Lesson 1 ~35 min Circle Geometry · Path +85 XP

The Language of Circles

Circle geometry has more vocabulary than any other topic at this level, and every theorem is stated in it. The single most useful skill is naming the arc an angle stands on, because almost every theorem is a claim about angles standing on the same arc.

Today's hook: "The angle at the centre is twice the angle at the circumference standing on the same arc." Every word of that is doing work. Get the arc wrong and the theorem gives you the wrong angle, with no warning, because the diagram will look perfectly reasonable either way.

0/5QUESTS

Think First

warm-up

Draw a circle and mark two points $A$ and $B$ on it. You have created two arcs, not one. Now mark a third point $P$ somewhere on the circle and join $PA$ and $PB$. Which of the two arcs does the angle at $P$ stand on, and how would you describe the difference to someone who could not see your drawing?

Record your answer in your workbook.

The Big Idea

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An angle in a circle is defined by three things: its vertex, the two points its arms pass through, and therefore the arc between those two points that does not contain the vertex. That last arc is what the angle stands on, and it is the piece of information every circle theorem is really about.

An angle stands on the arc between its arms that does NOT contain its vertex.

Two points on a circle always create two arcs, a minor and a major one. So "the arc $AB$" is ambiguous until you say which, and the vertex settles it: the angle stands on the arc on the far side. Saying which arc, every time, is what stops a theorem being applied to the wrong pair of angles.

Two points, two arcs

Always say minor or major, or name a third point on the arc you mean.

The vertex is not on it

An angle stands on the arc on the opposite side from its vertex.

Same arc, same theorem

Most circle theorems compare angles standing on the SAME arc.

What You'll Master

objectives

Know

The terms centre, radius, diameter, chord, arc, segment, sector, tangent and secant
That two points on a circle create a minor and a major arc
What it means for an angle to stand on an arc

Understand

Why an angle stands on the arc NOT containing its vertex
Why naming the arc is necessary before any circle theorem can be applied

Can Do

Label the parts of a circle correctly in a diagram
Identify the arc on which a given angle at the centre or circumference stands
Distinguish angles that stand on the same arc from angles that do not

Words You Need

vocabulary

ChordA line segment joining two points on the circle. A diameter is the longest chord.

ArcPart of the circumference between two points. Two points give a minor arc and a major arc.

SegmentThe region between a chord and one of its arcs.

SectorThe region between two radii and the arc between them.

TangentA line touching the circle at exactly one point.

SecantA line crossing the circle at two points. A chord is the part of a secant inside the circle.

The Parts, and What Distinguishes Them

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Several pairs of terms are easy to confuse because they describe the same picture from different angles.

Chord and secant. A chord is a segment with both endpoints on the circle. A secant is the whole infinite line through those points. The chord is the part of the secant inside the circle, so every chord lies on a secant.

Segment and sector. A segment is cut off by a chord; a sector is cut off by two radii. A sector has a corner at the centre; a segment does not touch the centre at all unless the chord is a diameter.

Tangent and secant. A tangent meets the circle exactly once, a secant exactly twice. There is no line meeting a circle three times, and the only remaining case is a line that misses it entirely.

A useful check

Count intersections. Two means secant, one means tangent, none means the line misses. That is the complete list.

Two Points Make Two Arcs

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Mark $A$ and $B$ on a circle and you have divided the circumference into two pieces. The shorter is the minor arc, the longer the major arc, and they are equal only when $AB$ is a diameter.

This makes "arc $AB$" ambiguous, and the ambiguity is not pedantic: the two arcs subtend very different angles. There are two standard ways to remove it.

Say which. "The minor arc $AB$" or "the major arc $AB$".

Name a third point. "Arc $APB$" means the arc from $A$ to $B$ that passes through $P$. This is the more precise convention and the one used when both arcs are in play.

Once you are naming arcs properly, the theorems become readable. "Angles in the same segment" and "angles standing on the same arc" are the same condition stated two ways, and neither means anything until you can say which arc.

