If there is one prayer that you should

- Samuel Dominic Chukwuemeka
**pray/sing** every day and every hour, it is the
LORD's prayer (Our FATHER in Heaven prayer)

It is the **most powerful prayer**.
A **pure heart**, a **clean mind**, and a **clear conscience** is necessary for it.

For in GOD we live, and move, and have our being.

- Acts 17:28

The

- Samuel Chukwuemeka**Joy** of a **Teacher** is the **Success** of his **Students**.

I greet you this day,

__First:__ read the notes.

__Second:__ view the videos.

__Third:__ solve the questions/solved examples.

__Fourth:__ check your solutions with my **thoroughly-explained** examples.

Comments, ideas, areas of improvement, questions, and constructive criticisms are welcome. You may contact me.

If you are my student, please do not contact me here. Contact me via the school's system.

Thank you for visiting.

**Samuel Dominic Chukwuemeka** (Samdom For Peace)
B.Eng., A.A.T, M.Ed., M.S

Students will:

(1.) Discuss the terms used in circle theorems.

(2.) State circle theorems.

(3.) Prove circle theorems.

(4.) Solve problems on circle theorems.

locus, point, equidistant (equal distance), radius, diameter, chord, secant, tangent,
*
Make the connection: equilateral (equal sides), equiangular (equal angles), equity (equality)
*

(1.) A **circle** is the **locus of points equidistant from a fixed point.**

The fixed point is known as the **center** of the circle.

The equal distance (equidistant) of the points from the fixed point is known as the **radius** of the circle.

$
C = center \\[3ex]
r = radius \\[3ex]
|CA| = |CB| = |CD| = |CE| = |CF| = r = radius \\[3ex]
$
(2.) The **circumference** of a circle is the **entire distance around the circle.**

It is also known as the **perimeter** of the circle.

(3.) The **chord** in a circle is the **line segment that joins two points on the circumference of the circle.**

(4.) The

It is the

It is

(5.) A **secant** to a circle is the line segment that ** intersects (cuts) two points** on the circle.

We have 2 chords and 1 secant line

The first chord joins the two points (Point A and Point B) on the circumference of the circle.

The second chord joins the two points (Point D and Point E) on the circumference of the circle.

The secant cuts the 2 points (Points D and E) on the circumference of the circle

$
chords = |AB| \;\;and\;\; |DE| \\[3ex]
secant = |CF| \\[3ex]
$
*
Student: Mr. C
Teacher: What's good?
Student: What if I write secant AB rather than chord AB?
Is that going to be a problem?
Teacher: If you wrote it as part of a reason in solving a question, I may overlook it.
However, if I wanted to assess your knowledge on the difference between a secant and a chord and you wrote it
that way, I may deduct points.
Notice the terms: joins (chord), cuts (secant), intersects (secant)
*

(6.) A **tangent line** to a circle is the line that ** touches only one point** on the curve.

$ AB = chord \\[3ex] CD = diameter\;(longest\;\;chord) \\[3ex] EFGH = secant:\;\;points\;\;cut = Point\;F\;\;and\;\;Point\;G \\[3ex] JKM = tangent:\;\;point\;\;touched = Point\;K \\[3ex] $

(7.) The **central angle** is the **angle formed at the center** of the circle by **two radii**

The **intercepted arc** is **part of the circumference** of a circle that is formed by a chord or a secant.

The measure of the **angle at the center** (central angle) is equal to the measure of it's **intercepted arc** and vice versa.

$ Center = O \\[3ex] m\overset{\huge\frown}{AB} = \angle\theta \\[3ex] m\overset{\huge\frown}{CD} = \angle\phi \\[3ex] $

(8.) The **inscribed angle** is the **angle whose vertex is on the circumference of the circle.**

## Symbols and Meanings

- $r$ = radius
- $d$ = diameter
- $\pi = pi = \dfrac{22}{7}$
- $A$ = area
- $C$ = circumference

## Formulas

- To solve for a specified variable for each formula, please review Solved Examples: Literal Equations
- $d = 2r$
- $r = \dfrac{d}{2}$
- $C = \pi d$
- $C = 2\pi r$
- $A = \pi r^2$
- $A = \dfrac{\pi d^2}{4}$

For all the circles here (here only): **O** is the **center of the circle**

(1.) The angle in a semicircle is a right angle (an angle of 90°).

The diameter of a circle divides the circle into two equal parts known as semicircles.

So, if you are given the center of a circle and/or the diameter of the circle, check to see if you can apply this theorem.

$
\underline{Diameters} \\[3ex]
AOB \\[3ex]
EOF \\[3ex]
GOJ \\[3ex]
\underline{\angle s\;\;in\;\;a\;\;semicircle} \\[3ex]
\angle ACB = 90^\circ \\[3ex]
\angle EDF = 90^\circ \\[3ex]
\angle EMF = 90^\circ \\[3ex]
\angle GHJ = 90^\circ \\[3ex]
$
__NOTE:__ $GNJ$ is __not__ a semicircle

Hence, $\angle GNJ \ne 90^\circ$ unless specified otherwise

The angle in a semicircle must be the angle formed by the intersection of two chords, where one end of each chord intersects
with the diameter of the circle, and the other end of each chord touches the circumference of the circle.

(2.) Angles in the same segment of a circle are equal.

OR

Angles subtended by a chord of a circle in the same segment of the circle are equal.

In this theorem:

(I.) The first thing is to understand the segment of a circle. (*as shown in the diagram above*)

The segment consists of both the chord and the circumference (perimeter of the segment) including the
area in-between the chord and the circumference (area of the segment)

(II.) Usually, we have the major segment and the minor segment.

However, we can also two equal segments (also known as semicircles)

*As shown in the diagram above*

(III.) The **main chord** (chord separating the major segment from the minor segment) __must not be indicated__
(*as shown in the second diagram*)

The theorem still holds even if that chord is not indicated in the segment.

So, what should you do in that case?

Try to draw a dotted main chord (visualize it and/or draw it on the paper) and find the angles in that segment.

Let us do some examples.

