Circle Relationships: Angles, Radii, and Chords - Geometry

1. What is the measure of the central angle if an inscribed angle intercepting the same arc is $28^\circ$?

   $56^\circ$. The central angle is twice the inscribed angle: $2 \times 28° = 56°$.

2. What is the intercepted arc measure if a tangent-chord angle measures $41^\circ$?

   $82^\circ$. The arc is twice the angle: $2 \times 41° = 82°$.

3. What is the relationship between congruent chords and their intercepted arcs in the same circle?

   Congruent chords intercept congruent arcs. Equal chords create equal arcs in the same circle.

4. What conclusion can you make if two chords in the same circle are congruent?

   They are equidistant from the center. Equal chords are always equidistant from the center.

5. What conclusion can you make if a chord is closer to the center than another chord (same circle)?

   The closer chord is longer. Shorter distance from center means longer chord length.

6. What is the measure of an exterior angle formed by two tangents from an external point?

   $m\angle = \frac{1}{2}(m\overset{\frown}{\text{major}}-m\overset{\frown}{\text{minor}})$. Exterior angles equal half the difference of intercepted arcs.

7. What is the measure of an inscribed angle that intercepts a semicircle (a diameter)?

   $90^\circ$. A semicircle (diameter) creates a $180^\circ$ arc, so inscribed angle is $\frac{180^\circ}{2} = 90^\circ$.

8. What is the exterior angle measure if two secants intercept arcs of $210^\circ$ and $130^\circ$?

   $40^\circ$. Use $\frac{1}{2}(210° - 130°) = \frac{80°}{2} = 40°$.

9. What is the measure of an inscribed angle that intercepts an arc of $180^\circ$?

   $90^\circ$. A $180°$ arc (semicircle) creates a $90°$ inscribed angle.

10. What theorem states that an angle inscribed in a semicircle is a right angle?

    Thales' theorem: an inscribed angle subtending a diameter is $90^\circ$. This fundamental theorem about semicircles is named after Thales.

11. What is the relationship between a radius and a tangent line at the point of tangency?

    They are perpendicular: $\text{radius} \perp \text{tangent}$. A radius to the tangent point creates a right angle.

12. What is the name for a chord that passes through the center of the circle?

    A diameter. A diameter is a special chord passing through the center.

13. What is the relationship between congruent arcs and their corresponding chords in the same circle?

    Congruent arcs have congruent chords. Equal arcs are created by equal chords in the same circle.

14. What is the exterior angle measure if two secants intercept arcs of $250^\circ$ and $150^\circ$?

    $50^\circ$. Use $\frac{1}{2}(250° - 150°) = \frac{100°}{2} = 50°$.

15. What is the relationship between a diameter perpendicular to a chord and that chord?

    It bisects the chord (and its intercepted arc). A perpendicular diameter divides the chord into equal parts.

16. What is the relationship between a larger chord and its distance from the center (same circle)?

    A larger chord is closer to the center. Longer chords are positioned closer to the center.

17. What is the measure of an inscribed angle that intercepts a diameter $\overline{AB}$?

    $90^\circ$. By Thales' theorem, any inscribed angle on a diameter is $90°$.

18. What is the measure of an inscribed angle if the corresponding central angle is $110^\circ$?

    $55^\circ$. The inscribed angle is half the central angle: $\frac{110°}{2} = 55°$.

19. What is the measure of an exterior angle formed by a tangent and a secant from an external point?

    $m\angle = \frac{1}{2}(m\overset{\frown}{\text{far}}-m\overset{\frown}{\text{near}})$. Exterior angles equal half the difference of intercepted arcs.

20. What is the angle measure if two chords intersect inside a circle and the intercepted arcs are $100^\circ$ and $60^\circ$?

    $80^\circ$. Use $\frac{1}{2}(100° + 60°) = \frac{160°}{2} = 80°$.

21. What is the relationship between equal inscribed angles and their intercepted arcs?

    Equal inscribed angles intercept congruent arcs. Equal inscribed angles create equal arcs in the same circle.

22. What is the measure of a central angle that intercepts an arc of $35^\circ$?

    $35^\circ$. A central angle equals its intercepted arc measure.

23. What is the measure of an inscribed angle that intercepts a major arc of $240^\circ$?

    $120^\circ$. Use $\frac{1}{2} \times 240° = 120°$.

24. What is the measure of an inscribed angle that intercepts an arc of $86^\circ$?

    $43^\circ$. Use $\frac{1}{2} \times 86° = 43°$.

