Circle Theorems Flashcards

All 36 cards in this deck

State the circle theorem about the angle at the centre.

The angle at the centre is twice the angle at the circumference when both are subtended by the same arc.

What does it mean to say two angles are 'subtended by the same arc'?

e.g. $\angle AOB$ and $\angle ACB$

Both angles are formed by lines drawn to the same two endpoints of that arc.

e.g. both stand on arc $AB$.

Does the 'angle at the centre' theorem still work when the angle at the centre is reflex?

Yes — the reflex angle at the centre is twice the angle at the circumference in the opposite segment.

True or false? Any angle at the centre of a circle is twice any angle at the circumference.

False. The two angles must be subtended by the same arc (stand on the same chord).

State the circle theorem about the angle in a semicircle.

The angle in a semicircle is $90^\circ$ (the angle at the circumference subtended by a diameter is a right angle).

The angle in a semicircle being $90^\circ$ is a special case of which circle theorem?

The angle at the centre is twice the angle at the circumference — the angle at the centre is a straight line, $180^\circ$, so the angle at the circumference is $90^\circ$.

What must you check before using the 'angle in a semicircle' theorem?

That the side opposite the angle is a diameter, i.e. it passes through the centre.

True or false? In a triangle drawn in a semicircle, the right angle is the one at the centre.

False. The right angle is at the vertex on the circumference, opposite the diameter.

What is a chord of a circle?

A straight line joining two points on the circumference.

What is the difference between a sector and a segment of a circle?

A sector is bounded by two radii and an arc; a segment is bounded by a chord and an arc.

What is the angle between a tangent and the radius at the point of contact?

$90^\circ$ — the tangent to a circle is perpendicular to the radius.

What is the equal-tangents property?

The two tangents drawn to a circle from the same external point are equal in length.

What does the perpendicular from the centre of a circle to a chord do?

It bisects the chord (cuts it into two equal halves).

True or false? The angle between a tangent and a chord at the point of contact is always $90^\circ$.

False. It is $90^\circ$ only between a tangent and the radius (or diameter) at that point.

What is a cyclic quadrilateral?

A quadrilateral with all four vertices on the circumference of a circle.

State the circle theorem about the angles of a cyclic quadrilateral.

Opposite angles of a cyclic quadrilateral sum to $180^\circ$.

In a cyclic quadrilateral, what is the exterior angle equal to?

The opposite interior angle.

True or false? Opposite angles of a cyclic quadrilateral are equal.

False. They sum to $180^\circ$.

True or false? The opposite-angles theorem works for any quadrilateral drawn inside a circle.

False. All four vertices must lie on the circumference.

State the circle theorem about angles in the same segment.

Angles in the same segment, subtended by the same arc (chord), are equal.

Which angle equals $\angle ACB$ by 'angles in the same segment'?

e.g. chord $AB$, with $C$ and $D$ both on the major arc

$\angle ADB$ — angles in the same segment are equal, so $\angle ADB = \angle ACB$.

True or false? Two angles at the circumference in the same segment are equal even if they stand on different chords.

False. They must be subtended by the same chord (arc).

True or false? An angle in the major segment equals the angle in the minor segment standing on the same chord.

False. Those two angles sum to $180^\circ$ (opposite angles of a cyclic quadrilateral).

State the alternate segment theorem.

The angle between a tangent and a chord equals the angle in the alternate segment.

What two features must a diagram have before you can use the alternate segment theorem?

A tangent, and a chord drawn from the point of contact.

Which angle equals the angle between the tangent and the chord?

e.g. tangent $DAE$ at $A$, chord $AB$, $\angle BAE = 56^\circ$, with $C$ on the other side of $AB$

$\angle ACB = 56^\circ$ — the angle in the alternate segment.

True or false? The angle between a tangent and a chord equals the angle in the segment on the same side of the chord.

False. It equals the angle in the alternate (opposite) segment.

True or false? The alternate segment theorem only works if the chord passes through the centre.

False. It works for any chord drawn from the point of contact.

Which triangle fact is used in nearly every circle theorem proof?

Two radii form an isosceles triangle, so its base angles are equal.

What are the steps to prove that the angle at the centre is twice the angle at the circumference?

e.g. $\angle ACB$ at the circumference, $\angle AOB$ at the centre

1. Join $CO$ and extend it past $O$; let $\angle OCA = x$, $\angle OCB = y$.
2. $OA = OC$ and $OB = OC$ (radii), so base angles of isosceles triangles give $\angle OAC = x$, $\angle OBC = y$.
3. Exterior angle of a triangle: $\angle AOB = 2x + 2y = 2\angle ACB$.

What are the steps to prove that the angle in a semicircle is $90^\circ$ without using circle theorems?

e.g. $AOB$ a diameter, $C$ on the circumference

1. Join $OC$: $OA = OB = OC$ (radii), giving two isosceles triangles.
2. Base angles equal: $\angle OAC = \angle OCA = x$ and $\angle OBC = \angle OCB = y$.
3. Angles in triangle $ABC$: $x + x + y + y = 180$, so $\angle ACB = x + y = 90^\circ$.

What are the steps to prove that opposite angles of a cyclic quadrilateral sum to $180^\circ$?

e.g. cyclic quadrilateral $ABCD$, centre $O$, with $\angle A = a$ and $\angle C = c$

1. Angle at the centre is twice the angle at the circumference: the two angles at $O$ are $2a$ and $2c$.
2. Angles around a point: $2a + 2c = 360^\circ$.
3. Divide by 2: $a + c = 180^\circ$.

Which circle theorem is normally used to prove that two triangles formed by two chords crossing inside a circle are similar?

Angles in the same segment are equal (together with vertically opposite angles), giving equal angles in both triangles.

In a question saying 'you must give a reason for each stage of your working', what counts as a reason?

The full name of the circle theorem or angle fact used at that step, e.g. 'the tangent to a circle is perpendicular to the radius'.

True or false? A triangle formed by two radii and a chord of a circle is always isosceles.

True. The two radii are equal in length, so the base angles at the ends of the chord are equal.

What are the steps to prove two triangles in a circle are similar?

e.g. chords $AC$ and $BD$ cross at $P$; prove triangle $APB$ is similar to triangle $DPC$

1. Find a pair of equal angles from angles in the same segment: $\angle BAP = \angle CDP$.
2. Find a second pair: $\angle APB = \angle DPC$ (vertically opposite angles).
3. Two pairs of equal angles, so the triangles are similar (AA).