The document discusses theorems related to circles, secants, tangents, and segments. It begins by defining theorems and postulates. It then presents several theorems about angles formed between secants and tangents, relationships between intercepted arcs and angles, congruent tangent segments, and properties of secant segments drawn from an exterior point. Examples are provided to demonstrate how to use the theorems to solve problems involving lengths and angle measures in circle geometry.

1. Sir Pee Jay

4. E
B
A
C
I am a line that intersects a circle in exactly one point
WHAT AM I?
Tangent
Line

5. I am a line that intersects a circle in exactly two point.
WHAT AM I?
K
Y
A
B
X
Secant
Line

6. I am a part of secant segment that is outside the circle.
WHAT AM I?
A
B
M
C
External
Secant
Segment

7. B
C
A
Sector of A
Circle

8. Direction: This game is called word search and
all you have to do is search a word that is
related to mathematics and give some insight
about the word you search
WARM-UP
Exercise

10. Objectives
At the end of the lesson, the students can:
 Understand the theorems on secants, tangents and segments
of a circle;
 Value Accumulated knowledge as means of new
understanding;
 Solve and proves problems involving secant segment, tangent
segment and external secant segment theorems
 Solve and proves theorems on angle formed by secants and
tangents

12. THEOREMS
is a true
statement that
can be proven.
POSTULATES
is a statement
that is assumed
true without
proof.

13. If X is a given point on the circle, there is only a single line
which can be drawn through X that is tangent to the circle.
Postulate on Tangents

14. Theorems on Tangents
OR is a tangent line and point U is the point of tangency. If OR is
tangent to Circle T at point U, then it is perpendicular to Radius TU.
↔ ↔
1

15. Theorems on Tangents
2

16. Theorems on Tangents
3

17. C
A
B
D
If AC = 10cm, then what
is the length of BC ?
Solution: AC = 10cm
AC ≅ BC
 therefore BC = 10cm
If two tangents segments is drawn from the point
outside the circle, then the segments are congruent

18. Theorems on
angles formed by
tangents and
secants

19. The measure of the angle formed by two secants that intersects
outside the circle is one-half the positive difference of the two
intercepted arcs
1

20. The measure of the angle formed by two
secants that intersects outside the circle is
one-half the positive difference of the two
intercepted arcs
1. If m FC = 96° and m EB = 32°, what is m ∠FDC?
⌒ ⌒
m ∠FDC=?
m ∠FDC= ½ (mFC –mEB )
⌒ ⌒
m ∠FDC= ½ (96° – 32° )
m ∠FDC= ½ (64°)
m ∠FDC= 32°

21. The measure of the angle formed by a secant and a tangent that
intersect outside the circle is one-half the positive difference of
the two intercepted arcs.
2

22. The measure of the angle formed by a
secant and a tangent that intersect outside
the circle is one-half the positive difference
of the two intercepted arcs.
1. If mDFC = 220° and mDB = 80°, what is m ∠DEC?
⌒
⌒
m ∠DEC=?
m ∠DEC= ½ (mDFC –mDB)
⌒
⌒
m ∠DEC= ½ (220° – 80°)
m ∠DEC= ½ (140°)
m ∠DEC= 70°

23. The measure of angle formed by two tangents that intersects
outside the circle is one-half the positive difference of two
intercepted arcs.
Figure 7
3
In Figure 7 at the right, EP and DP are two
tangents that intersects outside the circle
at point P, EFD and ED are the two
intercepted arcs of ∠EPD
m∠EPD= ½(mEFD – mED)
If mEPD = 214 and mED = 46°, then
m∠EPD=?
m∠EPD= ½(214° - 46°)
m∠EPD= ½(168°)
m∠EPD = 84°
°

24. The measure of angle formed by two
tangents that intersects outside the circle
is one-half the positive difference of two
intercepted arcs.
1. If mHOD = 216° and mHD = 66°, what is m ∠HFD?
⌒
⌒
m ∠HFD=?
m ∠HFD= ½ (mHOD –mHD)
⌒
⌒
m ∠HFD= ½ (216° – 66°)
m ∠HFD= ½ (150°)
m ∠HFD= 75°

25. The measure angle formed by two secants that intersects inside
the circle is one-half the sum of the measures of the two
intercepted arcs and its vertical angle
In Figure 8 at the right, EC and PY are two secants that
intersects inside the circle at point A, EY and PC are the
two intercepted arcs of ∠EAY and ∠PAC. EP and YC are
the two intercepted arcs of ∠EAP and ∠YAC
m∠EAY = ½ (mEY+mPC)
if mEY=92° and mPC = 196 ,
What is m∠EAY and ∠YAC?
m∠EAY = ½ (mEY+mPC)
m∠EAY = ½ (92°+196°)
m∠EAY = ½ (288°)
m∠EAY= 144°
°
4
Figure 8
m∠YAC=?
if two angles formed a linear
pair, the angles are
supplementary
m∠EAY + m∠YAC = 180°
144° + m∠YAC = 180°
m∠YAC = 180 ° - 144 °
m∠YAC =36 °