$ \underline{Main\;\;Chord:\;\;AB} \\[3ex] Indicated\;\;in\;\;the\;\;1st\;\;Circle \\[3ex] Not\;\;indicated\;\;in\;\;the\;\;2nd\;\;Circle\;\;(should\;\;have\;\;been\;\;drawn\;\;as\;\;FH)\\[3ex] a = b ...\angle s\;\;in\;\;the\;\;same\;\;major\;\;segment\;\;1st\;\;circle \\[3ex] c = d ...\angle s\;\;in\;\;the\;\;same\;\;major\;\;segment\;\;2nd\;\;circle \\[3ex] \underline{Main\;\;Chord:\;\;CD} \\[3ex] Indicated\;\;in\;\;the\;\;1st\;\;Circle \\[3ex] Not\;\;indicated\;\;in\;\;the\;\;2nd\;\;Circle\;\;(should\;\;have\;\;been\;\;drawn\;\;as\;\;EG)\\[3ex] g = h ...\angle s\;\;in\;\;the\;\;same\;\;major\;\;segment\;\;1st\;\;circle \\[3ex] e = f ...\angle s\;\;in\;\;the\;\;same\;\;major\;\;segment\;\;2nd\;\;circle \\[3ex] \underline{Main\;\;Chord:\;\;BD} \\[3ex] Indicated\;\;in\;\;the\;\;1st\;\;Circle \\[3ex] Does\;\;not\;\;apply\;\;in\;\;the\;\;2nd\;\;Circle \\[3ex] m = k ...\angle s\;\;in\;\;the\;\;same\;\;major\;\;segment \\[3ex] \underline{Main\;\;Chord:\;\;AC} \\[3ex] Indicated\;\;in\;\;the\;\;1st\;\;Circle \\[3ex] Does\;\;not\;\;apply\;\;in\;\;the\;\;2nd\;\;Circle \\[3ex] n = p ...\angle s\;\;in\;\;the\;\;same\;\;major\;\;segment $

(IV.) Angles in the same segment (major segment or minor segment or equal segment/semicircle) are equal.

**Angles in the same segment** means that the
**angles formed by the intersection of any two chords at the circumference of the circle** __must__ **be in the same segment.**

If there is any angle that is formed by the two chords where both chords do not touch the circumference of the circle, then
this theorem does not apply.

Both chords that formed that angle must touch the circumference of the circle and must be in the same segment.

Let us do more examples.

$ a = b ...\angle s\;\;in\;\;the\;\;same\;\;segement\;\;are\;\;equal \\[3ex] c = d = e ...\angle s\;\;in\;\;the\;\;same\;\;segement\;\;are\;\;equal \\[3ex] f = g ...\angle s\;\;in\;\;the\;\;same\;\;segement\;\;are\;\;equal \\[3ex] f = g = 90^\circ ... \angle\;\;in\;\;a\;\;semicircle \\[3ex] h = 90^\circ ... \angle\;\;in\;\;a\;\;semicircle \\[3ex] \therefore f = g = h \\[3ex] k = n ...\angle s\;\;in\;\;the\;\;same\;\;segement\;\;are\;\;equal \\[3ex] m = p ...\angle s\;\;in\;\;the\;\;same\;\;segement\;\;are\;\;equal \\[3ex] t = q ...\angle s\;\;in\;\;the\;\;same\;\;segement\;\;are\;\;equal \\[3ex] v = w ...\angle s\;\;in\;\;the\;\;same\;\;segement\;\;are\;\;equal $

(3.) The angle which an arc of a circle subtends at the center is twice the angle which the same
arc of the circle subtends at the circumference.

OR

The measure of any angle inscribed in a circle is half the measure of the intercepted arc.

In the second case:

The ** inscribed angle** is the

The

This is another theorem to watch out for when you are given the center of the circle.

It deals with the relationship between the angle at the center of the circle (central angle) and the

Check out the way these angles resemble in these diagrams.

The angle at the center must be

$ \boldsymbol{\angle \;\;at\;\;center = 2 * \angle\;\;at\;\;circumference} \\[3ex] x = 2y \\[3ex] n = 2m \\[3ex] p = 2k \\[3ex] e = 2f \\[3ex] \theta = 2\phi \\[3ex] t = 2w \\[3ex] \psi = 2\tau $

(4.) The sum of the interior opposite angles of a cyclic quadrilateral is 180°

OR

The interior opposite angles of a cyclic quadrilateral are supplementary

A Quadrilateral is any four-sided polygon.

It has four vertices/corners.

A **Cyclic Quadrilateral** is a **quadrilateral** whose four vertices lie on the circumference of the circle.

It is also known as an **Inscribed Quadrilateral**

Because it is a quadrilateral, the sum of the four interior angles is equal to $360^\circ$

Because it is a cyclic quadrilateral, the sum of the two interior opposite angles is $180^\circ$

__NOTE:__ A cyclic quadrilateral __must__ have all four chords that connects all four points on the circumference of the
circle.

Because the four vertices touch the circumference of the circle, those vertices are points on the circumference of the circle.

Those vertices must be joined by chords.

So, those four vertices must be joined by four chords when each chord joins two vertices/points.

*Can you spot which of these circles is not a cyclic quadrilateral?*

$
\underline{Circles\;\;1\;\;through\;\;5} \\[3ex]
\boldsymbol{Interior\;\;opposite\;\;\angle s\;\;of\;\;a\;\;Cyclic\;\;Quadrilateral\;\;are\;\;supplementary} \\[3ex]
a + d = 180^\circ \\[3ex]
c + b = 180^\circ \\[3ex]
e + g = 180^\circ \\[3ex]
f + h = 180^\circ \\[3ex]
\theta + \tau = 180^\circ \\[3ex]
\psi + \phi = 180^\circ \\[3ex]
k + n = 180^\circ \\[3ex]
p + m = 180^\circ \\[3ex]
r + w = 180^\circ \\[3ex]
u + t = 180^\circ \\[3ex]
$
The diameter is also a chord. Hence, Circle 4 is a cyclic quadrilateral.

However, Circle 6 is not a cyclic quadrilateral because there is no chord joining Points B and C

Hence this theorem does not apply to Circle 6

*
But what theorem would apply to Circle 6?
Check out Theorem 2
*

(5.) The exterior angle of a cyclic quadrilateral is equal to the interior opposite angle.

