theorems on circles

1. POSTULATES, THEOREMS,
COROLLARIES AND LAWS
INVOLVING CIRCLES

2. CENTRAL ANGLE THEOREM
• The measure of the central angle is equal to its intercepted
arc.
O
A
B
• m∠AOB = m𝐴𝐵
• 60° = 60°
60°
60°

3. VERTICAL ANGLE THEOREM
• Vertical angles are congruent.
A
C
B
•m∠CAB = m∠MAP
•60° = 60°
60°
60°
P
M

4. INSCRIBED ANGLE THEOREM
• The measure of the inscribed angle is equal to half of the
measure of its intercepted arc.
O
A
B
• m∠AOB =
1
2
m𝐴𝐵
• m∠AOB =
1
2
(60)
• m∠AOB = 30°
30°
60°

5. INVERSE OF THE INSCRIBED ANGLE
THEOREM
• The measure of the intercepted arc is equal to twice of the
measure of its inscribed angle
O
A
B
• m𝐴𝐵 = 2(m∠AOB )
• m𝐴𝐵 = 2(30°)
• m𝐴𝐵 = 60°
30°
60°

6. COROLLARY 1: ANGLE INTERCEPTING A SEMICIRCLE
• If an inscribed angle in a circle intercepts a
semicircle, then the angle formed is a right
angle. C
B
• ∠BAM intercepts 𝐵𝑀
which is a semicircle,
therefore it is a right
angle.
M
A

7. COROLLARY 2: INSCRIBED POLYGON
• If a polygon is inscribed in a circle, all its
vertices lie on the circle.
B • ∠BAM intercepts 𝐵𝑀
which is a semicircle,
therefore it is a right
angle.
M
A
X

8. COROLLARY 3: INSCRIBED QUADRILATERAL
• If a quadrilateral is inscribed in a circle, then its
opposite angles are supplementary.
P
• Quadrilateral HOPE is inscribed in Circle S.
• 𝑚∠H + 𝑚∠P = 180°
• 𝑚∠O + 𝑚∠E = 180°
H
E
S
O

9. COROLLARY 4: ANGLES INTERSECTING
THE SAME ARC
• The measure of an angle is half the measure of the central
angle with the same intercepted arc in a circle.
O
A
S
• ∠MBA and ∠MOA both
intercepts 𝐴𝐵
• m ∠MBA =
1
2
m ∠MOA
• ∠MBA =
1
2
(120°)
• m ∠MBA = 60°
120°
60° B
M

10. TANGENT LINE TO THE CIRCLE
• If a line is tangent to a circle, then it is perpendicular to the
radius drawn to the point of tangency.
O
A
B
• 𝑃𝐴 is tangent to Circle O at
point A, then it is
perpendicular to 𝑂𝐴 which is
the radius of the circle.
• ∠PAO is a right angle
P

11. ANGLE FORMED BY A TANGENT AND A
CHORD ON THE CIRCLE
• The measure of an angle formed by a tangent and a chord
that intersect on a circle is one-half the measure of the
intercepted arc.
O
A
B
• 𝑃𝐴 is tangent, 𝑂𝐴 is a chord to
Circle O, then m∠PAB is one-half
m𝐴𝐵.
m∠PAB =
1
2
𝑚𝐴𝐵
=
1
2
(150°)
= 75°
P
150°
75°

12. DIAMETER AND A CHORD
• A diameter that is perpendicular to a chord bisects the
chord and its arc.
O
A
B
• 𝑀𝑃 is a diameter perpendicular to
𝐴𝐵. Therefore, 𝐴𝐵 is divided into
2 equal parts.
P
3cm
3cm
M

13. ANGLE FORMED BY A TWO CHORDS INTERSECTING
INSIDE THE CIRCLE
• The measure of an angle formed by two chords
intersecting inside the circle is one-half the sum of the
measure of the intercepted arc.
O
A
B
m∠BOP =
1
2
(𝑚𝑀𝐴 + 𝑚𝐵𝑃)
=
1
2
(75° + 125°)
=
1
2
(200°)
= 100°
P
75°
M
125°

14. ANGLE FORMED BY A TWO LINES INTERSECTING
OUTSIDE THE CIRCLE
• The measure of an angle formed by two lines intersecting
outside the circle is one-half the difference of the measure
of the intercepted arcs.
O
A
B
m∠NOD =
1
2
(𝑚𝑀𝐵 − 𝑚𝐴𝑃)
=
1
2
(110° - 40°)
=
1
2
(70°)
m∠NOD = 35°
P
M
110°
40°
N
D

15. SEGMENT LENGTHS OF TWO CHORDS
INTERSECTING INSIDE THE CIRCLE
• If two chords intersect inside a circle, the product of the
segment lengths on one chord is equal to the product of
the segment lengths of the other chord.
O
A
B
𝑚 𝑀𝑂 x 𝑚 𝑂𝑃 = 𝑚 𝐴𝑂 x 𝑚 𝑂𝐵
8 x 6 = 3 x 𝑚 𝑂𝐵
8 𝑥 6
3
= 𝑚 𝑂𝐵
16 = 𝑚 𝑂𝐵
P
M
3
?
8
6

16. SEGMENT LENGTHS OF TWO SECANTS
INTERSECTING OUTSIDE THE CIRCLE
• If two secants intersect to a point outside a circle, the product of length of one
secant segment and its external segment is equal to the product of length of
the other secant segment and its external segment
O
A
B
𝑚 𝐴𝑂 (𝑚 𝑀𝐴 + 𝑚 𝐴𝑂) = 𝑚 𝑃𝑂 (𝑚𝐵𝑃 + 𝑚 𝑃𝑂)
5 (8 + 5) = 4 (x + 4)
5(13) = 4x + 16
65 = 4x + 16
65 – 16 = 4x
49 = 4x
𝑚𝐵𝑃 = x = 12.25
P
M
5
x
8
4

17. SEGMENT LENGTHS OF A TANGENT AND A SECANT
INTERSECTING OUTSIDE THE CIRCLE
• If a tangent and a secant intersect to a point outside a circle, the product of
length of the secant segment and its external segment is equal to the square of
length of the tangent segment.
O
A
𝑚 𝐴𝑂 (𝑚 𝑀𝐴 + 𝑚 𝐴𝑂) = 𝑚 (𝑃𝑂)2
4 (21+ 4) = x 2
100 = x2
𝑚𝑃𝑂 = x = 10
P
M
4
x
21

18. PRACTICE:
FIND THE VALUE OF EACH VARIABLE
28
x
16
21
18
15
21
x
x
15
24
n
15 35
12
42
m
x= 12 x= 12 x=23.4
m=12.5
n=43.75

19. PRACTICE:
FIND THE VALUE OF EACH VARIABLE
64°
172°
e°
a°
b°
85°
43°
32°
173°
38°
c°
44°
a°= 64 ° e°=54°
b°=126°
c°=109°

20. PRACTICE:
FIND THE VALUE OF EACH VARIABLE
g°
168°
52°
m°
k°
214°
40°
94°
79°
z°
48°
h°
d°
100°
y°
x°
m=34° g=64°
h=64°
k=110°
d=31°
y= 83°
z=20°
x=27°

21. • Thank you for actively participating.