Questions: Angles Formed by Chords, Secants, and Tangents
4 questions to test your understanding
Score: 0 / 4
Question 1 Multiple Choice
Two chords intersect inside a circle. The angle and its vertical angle intercept arcs of 70° and 110°. What is the measure of the angle?
A20°
B70°
C90°
D180°
For two chords crossing inside a circle, the angle equals half the sum of the intercepted arcs: (70° + 110°)/2 = 90°. 20° would come from wrongly using half the difference, which is the rule for vertices outside the circle.
Question 2 Short Answer
Two secants meet at a point outside a circle. They intercept a far arc of 130° and a near arc of 50°. Find the angle between the secants.
Think about your answer, then reveal below.
Model answer: 40°
For a vertex outside the circle, the angle is half the difference of the intercepted arcs: (130° - 50°)/2 = 40°.
Question 3 True / False
An angle whose vertex is inside a circle (but not at the center) is always half of the arc between its sides.
TTrue
FFalse
Answer: False
Half the arc is the rule for vertices on the circle. For a vertex inside, the angle is half the sum of the arc between its sides and the arc between the extensions of its sides. Only when the vertex is the center do both arcs equal the angle itself.
Question 4 Multiple Choice
A tangent and a chord meet at the point of tangency, and the chord cuts off an arc of 140° on the side of the angle. What is the angle between the tangent and the chord?
A70°
B140°
C40°
D110°
A tangent-chord angle has its vertex on the circle, so it behaves like an inscribed angle: it measures half its intercepted arc, 140°/2 = 70°.