An easy way to recognize and apply this theorem is this:

(1st.) Locate the interior angle on the same straight line as the exterior angle.

That angle is known as the interior adjacent angle to the exterior angle.

Please note that it must be on the same straight line as the exterior angle.

(2nd.) Then, locate the interior angle directly opposite that interior adjacent angle.

That angle is known as the interior opposite angle to the exterior angle.

That interior opposite angle is equal to the exterior angle.

__Recap:__

(1.) The circle must be a cyclic quadrilateral.

This theorem applies only to cyclic quadrilaterals.

(2.) The interior adjacent angle must be on the same straight line as the exterior angle.

Those are the two angles that must be on the same straight line.

(3.) The interior opposite angle is the angle that is directly opposite the interior adjacent angle to the exterior angle.

That interior opposite angle is equal to the exterior angle.

$
Exterior\;\;\angle\;\;of\;\;a\;\;cyclic\;\;quad = interior\;\;opposite\;\;\angle \\[3ex]
e = c \\[3ex]
k = f \\[3ex]
t = r \\[3ex]
y = v \\[5ex]
$

$ Exterior\;\;\angle\;\;of\;\;a\;\;cyclic\;\;quad = interior\;\;opposite\;\;\angle \\[3ex] \lambda = \beta \\[3ex] \mu = \beta \\[5ex] B = \phi \\[3ex] X = \phi \\[5ex] \psi = \theta \\[3ex] N = \theta \\[5ex] \epsilon = \alpha \\[3ex] T = \alpha \\[5ex] Vertical\;\;\angle s\;\;are\;\;equal \\[3ex] \mu = \lambda \\[3ex] \tau = \alpha \\[5ex] B = X \\[3ex] A = \theta \\[5ex] P = \beta \\[3ex] \epsilon = T \\[5ex] \phi = K \\[3ex] \psi = N \\[5ex] \angle s\;\;on\;\;a\;\;straight\;\;line \\[3ex] \alpha + \mu = 180^\circ \\[3ex] \alpha + \lambda = 180^\circ \\[3ex] \mu + \tau = 180^\circ \\[3ex] \tau + \lambda = 180^\circ \\[5ex] A + B = 180^\circ \\[3ex] B + \theta = 180^\circ \\[3ex] \theta + X = 180^\circ \\[3ex] X + A = 180^\circ \\[5ex] P + T = 180^\circ \\[3ex] T + \beta = 180^\circ \\[3ex] \beta + \epsilon = 180^\circ \\[3ex] \epsilon + P = 180^\circ \\[5ex] \psi + K = 180^\circ \\[3ex] K + N = 180^\circ \\[3ex] N + \phi = 180^\circ \\[3ex] \phi + \psi = 180^\circ \\[5ex] $

(6.) The radius of a circle is perpendicular to the tangent of the circle at the point of contact.

This implies that the angle between the radius of a circle and the tangent to the circle at the point of contact is 90°

$ \underline{Theorem\;6} \\[3ex] 1st\;\;Circle:\;\;radius\;OB \perp tangent\;ABC \;\;at\;\;point\;\;of\;\;contact\; B \\[3ex] 2nd\;\;Circle:\;\;radius\;OE \perp tangent\;DEF \;\;at\;\;point\;\;of\;\;contact\; E \\[3ex] 3rd\;\;Circle:\;\;radius\;OH \perp tangent\;GHJ \;\;at\;\;point\;\;of\;\;contact\; H \\[3ex] 4th\;\;Circle:\;\;radius\;OM \perp tangent\;KMN \;\;at\;\;point\;\;of\;\;contact\; M \\[3ex] $

$ Radius\;OP \perp tangent\;UPV \;\;at\;\;point\;\;of\;\;contact\; P \\[3ex] Radius\;OQ \perp tangent\;VQX \;\;at\;\;point\;\;of\;\;contact\; Q \\[3ex] Radius\;OT \perp tangent\;WTX \;\;at\;\;point\;\;of\;\;contact\; T \\[3ex] Radius\;OR \perp tangent\;URW \;\;at\;\;point\;\;of\;\;contact\; R \\[3ex] $

(7.) **Intersecting Tangents Theorem** or **Intersecting Tangent-Tangent Theorem** __and__
**Angle of Intersecting Tangents Theorem**

If two tangents are drawn from the same external point:

(a.) the two tangents are equal in length

(b.) the line joining the external point and the centre of the circle bisects the angle formed by the two tangents.

(c.) the line joining the external point and the centre of the circle bisects the angle formed by the two radii.

$ Radius = r \\[3ex] Circle\;\;Centers = OA \;\;and\;\; OB \\[3ex] External\;\;Point = C \\[3ex] Tangents = AC \;\;and\;\; BC \\[3ex] Points\;\;of\;\;Contact = A \;\;and\;\; B \\[3ex] \angle OCA = \alpha \\[3ex] \angle OCB = \beta \\[3ex] \angle AOC = \theta \\[3ex] \angle BOC = \phi \\[5ex] \underline{Theorem\;7} \\[3ex] AC = BC...tangents\;\;drawn\;\;from\;\;the\;\;same\;\;external\;\;point\; C \\[3ex] \alpha = \beta \\[3ex] ... the\;\;line\;\;joining\;\;the\;\;external\;\;point\;C \;\;and\;\;the\;\;centre\;\;O \;\;bisects\;\;the\;\;\angle\;\;formed\;\;by\;\;the\;\;two\;\;tangents \\[5ex] Because: \\[3ex] \angle OAC = \angle OBC = 90^\circ ...radius \perp tangent \;\;at\;\;point\;\;of\;\;contact ...Theorem\;6 \\[3ex] and \\[3ex] \angle OCA = \angle OCB ...Theorem\;7 \\[3ex] and \\[3ex] radius = radius \\[3ex] \implies \\[3ex] \angle AOC = \angle BOC ... AAS\;\;Congruency\;\;(Angle-Angle-Side\;\;Congruency) \\[3ex] \therefore \theta = \phi \\[3ex] $

(8.) **Alternate Segment Theorem**

The angle between a tangent to a circle and a chord drawn from the point of contact, is equal to the angle in the
alternate segment.

