Let θ be the acute angle formed between the tangent at point A to a circle and one of its chords, AB. Considering any point D on the larger of the arcs AB, denoted by ϕ as the angle ADB, what relationship exists between the angles ϕ and θ ?
A) They are explementary.
B) There is no relationship between the two angles.
C) They are supplementary.
D) They are equal.
E) They are complementary.
IMAT 2024 · Question 52
Q52 - Circle and circumference
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The correct answer is D) They are equal.
This is the tangent–chord theorem: the angle between a tangent and a chord equals the inscribed angle on the other side of the chord. D is on the larger arc, which is exactly the "other side" for the acute angle θ, so ϕ = θ.
Both are half the central angle on AB, which is why they come out equal.
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