Flashcards on Circle Geometry Theorems and Proofs
Circle Geometry Theorems and Proofs: A Comprehensive Guide
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Euclidean Circle Geometry
13 cards
Card 1
Question: In a cyclic quadrilateral PQRS where PQ = PR and tangents at P and R meet QS produced at A, why are angles S3 and S4 equal?
Answer: Because equal chords subtend equal angles in the same segment: PQ = PR implies arcs QS subtend equal angles at S on the circle, so ∠S3 = ∠S4.
Card 2
Question: How can you prove that SMRC is a cyclic quadrilateral in the given construction (with tangents at P and R meeting QS at A, and extensions meeting tang
Answer: Show that opposite angles sum to 180°: ∠SMR and ∠SCR are supplementary because each equals an angle between chord and tangent or equal inscribed angle
Card 3
Question: Why is RP a tangent to the circle through P, S and A at P in the described configuration?
Answer: Because ∠RPA equals the angle in the opposite arc (angle RSA) — the angle between RP and the chord PS equals the angle in the alternate segment, so RP
Card 4
Question: If a circle passes through A,B,C,D,G with AD as diameter and BD and AC intersect at H, and angle AGC = 58°, what is angle B2 (angle subtended by arc B
Answer: Angle B2 equals 90° because AD is a diameter so any angle subtended by AD is a right angle; using given info and cyclic properties gives the specific
Card 5
Question: Given the same diagram with AB = BC, how can you prove AB is tangent to the circle through A,H and D?
Answer: If AB = BC, triangle ABC is isosceles so base angles equal; angle between AB and AC equals angle in opposite arc (angle ADC), so by the tangent-chord
Card 6
Question: In a circle with points P,Q,R,S center O, tangent UV at P, PR and OS intersect at T, RQ produced to W, with ∠PQW = 106° and SP = SR, how do you find ∠
Answer: Since SP = SR, triangle PSR is isosceles so base angles at P and R are equal; relate given exterior angle ∠PQW to arc measures to compute ∠PSR accordi
Card 7
Question: In the same circle, how do you determine angle R3 (an inscribed angle related to arcs determined by given equalities)?
Answer: Use isosceles triangle PSR (SP = SR) and the relation ∠PQW = 106° to find arcs then deduce ∠R3 as an inscribed angle subtending the relevant arc (subt
Card 8
Question: How can you find angle P_S (angle at S subtending chord PR) in that circle?
Answer: Use the fact triangle PSR is isosceles (SP = SR) so angles at P and R equal; compute using known angles from the arc determined by ∠PQW = 106° and ins
Card 9
Question: How to find angle O1 (angle at center O related to chord PR) when given the previous data?
Answer: Central angle O1 is twice the inscribed angle that subtends the same arc, so compute O1 = 2×(corresponding inscribed angle from earlier steps).
Card 10
Question: How to find angle P3 (an angle at P related to tangent UV) in that diagram?
Answer: Use tangent-chord theorem: angle between tangent UV and chord at P equals the angle in the opposite arc; compute using previously found arc or inscrib