26. The measure angle formed by two
secants that intersects inside the circle
is one-half the sum of the measures of
the two intercepted arcs and its vertical
angle
1. If mEB = 45° and mCD = 49°, what is m ∠EFB? m ∠BFD?
⌒
⌒
m ∠EFB=?
m ∠EFB= ½ (mEB +mCD)
⌒
⌒
m ∠EFB= ½ (94°)
m ∠EFB= 47°
m ∠EFB= ½ (45° + 49°)
m∠BFD=?
if two angles formed a linear pair, the angles are
supplementary
m∠EFB + m∠BFD = 180°
47° + m∠BFD = 180°
m∠BFD = 180 ° - 47 °
m∠BFD =133°

27. The measure of the angle formed by a secant and tangent that
intersect at the point of tangency is half the measure of its
intercepted arc.
In Figure 9 at the right, IA is a tangent and
GH is a secant intersect at point G which is
the point of tangency. GOH is the intercepted
arc of ∠IGH
m∠IGH= ½(mGOH )
If mGOH = 232 , what is the m∠IGH?
m∠IGH = ½ (mGOH )
m∠IGH = ½ (232°)
m∠IGH = 116°
°
Figure 9
5

28. The measure of the angle formed by a
secant and tangent that intersect at the
point of tangency is half the measure of its
intercepted arc.
1. If mBFD = 216° , what is m ∠DBE?
m ∠DBE=?
m ∠DBE= ½ (mBDF )
⌒
⌒
m ∠DBE= ½ (216° )
m ∠DBE= 108°

30. If two secant segments are drawn to a circle from the same exterior point,
then the product of the lengths is of one secant segment and its external
secant segment is equal to the product of the lengths of the other secant
segment and its external secant segment.
In Figure 10 at the right, AE and CE are a
secant segment drawn from exterior point E.
Therefore, AE ● BE = CE ● DE.
If the lengths of AE=10, BE=4 and CE= 8
DE=x, What is the length of DE?
AE ● BE = CE ● DE
10 ● 4 = 8 ● x
40 = 8x
5 = x
1
Figure 10
40
8
=
8𝑥
8 Therefore DE = 5

31. If two secant segments are drawn to a circle
from the same exterior point, then the product
of the lengths is of one secant segment and its
external secant segment is equal to the product
of the lengths of the other secant segment and
its external secant segment.
1. If the lengths of DC=16, EC=5 and BC= 10
FC=x, What is the length of FC?
DC ● EC = BC ● FC
Therefore the
length of FC = 8
16 ● 5 = 10 ● x
80 = 10x
80
10
=
10𝑥
10
8 = x

32. If tangent segment and secant segment are drawn to a circle from the
same exterior point, then the square of the length of the tangent
segment is equal to the product of the lengths of the secant segment
and its external segment.
In Figure 11 at the right, ML is a tangent
segment and KL is a secant segment drawn
from the same exterior point which is point L.
Therefore 𝑀𝐿2
= KL ● NL
If KL = 9 and NL = 5, Find ML
𝑀𝐿2 = KL ● NL
𝑀𝐿2
= 9 ● 5
𝑀𝐿2 = 45
ML = 9●5
ML = 3 𝟓
2
Figure 11

33. If tangent segment and secant segment are
drawn to a circle from the same exterior point,
then the square of the length of the tangent
segment is equal to the product of the lengths of
the secant segment and its external segment.
If the lengths of BD=6, CD=9 and ED= x, What is
the length of ED?
𝐵𝐷2
= CD ● ED
Therefore the
length of ED = 4
(6)2
= 9 ● x
36 = 9x
36
9
=
9𝑥
9
4 = x

34. Formative
Assessment

35. C
A
B
D
If BC = 15cm, then what
is the length of AC ?
Solution: BC = 15cm
BC ≅ AC
 therefore AC = 15cm
If two tangents segments is drawn from the point
outside the circle, then the segments are congruent

36. The measure of the angle formed by two
secants that intersects outside the circle is
one-half the positive difference of the two
intercepted arcs
If m DB = 80° and mEF = 30°, what is m ∠DCB?
⌒ ⌒
m ∠DCB=?
m ∠DCB= ½ (mDB –mEF )
⌒ ⌒
m ∠DCB= ½ (80° – 30° )
m ∠DDB= ½ (50°)
m ∠DCB= 25°

37. The measure of the angle formed by a
secant and a tangent that intersect outside
the circle is one-half the positive
difference of the two intercepted arcs.
If mHOD = 160° and mHE = 50°, what is m ∠HLD?
⌒
⌒
m ∠HLD=?
m ∠HLD= ½ (mHOD –mHE)
⌒
⌒
m ∠HLD= ½ (160° – 50°)
m ∠HLD= ½ (110°)
m ∠HLD= 55°

38. The measure of the angle formed by a
secant and a tangent that intersect outside
the circle is one-half the positive difference
of the two intercepted arcs.
If mDFC = 200° and mDB = 50°, what is m ∠DEC?
⌒
⌒
m ∠DEC=?
m ∠DEC= ½ (mDFC –mDB)
⌒
⌒
m ∠DEC= ½ (200° – 50°)
m ∠DEC= ½ (150°)
m ∠DEC= 75°

39. The measure of angle formed by two
tangents that intersects outside the circle
is one-half the positive difference of two
intercepted arcs.
If mCEB = 200° and mCB = 54°, what is m ∠CDB?
⌒
⌒
m ∠CDB=?
m ∠CDB= ½ (mCEB –mCB)
⌒
⌒
m ∠CDB= ½ (200° – 54°)
m ∠CDB= ½ (146°)
m ∠CDB= 73°