Whenever you have:

(a.) a tangent and

(b.) a chord drawn from that tangent

then at the point of contact between the tangent and the chord, you may want to watch out for this theorem.

Remember that a chord divides a circle into two segments.

When a chord is drawn from the tangent at the point of contact, one of the segments is close to the tangent, and the
other segment (alternate segment) is not close to the tangent.

The angle in the alternate segment means the angle in the segment that is not close to the tangent.

Watch out for the position/location of the angles.

$ \underline{1st\;\;Circle} \\[3ex] Tangent = ABC \\[3ex] Chord = BD \\[3ex] Angle\;\;between\;\;tangent\;ABC \;\;and\;\; chord\;BD = \theta \\[3ex] Alternate\;\;Segment = BED \\[3ex] Angle\;\;in\;\;the\;\;alternate\;\;segment = \theta \\[3ex] Also: \\[3ex] Tangent = ABC \\[3ex] Chord = EB \\[3ex] Angle\;\;between\;\;tangent\;ABC \;\;and\;\; chord\;EB = \phi \\[3ex] Alternate\;\;Segment = EDB \\[3ex] Angle\;\;in\;\;the\;\;alternate\;\;segment = \phi \\[5ex] \underline{2nd\;\;Circle} \\[3ex] Tangent = FGH \\[3ex] Chord = GM \\[3ex] Angle\;\;between\;\;tangent\;FGH \;\;and\;\; chord\;GM = e \\[3ex] Alternate\;\;Segment = MKG \\[3ex] Angle\;\;in\;\;the\;\;alternate\;\;segment = e \\[3ex] Also: \\[3ex] Tangent = FGH \\[3ex] Chord = GK \\[3ex] Angle\;\;between\;\;tangent\;FGH \;\;and\;\; chord\;GK = d \\[3ex] Alternate\;\;Segment = KJG \\[3ex] Angle\;\;in\;\;the\;\;alternate\;\;segment = d \\[3ex] Also: \\[3ex] Tangent = FGH \\[3ex] Chord = GK \\[3ex] Angle\;\;between\;\;tangent\;FGH \;\;and\;\; chord\;GK = c \\[3ex] Alternate\;\;Segment = GMK \\[3ex] Angle\;\;in\;\;the\;\;alternate\;\;segment = c \\[3ex] Also: \\[3ex] Tangent = FGH \\[3ex] Chord = GJ \\[3ex] Angle\;\;between\;\;tangent\;FGH \;\;and\;\; chord\;GJ = a \\[3ex] Alternate\;\;Segment = GKJ \\[3ex] Angle\;\;in\;\;the\;\;alternate\;\;segment = a \\[3ex] $

(9.) If a line drawn from the center of the circle bisects a chord, then:

(a.) it bisects its arc (the angle opposite the chord) __and__

(b.) it is perpendicular to the chord.

(10.) If a line drawn from the center of the circle is perpendicular to a chord, then:

(a.) it bisects the chord __and__

(b.) it bisects its arc (the angle opposite the chord).

(11.) **Intersecting Chords Theorem**

When two chords intersect, the product of the lengths of the segments of one chord is equal to the product of the
lengths of the segments of the other chord.

$ RT * TS = PT * TQ ...Intersecting\;\;Chords\;\;Theorem \\[3ex] $

$ c * d = e * f ...Intersecting\;\;Chords\;\;Theorem \\[3ex] $

(12.) **Angle of Intersecting Chords Theorem**

The angle formed when two chords intersect is equal to half the sum of the intercepted arcs.

Example 1:

$ Center:\;\; O \\[3ex] Chords:\;\; |RS|\;\;and\;\;|PQ| \\[3ex] Arcs:\;\; \overset{\huge\frown}{PR}\;\;and\;\;\overset{\huge\frown}{SQ};\;\;\overset{\huge\frown}{PS}\;\;and\;\;\overset{\huge\frown}{RQ} \\[5ex] \underline{Similar\;\;Arcs:}\;\; \overset{\huge\frown}{PR}\;\;and\;\;\overset{\huge\frown}{SQ} \\[3ex] \angle\;\;between\;\;Similar\;\;Arcs:\;\; \angle PTR \;\;and\;\; \angle STQ \\[3ex] \angle PTR = \angle STQ ...Vertical\;\;\angle s\;\;are\;\;equal \\[3ex] \underline{Angle\;\;of\;\;Intersecting\;\;Chords\;\;Theorem} \\[3ex] \angle PTR = \dfrac{\overset{\huge\frown}{PR} + \overset{\huge\frown}{SQ}}{2} \\[5ex] \angle STQ = \dfrac{\overset{\huge\frown}{PR} + \overset{\huge\frown}{SQ}}{2} \\[7ex] Also: \\[5ex] \underline{Similar\;\;Arcs:}\;\; \overset{\huge\frown}{PS}\;\;and\;\;\overset{\huge\frown}{RQ} \\[3ex] \angle\;\;between\;\;Similar\;\;Arcs:\;\; \angle PTS \;\;and\;\; \angle RTQ \\[3ex] \angle PTS = \angle RTQ ...Vertical\;\;\angle s\;\;are\;\;equal \\[3ex] \underline{Angle\;\;of\;\;Intersecting\;\;Chords\;\;Theorem} \\[3ex] \angle PTS = \dfrac{\overset{\huge\frown}{PS} + \overset{\huge\frown}{RQ}}{2} \\[5ex] \angle RTQ = \dfrac{\overset{\huge\frown}{PS} + \overset{\huge\frown}{RQ}}{2} \\[7ex] $ Example 2:

$ Center:\;\; A \\[3ex] Chords:\;\; |CB|\;\;and\;\;|DE| \\[3ex] Arcs:\;\; \overset{\huge\frown}{CD}\;\;and\;\;\overset{\huge\frown}{BE};\;\;\overset{\huge\frown}{DB}\;\;and\;\;\overset{\huge\frown}{CE} \\[5ex] Diameter:\;\; \overline{DAE} \\[3ex] \implies \\[3ex] \overset{\huge\frown}{CD} + \overset{\huge\frown}{CE} = 180^\circ ...arcs\;\;of\;\;semicircle \\[3ex] \overset{\huge\frown}{DB} + \overset{\huge\frown}{BE} = 180^\circ ...arcs\;\;of\;\;semicircle \\[5ex] \underline{Given\;\;only:}\;\; \overset{\huge\frown}{CD}\;\;and\;\;\overset{\huge\frown}{DB} \\[3ex] \overset{\huge\frown}{CE} = 180 - \overset{\huge\frown}{CD} \\[3ex] \overset{\huge\frown}{BE} = 180 - \overset{\huge\frown}{DB} \\[5ex] \underline{Similar\;\;Arcs:}\;\; \overset{\huge\frown}{CD}\;\;and\;\;\overset{\huge\frown}{BE} \\[3ex] \angle\;\;between\;\;Similar\;\;Arcs:\;\; \angle CFD \;\;and\;\; \angle BFE \\[3ex] \underline{Angle\;\;of\;\;Intersecting\;\;Chords\;\;Theorem} \\[3ex] \angle CFD = \dfrac{\overset{\huge\frown}{CD} + \overset{\huge\frown}{BE}}{2} \\[5ex] \angle CFD = \dfrac{\overset{\huge\frown}{CD} + (180 - \overset{\huge\frown}{DB})}{2} \\[5ex] \angle BFE = \angle CFD ...Vertical\;\;\angle s\;\;are\;\;equal \\[7ex] Also: \\[5ex] \underline{Similar\;\;Arcs:}\;\; \overset{\huge\frown}{CE}\;\;and\;\;\overset{\huge\frown}{DB} \\[3ex] \angle\;\;between\;\;Similar\;\;Arcs:\;\; \angle CFE \;\;and\;\; \angle DFB \\[3ex] \underline{Angle\;\;of\;\;Intersecting\;\;Chords\;\;Theorem} \\[3ex] \angle CFE = \dfrac{\overset{\huge\frown}{DB} + \overset{\huge\frown}{CE}}{2} \\[5ex] \angle CFE = \dfrac{\overset{\huge\frown}{DB} + (180 - \overset{\huge\frown}{CD})}{2} \\[5ex] \angle DFB = \angle CFE ...Vertical\;\;\angle s\;\;are\;\;equal $

$ Center:\;\; O \\[3ex] Chords:\;\; |RS|\;\;and\;\;|PQ| \\[3ex] Arcs:\;\; \overset{\huge\frown}{PR}\;\;and\;\;\overset{\huge\frown}{SQ};\;\;\overset{\huge\frown}{PS}\;\;and\;\;\overset{\huge\frown}{RQ} \\[5ex] \underline{Similar\;\;Arcs:}\;\; \overset{\huge\frown}{PR}\;\;and\;\;\overset{\huge\frown}{SQ} \\[3ex] \angle\;\;between\;\;Similar\;\;Arcs:\;\; \angle PTR \;\;and\;\; \angle STQ \\[3ex] \angle PTR = \angle STQ ...Vertical\;\;\angle s\;\;are\;\;equal \\[3ex] \underline{Angle\;\;of\;\;Intersecting\;\;Chords\;\;Theorem} \\[3ex] \angle PTR = \dfrac{\overset{\huge\frown}{PR} + \overset{\huge\frown}{SQ}}{2} \\[5ex] \angle STQ = \dfrac{\overset{\huge\frown}{PR} + \overset{\huge\frown}{SQ}}{2} \\[7ex] Also: \\[5ex] \underline{Similar\;\;Arcs:}\;\; \overset{\huge\frown}{PS}\;\;and\;\;\overset{\huge\frown}{RQ} \\[3ex] \angle\;\;between\;\;Similar\;\;Arcs:\;\; \angle PTS \;\;and\;\; \angle RTQ \\[3ex] \angle PTS = \angle RTQ ...Vertical\;\;\angle s\;\;are\;\;equal \\[3ex] \underline{Angle\;\;of\;\;Intersecting\;\;Chords\;\;Theorem} \\[3ex] \angle PTS = \dfrac{\overset{\huge\frown}{PS} + \overset{\huge\frown}{RQ}}{2} \\[5ex] \angle RTQ = \dfrac{\overset{\huge\frown}{PS} + \overset{\huge\frown}{RQ}}{2} \\[7ex] $ Example 2:

$ Center:\;\; A \\[3ex] Chords:\;\; |CB|\;\;and\;\;|DE| \\[3ex] Arcs:\;\; \overset{\huge\frown}{CD}\;\;and\;\;\overset{\huge\frown}{BE};\;\;\overset{\huge\frown}{DB}\;\;and\;\;\overset{\huge\frown}{CE} \\[5ex] Diameter:\;\; \overline{DAE} \\[3ex] \implies \\[3ex] \overset{\huge\frown}{CD} + \overset{\huge\frown}{CE} = 180^\circ ...arcs\;\;of\;\;semicircle \\[3ex] \overset{\huge\frown}{DB} + \overset{\huge\frown}{BE} = 180^\circ ...arcs\;\;of\;\;semicircle \\[5ex] \underline{Given\;\;only:}\;\; \overset{\huge\frown}{CD}\;\;and\;\;\overset{\huge\frown}{DB} \\[3ex] \overset{\huge\frown}{CE} = 180 - \overset{\huge\frown}{CD} \\[3ex] \overset{\huge\frown}{BE} = 180 - \overset{\huge\frown}{DB} \\[5ex] \underline{Similar\;\;Arcs:}\;\; \overset{\huge\frown}{CD}\;\;and\;\;\overset{\huge\frown}{BE} \\[3ex] \angle\;\;between\;\;Similar\;\;Arcs:\;\; \angle CFD \;\;and\;\; \angle BFE \\[3ex] \underline{Angle\;\;of\;\;Intersecting\;\;Chords\;\;Theorem} \\[3ex] \angle CFD = \dfrac{\overset{\huge\frown}{CD} + \overset{\huge\frown}{BE}}{2} \\[5ex] \angle CFD = \dfrac{\overset{\huge\frown}{CD} + (180 - \overset{\huge\frown}{DB})}{2} \\[5ex] \angle BFE = \angle CFD ...Vertical\;\;\angle s\;\;are\;\;equal \\[7ex] Also: \\[5ex] \underline{Similar\;\;Arcs:}\;\; \overset{\huge\frown}{CE}\;\;and\;\;\overset{\huge\frown}{DB} \\[3ex] \angle\;\;between\;\;Similar\;\;Arcs:\;\; \angle CFE \;\;and\;\; \angle DFB \\[3ex] \underline{Angle\;\;of\;\;Intersecting\;\;Chords\;\;Theorem} \\[3ex] \angle CFE = \dfrac{\overset{\huge\frown}{DB} + \overset{\huge\frown}{CE}}{2} \\[5ex] \angle CFE = \dfrac{\overset{\huge\frown}{DB} + (180 - \overset{\huge\frown}{CD})}{2} \\[5ex] \angle DFB = \angle CFE ...Vertical\;\;\angle s\;\;are\;\;equal $