Which Arc Does an Angle Stand On

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Take $\angle APB$, with vertex $P$ on the circle and arms through $A$ and $B$. The angle stands on the arc $AB$ that does not contain $P$.

The reason is worth seeing rather than memorising. The arms of the angle cut the circle at $A$ and $B$, and they open away from $P$ across the circle. The arc they open onto is the far one.

The same applies at the centre. $\angle AOB$, with $O$ the centre, stands on the arc $AB$ on the far side of the centre from the angle's opening. When the angle at the centre is reflex, it stands on the major arc; when it is not, it stands on the minor arc.

This is precisely where the angle-at-the-centre theorem goes wrong for students. If $P$ is on the minor arc, $\angle APB$ stands on the major arc, and it must be compared with the reflex angle at the centre, not the ordinary one.

Reading a Theorem as a Sentence About Arcs

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Every theorem in this area has the same shape once you can see it. Two examples, both proved in later lessons.

"The angle at the centre is twice the angle at the circumference standing on the same arc." The condition is the same arc. Two angles standing on different arcs have no relationship of this kind, and the theorem simply does not apply.

"Angles in the same segment are equal." A segment is one side of a chord, so all vertices in that segment are on the same arc, and their angles all stand on the other arc. Same arc again, differently worded.

So before applying any theorem, answer two questions in order: where is the vertex, and which arc does the angle stand on. If two angles do not stand on the same arc, no theorem comparing them applies, however similar the diagram looks.

Common Pitfalls

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Writing "arc $AB$" without saying which of the two arcs is meant.

Fix: say minor or major, or name a third point on the arc, as in arc $APB$. In a proof the marker cannot tell which you meant.

Assuming an angle stands on the arc nearest to it.

Fix: it stands on the FAR arc, the one not containing its vertex. The arms open away from the vertex.

Confusing a segment with a sector.

Fix: a sector is bounded by two radii and has a corner at the centre. A segment is bounded by a chord and does not reach the centre.

Watch Me Solve It · 3 examples

Watch Me Solve It · Naming the parts

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PROBLEM

In a circle with centre $O$, the points $A$, $B$ and $C$ lie on the circumference. The line $AB$ is drawn, the line $OC$ is drawn, and a line touches the circle at $C$ only. Name each of these three lines precisely.

1

The line AB

Both endpoints lie on the circle, so it is a chord. If it were extended beyond the circle it would be a secant, and the chord is the part inside.
2

The line OC

It joins the centre to a point on the circle, so it is a radius.
3

The line touching at C only

One point of contact means it is a tangent. Two would make it a secant, and there is no other possibility.
4

Note what the diagram now guarantees

Because $OC$ is a radius and the tangent touches at $C$, those two lines are perpendicular. That fact is proved in Lesson 4, and naming the parts correctly is what makes it available.

AnswerA chord, a radius, and a tangent

Watch Me Solve It · Identifying the arc

+15 XP per step

PROBLEM

$A$, $B$ and $P$ lie on a circle, with $P$ on the major arc $AB$. State the arc on which $\angle APB$ stands, and the arc on which the non-reflex $\angle AOB$ stands.

1

Locate the vertex of the first angle

$P$ is on the major arc, so the angle at $P$ cannot stand on the arc containing $P$.
2

Name the arc it stands on

$\angle APB$ stands on the minor arc $AB$, the arc not containing $P$.
3

Consider the angle at the centre

The non-reflex $\angle AOB$ opens across the shorter side, so it also stands on the minor arc $AB$.
4

State what follows

Both angles stand on the SAME arc, so the angle-at-the-centre theorem applies and $\angle AOB = 2 \times \angle APB$.

AnswerBoth stand on the minor arc $AB$, so the theorem applies

Watch Me Solve It · When the theorem does not apply

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PROBLEM

$A$, $B$ and $Q$ lie on a circle, with $Q$ on the MINOR arc $AB$. Explain why $\angle AOB = 2 \times \angle AQB$ is false for the non-reflex angle at the centre, and state the correct relationship.