25. What is the measure of an angle formed by two chords intersecting inside a circle?

    $m\angle = \frac{1}{2}(m\overset{\frown}{\text{arc}}_1+m\overset{\frown}{\text{arc}}_2)$. Interior chord angles equal half the sum of intercepted arcs.

26. What is the name for a segment from the center of a circle to a point on the circle?

    A radius. A radius connects the center to any point on the circle.

27. What is the definition of a tangent line to a circle?

    A line that touches the circle at exactly one point. A tangent touches at exactly one point without crossing.

28. What is the measure of an inscribed angle if the intercepted arc is $35^\circ$?

    $17.5^\circ$. Use $\frac{1}{2} \times 35° = 17.5°$.

29. What is the exterior angle measure if a tangent and secant intercept arcs of $190^\circ$ and $70^\circ$?

    $60^\circ$. Use $\frac{1}{2}(190° - 70°) = \frac{120°}{2} = 60°$.

30. What is the measure of a tangent-chord angle if its intercepted arc measures $148^\circ$?

    $74^\circ$. Use $\frac{1}{2} \times 148° = 74°$.

31. What is the name for a segment whose endpoints lie on a circle?

    A chord. A chord connects two points on the circle's circumference.

32. What is the relationship between equal central angles and their intercepted chords?

    Equal central angles intercept congruent chords. Equal central angles create equal chords in the same circle.

33. What is the measure of the intercepted arc if an inscribed angle is $17.5^\circ$?

    $35^\circ$. The arc is twice the inscribed angle: $2 \times 17.5° = 35°$.

34. What is the relationship between a central angle measure and its intercepted arc measure?

    They are equal: $m\angle AOB = m\overset{\frown}{AB}$. A central angle equals its intercepted arc by definition.

35. What is the angle measure if two chords intersect inside a circle and arcs are $140^\circ$ and $50^\circ$?

    $95^\circ$. Use $\frac{1}{2}(140° + 50°) = \frac{190°}{2} = 95°$.

36. What is the measure of the intercepted arc if an inscribed angle measures $37^\circ$?

    $74^\circ$. The arc is twice the inscribed angle: $2 \times 37° = 74°$.

37. What is the relationship between an inscribed angle measure and its intercepted arc measure?

    $m\angle ACB = \frac{1}{2}m\overset{\frown}{AB}$. An inscribed angle is half its intercepted arc.

38. What is the relationship between an inscribed angle and the central angle intercepting the same arc?

    $m\angle ACB = \frac{1}{2}m\angle AOB$. An inscribed angle is half the central angle on the same arc.

39. What conclusion can you make if two central angles intercept congruent arcs in the same circle?

    The central angles are congruent. Equal arcs create equal central angles in the same circle.

40. What is the measure of an angle formed by two tangents if the major arc is $260^\circ$?

    $80^\circ$. Use $\frac{1}{2}(260^\circ - 100^\circ) = \frac{160^\circ}{2} = 80^\circ$

41. What is the relationship between the perpendicular bisector of a chord and the circle's center?

    It passes through the center. The perpendicular bisector of any chord passes through the center.

42. What is the relationship between two chords equidistant from the center of the same circle?

    They are congruent. Chords at equal distances from center have equal lengths.

43. What is the relationship between equal inscribed angles and their intercepted chords?

    Equal inscribed angles intercept congruent chords. Equal inscribed angles create equal chords in the same circle.

44. What is the measure of the angle formed by a radius and a tangent at the point of tangency?

    $90^\circ$. This perpendicular relationship is fundamental to tangent properties.

45. What is the measure of an angle formed by a tangent and a chord, in terms of its intercepted arc?

    $m\angle = \frac{1}{2}m\overset{\frown}{\text{arc}}$. A tangent-chord angle equals half its intercepted arc.

46. What is the measure of a central angle that intercepts a semicircle?

    $180^\circ$. A semicircle is exactly half the circle, which is $180°$.

47. What is the definition of a secant line to a circle?

    A line that intersects the circle at two points. A secant crosses through the circle at two points.

48. What conclusion can you make if two inscribed angles intercept the same arc $\overset{\frown}{AB}$?

    They are congruent. Inscribed angles on the same arc are always equal.

49. What is the measure of an exterior angle formed by two secants from a point outside the circle?

    $m\angle = \frac{1}{2}(m\overset{\frown}{\text{far}}-m\overset{\frown}{\text{near}})$. Exterior angles equal half the difference of intercepted arcs.

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