(13.) **Intersecting Secants Theorem** or **Intersecting Secant-Secant Theorem**

In the intersection of two secants from the same external point:

the product of the distance between the first point and the external point __and__ the distance between the
second point and the external point **for the first secant** is equal to
the product of the distance between the first point and the external point __and__ the distance between the
second point and the external point **for the second secant**.

$ \underline{Intersecting\;\;Secants\;\;Theorem} \\[3ex] BC * AC = EC * DC \\[3ex] h * (h + k) = m * (m + n) \\[3ex] $

$ \underline{Intersecting\;\;Secants\;\;Theorem} \\[3ex] QR * PR = t * k \\[3ex] (y - w) * y = e * (e + j) \\[3ex] $

(14.) **Angle of Intersecting Secants (Inside the Circle) Theorem**

The angle formed when two secants intersect inside a circle is equal to half the sum of the intercepted arcs.

$ Center:\;\; O \\[3ex] Secants:\;\; |AEB|\;\;and\;\;|CGD| \\[3ex] Arcs:\;\; \overset{\huge\frown}{EF}\;\;and\;\;\overset{\huge\frown}{GH};\;\;\overset{\huge\frown}{EG}\;\;and\;\;\overset{\huge\frown}{FH} \\[5ex] \underline{Central\;\;\angle = Intercepted\;\; \overset{\huge\frown}{}} \\[3ex] \alpha = \overset{\huge\frown}{EF} \\[3ex] \psi = \overset{\huge\frown}{FH} \\[3ex] \beta = \overset{\huge\frown}{GH} \\[3ex] \lambda = \overset{\huge\frown}{EG} \\[5ex] $

Please note these important points as it will help us in writing the theorem:

(1.) $c$ has the same *shape* as $\alpha$

(2.) $d$ has the same *shape* as $\psi$

(3.) $\phi$ has the same *shape* as $\lambda$

(4.) $\theta$ has the same *shape* as $\beta$

(5.) $\overset{\huge\frown}{EF}$ is opposite to $\overset{\huge\frown}{GH}$

(5.) $\overset{\huge\frown}{EG}$ is opposite to $\overset{\huge\frown}{FH}$

$
\underline{\angle\;\;of\;\;Intersecting\;\;Secants\;(Inside\;\;the\;\;Circle)} \\[3ex]
c = \dfrac{\overset{\huge\frown}{EF} + \overset{\huge\frown}{GH}}{2} \\[5ex]
c = \dfrac{\alpha + \beta}{2} \\[5ex]
Similarly: \\[3ex]
\phi = \dfrac{\overset{\huge\frown}{EG} + \overset{\huge\frown}{FH}}{2} \\[5ex]
\phi = \dfrac{\lambda + \psi}{2} \\[5ex]
\underline{Vertical\;\angle s\;\;are\;\;equal} \\[3ex]
\theta = c = \dfrac{\alpha + \beta}{2} \\[5ex]
d = \phi = \dfrac{\lambda + \psi}{2} \\[5ex]
$

(15.) **Angle of Intersecting Secants (Outside the Circle) Theorem**

The angle formed when two secants intersect outside a circle is equal to half the difference of the intercepted arcs.

$ Center:\;\; O \\[3ex] Secants:\;\; |ABC|\;\;and\;\;|DEC| \\[3ex] Major\;\;Arc:\;\; \overset{\huge\frown}{AD} \\[3ex] Minor\;\;Arc:\;\; \overset{\huge\frown}{BE} \\[5ex] \underline{Central\;\;\angle = Intercepted\;\; \overset{\huge\frown}{}} \\[3ex] \phi = \overset{\huge\frown}{AD} = Major\;\;Arc \\[3ex] \theta = \overset{\huge\frown}{BE} = Minor\;\;Arc \\[5ex] \underline{\angle\;\;of\;\;Intersecting\;\;Secants\;(Outside\;\;the\;\;Circle)} \\[3ex] \psi = \dfrac{Major\;\;Arc - Minor\;\;Arc}{2} \\[5ex] \psi = \dfrac{\overset{\huge\frown}{AD} - \overset{\huge\frown}{BE}}{2} \\[5ex] \psi = \dfrac{\phi - \theta}{2} \\[5ex] $

(16.) **Intersecting Secant-Tangent Theorem** or **Intersecting Tangent-Secant Theorem**

In the intersection of a secant and a tangent from the same external point:

the product of the distance between the first point and the external point __and__ the distance between the
second point and the external point **for the secant** is equal to
the square of the distance between the point of contant and the external point **for the tangent.**

$ \underline{Intersecting\;\;Secant-Tangent\;\;Theorem} \\[3ex] \underline{Circle\;1} \\[3ex] CD * BD = FD * FD \\[3ex] \overline{CD} * \overline{BD} = \overline{FD}^2 \\[5ex] \underline{Circle\;2} \\[3ex] c * (c + d) = m * m \\[3ex] c(c + d) = m^2 \\[3ex] $

$ \underline{Intersecting\;\;Tangent-Secant\;\;Theorem} \\[3ex] \underline{Circle\;3} \\[3ex] p^2 = k(k + t) \\[5ex] \underline{Circle\;4} \\[3ex] (y - w) * y = r^2 \\[3ex] y(y - w) = r^2 \\[3ex] $

(17.) **Angle of Intersecting Secant-Tangent Theorem**

The angle formed when a secant and a tangent intersect outside a circle is equal to half the difference of the intercepted arcs.

