circle theorem

• What is the term for a straight line that goes from one side of a circle to the other?

  Chord
• What does a chord do to a circle?

  A chord splits a circle into two segments.
• What does the first Circle theorem state?

  • Angles in the same segment are equal.
  • If angles are formed using the same chord within the same segment, they are the same size.
• If one angle in a segment is 68 degrees, what are the other angles in that segment?

  68 degrees
• What is often referred to as the "bow tie theorem"?

  The theorem stating that angles in the same segment are equal.
• What is a diameter in a circle?

  A diameter is a chord that goes straight through the center of the circle.
• What does the second Circle theorem state?

  • The angle in a semicircle is always 90 degrees.
  • This occurs when the chord is the diameter of the circle.
• If the angle at the circumference is 50 degrees, what is the angle at the center using the same chord?

  100 degrees
• What is a cyclic quadrilateral?

  • A quadrilateral where all four corners touch the circumference of the circle.
  • Not all quadrilaterals are cyclic; for example, if one corner is not on the circumference, it is not cyclic.
• What does the theorem about cyclic quadrilaterals state?

  • Opposite angles in a cyclic quadrilateral add up to 180 degrees.
  • For example, if one angle is 86 degrees, the opposite angle must be 94 degrees.
• What is a tangent in relation to a circle?

  A tangent is a straight line that touches the circle at one point.
• What does the theorem about tangents and radii state?

  • A tangent meets a radius at 90 degrees.
  • This means the angle formed at the point of contact is a right angle.
• What is the theorem regarding tangents from the same point?

  • Tangents from the same point are equal in length.
  • This can be shown by measuring the distances from the point to where the tangents touch the circle.
• What does it mean to bisect an angle?

  • To bisect an angle means to divide it into two equal parts.
  • For example, if the whole angle is 30 degrees, each part would be 15 degrees.
• What does the alternate segment theorem state?

  • The angle that a tangent makes with a chord is equal to the angle that the chord makes at the circumference in the opposite segment.
  • This applies to both sides of the tangent.
• What is the significance of labeling angles in circle theorem problems?

  • Labeling angles helps clarify the relationships and theorems applied.
  • It aids in showing the reasoning behind the calculations.
• How do you find the angle ABD if angle ACD is known to be 75 degrees?

  Angle ABD is also 75 degrees because they are in the same segment.
• What is the relationship between angles in an isosceles triangle?

  Two angles in an isosceles triangle are equal.
• If angle ACD is 35 degrees in an isosceles triangle, what is angle ADC?

  110 degrees
• How do you determine the angle ABC in a cyclic quadrilateral?

  By subtracting the opposite angle from 180 degrees.
• What is the angle BCA if it is formed by a diameter?

  90 degrees
• How do you find the angle CAB in triangle ABC if angles BCA and ABC are known?

  By subtracting the sum of angles BCA and ABC from 180 degrees.
• If angle OCF is 90 degrees and angle OCA is 32 degrees, what is angle CEF?

  58 degrees
• What is the relationship between the angles in an isosceles triangle formed by two radii?

  The angles opposite the equal sides are equal.
• How do you find the angle OBC in triangle COB if angle OCF is known?

  By recognizing that triangle COB is isosceles and using the known angle to find the others.
• What is the final step to find angle ABC in triangle ABC?

  By subtracting the sum of the known angles from 180 degrees.
• What should you do when solving Circle theorem problems?

  • Clearly label all angles and segments.
  • State the theorems used to justify your answers.
  • Show all working out for clarity.