Circle Theorem Proofs Flashcards

All 7 cards in this deck

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'.

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).