
C
150
R
8
1
2
7
6
5
4
3
450
300
Q
450
N
M
Y
G
X
Z
P
O
C
150
R
8
1
2
7
6
5
4
3
450
300
Q
450
N
M
Y
L
X
Z
P
O
C
150
R
8
1
2
7
6
5
4
3
450
300
Q
450

55. “Life is a circle. The
end of one journey is
the beginning of the
next”
-Anonymous

56. References:
A. Books
Callanta, Melvin M., et al. 2015. Mathematics
Learner’s Module Philippines. Department of Education.
B. Government Publications
ADM Mathematics 10 Module 19: The Relations
among Chords, Arcs, Central Angles and Inscribed Angles.
Department of Education.
C. Online and Other Sources
freepik.com
pngtree.com