Geometric Proof Flashcards

All 14 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 stand on the same arc).

State the circle theorem about angles in the same segment.

Angles in the same segment are equal (angles at the circumference subtended by the same arc are equal).

What is the rule for the angles of a cyclic quadrilateral?

Opposite angles of a cyclic quadrilateral add up to $180°$.

State the alternate segment theorem.

The angle between a tangent and a chord equals the angle in the alternate segment (the angle subtended by that chord at the circumference on the other side).

What is the size of the angle in a semi-circle?

$90°$ — the angle subtended at the circumference by a diameter is a right angle.

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

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

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

$90°$.

What is true of the two tangents drawn to a circle from the same external point?

They are equal in length.

Which four conditions can be quoted to prove two triangles are congruent?

SSS, SAS, ASA (or AAS) and RHS.

What is the rule for co-interior (allied) angles between two parallel lines?

They add up to $180°$.

True or false? In a formal proof, 'the angles look equal in the diagram' is an acceptable reason.

False. Every step must be justified by a stated theorem or angle fact.

What are the steps to prove an angle result given in terms of letters?
e.g. in triangle $ABC$, $AB = AC$ and angle $BAC = y$; show each base angle is $\frac{180 - y}{2}$

1. Write each unknown angle in terms of the letters: base angles equal, each $= b$.
2. Form an equation using a theorem: $y + 2b = 180$ (angle sum of a triangle).
3. Rearrange to the required result: $b = \frac{180 - y}{2}$.

In a circle diagram, two lines are drawn from the centre to the circumference to make a triangle. What extra fact can you always use?

The triangle is isosceles, because both lines are radii and so are equal, making the two base angles equal.

What are the steps to decide which circle theorems to use when no theorem is named?
e.g. a tangent touches the circle at $A$ and chord $AB$ makes an angle of $60°$ with it

1. Name the features: tangent, chord, radius, diameter, cyclic quadrilateral. Here: tangent and chord meeting at $A$.
2. Quote the theorem that matches: alternate segment theorem, so the angle in the alternate segment $= 60°$.
3. Repeat with each new angle, giving a reason, until the required angle is reached.