1

Find the arc the angle at Q stands on

$Q$ is on the minor arc, so $\angle AQB$ stands on the MAJOR arc $AB$.
2

Find the arc the non-reflex angle at the centre stands on

The non-reflex $\angle AOB$ stands on the minor arc, which is a different arc.
3

Conclude that the theorem does not apply as written

The theorem requires both angles to stand on the same arc. These do not, so no relationship of that kind is available.
4

State the correct version

The angle at the centre standing on the MAJOR arc is the reflex $\angle AOB$, so reflex $\angle AOB = 2 \times \angle AQB$. Since the two angles at the centre sum to $360°$, this also gives the familiar cyclic-quadrilateral result.

AnswerThe reflex angle at the centre is twice $\angle AQB$, not the non-reflex one

Brain Trainer · 4 problems

Brain Trainer · Name it precisely

4 problems

Four quick problems. Work each one, then reveal the answer.

1 A line meets a circle at exactly one point. What is it?

One intersection means it touches rather than crosses.A tangent
2 A region is bounded by two radii and an arc. What is it?

A corner at the centre means it is not a segment.A sector
3 $P$ lies on the major arc $AB$. Which arc does $\angle APB$ stand on?

The arc not containing the vertex.The minor arc $AB$
4 How many arcs do two points on a circle create?

The circumference is divided into two pieces.Two, a minor and a major

Complete in your workbook.

Multiple Choice · 5 questions

MC1

Chord and secant

+10 XP

A line passing through a circle and extending beyond it on both sides is:

MC2

Segment and sector

+10 XP

The region bounded by a chord and one of its arcs is:

MC3

Which arc

+10 XP

$P$ lies on the minor arc $AB$. The angle $\angle APB$ stands on:

MC4

Ambiguity

+10 XP

Writing "arc $AB$" without further description is a problem because:

MC5

When a theorem applies

+10 XP

Two angles in a circle can be compared by a circle theorem only when they:

Short Answer · 3 questions

Label and describe

+15 XP

SHORT ANSWER

A circle has centre $O$. Points $A$, $B$, $C$ and $D$ lie on the circumference, in that order around the circle. The chord $AC$ is drawn, and a line touches the circle at $D$.
(a) Name the type of each of: $OA$, $AC$, the line through $A$ and $C$ extended, and the line at $D$.
(b) Name the two arcs created by $A$ and $C$, using a third point to remove any ambiguity.
(c) State which arc $\angle ABC$ stands on, and justify your answer.

Write your working in your book.

Decide whether the theorem applies

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SHORT ANSWER

In each case, state the arc each angle stands on and say whether the angle-at-the-centre theorem relates them.
(a) $\angle AOB$ (non-reflex) and $\angle APB$, with $P$ on the major arc.
(b) $\angle AOB$ (non-reflex) and $\angle AQB$, with $Q$ on the minor arc.
(c) Reflex $\angle AOB$ and $\angle AQB$, with $Q$ on the minor arc.

Write your working in your book.

Explain the rule

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SHORT ANSWER

(a) Explain, in your own words, why an angle stands on the arc that does not contain its vertex.
(b) A student says "it does not matter which arc you pick, as long as you are consistent". Give a specific counterexample.
(c) Explain why the two angles at the centre standing on the two arcs must sum to $360°$, and what that fact lets you do.

Write your working in your book.

Stretch Challenge · How many angles stand on one arc?

+25 XP

CHALLENGE

$A$ and $B$ are fixed on a circle, and the minor arc $AB$ subtends $70°$ at the centre.
(a) A point $P$ moves along the major arc. What happens to $\angle APB$ as it moves, and why?
(b) Find $\angle AQB$ for a point $Q$ on the minor arc, and explain why it differs from your answer to (a).
(c) Deduce a relationship between $\angle APB$ and $\angle AQB$, and name the standard result you have just derived.

Quick Review

recap

Two points

Make a minor arc and a major arc

Stands on

The arc NOT containing the vertex

Before any theorem

Name the vertex, then name the arc

At the centre

The two angles sum to $360°$

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