(1.)
**
The angle in a semicircle is a right angle (an angle of 90°).
**

Prove that angle

You must

So, we shall use triangle theorems here (first example)...GCSE students

Then, we shall use another approach (second example) where we shall use a circle theorem...for all those who would like to learn another approach.

Construction: Draw the radius from the centre

This will display two triangles: $\triangle AOC$ and $\triangle BOC$

Label the angles in the triangle.

$ \underline{\triangle AOC} \\[3ex] Because: \\[3ex] |AO| = |CO| ...same\;\;radius \\[3ex] \angle OAC = \angle OCA = \theta ...base\;\;\angle s\;\;of\;\;isosceles\;\;\triangle AOC \\[3ex] \angle OAC + \angle OCA + \angle AOC = 180^\circ ...sum\;\;of\;\;\angle s\;\;of\;\;\triangle AOC \\[3ex] \implies \\[3ex] \theta + \theta + \alpha = 180 \\[3ex] 2\theta + \alpha = 180 ...eqn.(1) \\[5ex] \underline{\triangle BOC} \\[3ex] Because: \\[3ex] |BO| = |CO| ...same\;\;radius \\[3ex] \angle OBC = \angle OCB = \phi ...base\;\;\angle s\;\;of\;\;isosceles\;\;\triangle BOC \\[3ex] \angle OBC + \angle OCB + \angle BOC = 180^\circ ...sum\;\;of\;\;\angle s\;\;of\;\;\triangle BOC \\[3ex] \implies \\[3ex] \phi + \phi + \beta = 180 \\[3ex] 2\phi + \beta = 180 ...eqn.(2) \\[5ex] \alpha + \beta = 180^\circ ...\angle s\;\;on\;\;a\;\;straight\;\;line \\[3ex] \alpha + \beta = 180...eqn.(3) \\[5ex] eqn.(1) + eqn.(2) \implies \\[3ex] (2\theta + \alpha) + (2\phi + \beta) = 180 + 180 \\[3ex] 2\theta + \alpha + 2\phi + \beta = 360 \\[3ex] 2\theta + 2\phi + \alpha + \beta = 360 \\[3ex] Substitute\;\;for\;\;eqn.(3) \\[3ex] 2\theta + 2\phi + 180 = 360 \\[3ex] 2\theta + 2\phi = 360 - 180 \\[3ex] 2\theta + 2\phi = 180 \\[3ex] 2(\theta + \phi) = 180 \\[3ex] \theta + \phi = \dfrac{180}{2} \\[5ex] \theta + \phi = 90^\circ \\[3ex] \angle ACB = \theta + \phi ...diagram \\[3ex] \therefore \angle ACB = 90^\circ \\[5ex] $

$ \angle AOB = 180^\circ ...\angle\;\;on\;\;a\;\;straight\;\;line \\[3ex] \angle AOB = 2 * \angle ACB ...\angle\;\;at\;\;centre = 2 * \angle\;\;at\;\;circumference \\[3ex] \implies \\[3ex] 180 = 2 * \angle ACB \\[3ex] 2 * \angle ACB = 180 \\[3ex] \angle ACB = \dfrac{180}{2} \\[5ex] \angle ACB = 90^\circ $

(2.) **Angles in the same segment of a circle are equal.**

OR

**Angles subtended by a chord of a circle in the same segment of the circle are equal.**

a) Prove that in a circle, the angles in the same segment are equal.

Construction: Draw the radii: from point P on the circumference to point O on the center; and from point Q on the circumference to point O on the center.

$ a) \\[3ex] \angle POQ = 2 * \angle PAQ ...\angle\;\;at\;\;centre = 2 * \angle\;\;at\;\;circumference \\[3ex] Also: \\[3ex] \angle POQ = 2 * \angle PBQ ...\angle\;\;at\;\;centre = 2 * \angle\;\;at\;\;circumference \\[3ex] \implies \\[3ex] 2 * \angle PAQ = 2 * \angle PBQ \\[3ex] Divide\;\;both\;\;sides\;\;by\;\;2 \\[3ex] \angle PAQ = \angle PBQ $

(3.)
**
The angle which an arc of a circle subtends at the center is twice the angle which the same
arc of the circle subtends at the circumference.
**

$E\hat{F}H = x$ and $G\hat{F}H = y$

Complete the proof of the following statement:

Tha angle at the centre is twice the angle at the circumference.

Proof:

$E\hat{F}G = x + y$

$F\hat{E}H = x$ (triangle

$F\hat{H}E = .................................$

$ (b.) \\[3ex] For\;\;this\;\;question: \\[3ex] \underline{Given}: \\[3ex] E\hat{F}G = x + y ...\angle \;\;at\;\;circumference \\[3ex] Obtuse\;\; \angle EHG = \angle \;\;at\;\;centre \\[3ex] \underline{To\;\;Prove}: \\[3ex] \angle EHG = 2(x + y) \\[3ex] \underline{\triangle FEH} \\[3ex] E\hat{F}H = x...as\;\;shown\;\;in\;\;the\;\;diagram \\[3ex] F\hat{E}H = E\hat{F}H = x ...base\;\;\angle s\;\;of\;\;isosceles\;\;\triangle FEH \\[3ex] F\hat{H}E + F\hat{E}H + H\hat{F}E = 180...sum\;\;of\;\;\angle s\;\;of\;\;\triangle FEH \\[3ex] F\hat{H}E + x + x = 180 \\[3ex] F\hat{H}E + 2x = 180 \\[3ex] F\hat{H}E = 180 - 2x ...eqn.(1) \\[3ex] \underline{\triangle FEG} \\[3ex] H\hat{F}G = y...as\;\;shown\;\;in\;\;the\;\;diagram \\[3ex] F\hat{G}H = H\hat{F}G = y ...base\;\;\angle s\;\;of\;\;isosceles\;\;\triangle FEH \\[3ex] F\hat{H}G + F\hat{G}H + H\hat{F}G = 180...sum\;\;of\;\;\angle s\;\;of\;\;\triangle FEH \\[3ex] F\hat{H}G + y + y = 180 \\[3ex] F\hat{H}G + 2y = 180 \\[3ex] F\hat{H}G = 180 - 2y ...eqn.(2) \\[3ex] Reflex\;\; \angle EHG = F\hat{H}E + F\hat{H}G \\[3ex] Reflex\;\; \angle EHG = (180 - 2x) + (180 - 2y) \\[3ex] Reflex\;\; \angle EHG = 180 - 2x + 180 - 2y \\[3ex] Reflex\;\; \angle EHG = 360 - 2x - 2y \\[3ex] Reflex\;\; \angle EHG + Obtuse\;\; \angle EHG = 360^\circ...sum\;\;of\;\;\angle s\;\;at\;\;a\;\;point \\[3ex] (360 - 2x - 2y) + Obtuse\;\; \angle EHG = 360 \\[3ex] Obtuse\;\;\angle EHG = 360 - (360 - 2x - 2y) \\[3ex] Obtuse\;\;\angle EHG = 360 - 360 + 2x + 2y \\[3ex] Obtuse\;\;\angle EHG = 2x + 2y \\[3ex] Obtuse\;\;\angle EHG = 2(x + y) \\[3ex] \implies \\[3ex] $ The angle at the centre is twice the angle at the circumference.

(4.)
**The interior opposite angles of a cyclic quadrilateral are supplementary**

Construction: Join the radii: from the centre O to point B, and from the centre O to point D

$ \color{blue}{\angle DOB} = 2 * \angle DCB ...\angle\;\;at\;\;centre = 2 * \angle\;\;at\;\;circumference \\[3ex] \color{green}{\angle DOB} = 2 * \angle DAB ...\angle\;\;at\;\;centre = 2 * \angle\;\;at\;\;circumference \\[3ex] \color{blue}{\angle DOB} + \color{green}{\angle DOB} = 360^\circ ...\angle s\;\;at\;\;a\;\;point \\[3ex] \implies \\[3ex] 2 * \angle DCB + 2 * \angle DAB = 360 \\[3ex] 2(\angle DCB + \angle DAB) = 360 \\[3ex] \angle DCB + \angle DAB = \dfrac{360}{2} \\[5ex] \angle DCB + \angle DAB = 180^\circ \\[3ex] \angle DCB = \angle C ...diagram \\[3ex] \angle DAB = \angle A ...diagram \\[3ex] \implies \\[3ex] \angle C + \angle A = 180^\circ $

(8.)
**
The angle between a tangent to a circle and the chord drawn from the point of contact, is equal to the angle in the
alternate segment.
**

O is the centre of the circle.

DE is a tangent to the circle.

Point A is where DE touches the circle.

Prove that angle

Construction: Join the radii: from centre O to the point A, and from centre O to point B

Radii: |OA| and |OB|

This forms a triangle.

Label the angles in the triangle.

$ \angle OAB = \theta \\[3ex] \angle OBA = \theta \\[3ex] \angle AOB = \phi \\[5ex] \angle OAB = \angle OBA = \theta ...base\;\;\angle s\;\;of\;\;isosceles\;\;\triangle AOB \\[3ex] \theta + \theta + \phi = 180^\circ ...sum\;\;of\;\;\angle s\;\;of\;\;\triangle AOB \\[3ex] 2\theta + \phi = 180 ...eqn.(1) \\[5ex] \theta + x = 90^\circ ...radius\;OA \perp tangent\;DAE\;\;at\;\;point\;\;contact\;A \\[3ex] \theta + x = 90 ...eqn.(2) \\[3ex] \implies \\[3ex] \theta = 90 - x ...eqn.(3) \\[5ex] \phi = 2 * y ...\angle \;\;at\;\;centre = 2 * \angle \;\;at\;\;circumference \\[3ex] \phi = 2y ...eqn.(4) \\[5ex] Substitute\;\;eqn.(3)\;\;and\;\;eqn.(4)\;\;into\;\;eqn.(1) \\[3ex] 2\theta + \phi = 180 \\[3ex] 2(90 - x) + 2y = 180 \\[3ex] 180 - 2x + 2y = 180 \\[3ex] 2y = 180 - 180 + 2x \\[3ex] 2y = 2x \\[3ex] Divide\;\;both\;\;sides\;\;by\;\;2 \\[3ex] y = x \\[3ex] x = y \\[3ex] $ Therefore, the angle between the tangent

KFC is a tangent to the circle at F.

Prove the theorem which states that $D\hat{F}K = \hat{E}$

Construction: Join the radii: from centre O to the point D, and from centre O to point F

Radii: |OD| and |OF|

This forms a triangle.

Label the angles in the triangle.

$ \angle ODF = \theta \\[3ex] \angle OFD = \theta \\[3ex] \angle DOF = \psi \\[5ex] \angle ODF = \angle OFD = \theta ...base\;\;\angle s\;\;of\;\;isosceles\;\;\triangle DOF \\[3ex] \theta + \theta + \psi = 180^\circ ...sum\;\;of\;\;\angle s\;\;of\;\;\triangle DOF \\[3ex] 2\theta + \psi = 180 \\[3ex] \psi = 180 - 2\theta ...eqn.(1) \\[5ex] \psi = 2 * \hat{E} ...\angle \;\;at\;\;centre = 2 * \angle \;\;at\;\;circumference \\[3ex] \psi = 2\hat{E} ...eqn.(2) \\[5ex] \psi = \psi \implies eqn.(2) = eqn.(1) \\[3ex] 2\hat{E} = 180 - 2\theta \\[3ex] 2\hat{E} = 2(90 - \theta) \\[3ex] \hat{E} = 90 - \theta \\[3ex] \theta = 90 - \hat{E} ... eqn.(3) \\[5ex] \theta + D\hat{F}K = 90^\circ ...radius\;OF \perp tangent\;KFC\;\;at\;\;point\;\;contact\;F \\[3ex] \theta + D\hat{F}K = 90 \\[3ex] \theta = 90 - D\hat{F}K ...eqn.(4) \\[5ex] \theta = \theta \implies eqn.(3) = eqn.(4) \\[3ex] 90 - \hat{E} = 90 - D\hat{F}K \\[3ex] D\hat{F}K = 90 - 90 + \hat{E} \\[3ex] D\hat{F}K = \hat{E} \\[3ex] $ Therefore, the angle between the tangent

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