Circle Theorems: Cyclic Quadrilaterals and the Same Segment · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Cyclic quadrilaterals and shared arcs

Mathematics · Geometry · ages 15-16
Name ______________________   Date ____________

  1. In cyclic quadrilateral PQRS, angle Q = 115°. Angle S is opposite angle Q. What is angle S, in degrees?

    Answer: ______________

  2. ABCD is a cyclic quadrilateral. Angle A = 70°. What is the size of angle C?

    • 110°
    • 70°
    • 180°
    • 35°

  3. ABCD is a cyclic quadrilateral. Angle A is 70 degrees. What is angle C?

    • 110 degrees
    • 70 degrees
    • 180 degrees

  4. In a cyclic quadrilateral, opposite angles always add to 180 degrees. True or false?

    Circle one:   True   False

  5. Two points on opposite sides of a chord always make equal angles with its endpoints. True or false?

    Circle one:   True   False

  6. True or false: if two points are on opposite sides of a chord, the angles they make with that chord's endpoints are always equal.

    Circle one:   True   False

  7. Points C and D lie on the same arc of a circle, on the same side of chord AB. Angle ACB = 42°. What is angle ADB?

    • 21°
    • 42°
    • 84°
    • 138°

  8. C and D sit on the same arc, on the same side of chord AB. Angle ACB is 42 degrees. What is angle ADB?

    • 21 degrees
    • 42 degrees
    • 84 degrees

  9. Quadrilateral WXYZ is cyclic and angle W = 80°. Points X and M both look at side WZ from the same side and share the same arc, with angle WMZ = 35°, so angle WXZ = 35° too. Which theorem justifies that step, and what is angle Y?

    • Same Segment Theorem for angle WXZ; angle Y = 100°
    • Cyclic Quadrilateral Theorem for angle WXZ; angle Y = 100°
    • Same Segment Theorem for angle WXZ; angle Y = 80°
    • Angle Bisector Theorem for angle WXZ; angle Y = 100°

  10. Cyclic ABCD has angle A of 3x degrees and opposite angle C of 2x plus 30 degrees. What is angle C, in degrees?

    Answer: ______________

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Answer key

For grown-ups. Fold this page away before handing over the rest.

Cyclic quadrilaterals and shared arcs W1-mt_11GlmkiqB1-s1

  1. 65 · Opposite angles in a cyclic quadrilateral sum to 180°, so angle S = 180° − 115° = 65°.
  2. 110° · Opposite angles in a cyclic quadrilateral add up to 180°, so angle C = 180° − 70° = 110°.
  3. 110 degrees · Opposite angles total 180, so 180 minus 70 is 110.
  4. True · That is exactly what the Cyclic Quadrilateral Theorem says.
  5. False · Equality needs both points in the same segment.
  6. False · The Same Segment Theorem only holds when both points are on the same side of the chord, in the same segment.
  7. 42° · Angles standing on the same arc AB are equal, so angle ADB matches angle ACB.
  8. 42 degrees · Same arc and same side means equal angles.
  9. Same Segment Theorem for angle WXZ; angle Y = 100° · Angle WXZ matches angle WMZ because X and M share the same arc, that's the Same Segment Theorem. Angle Y is opposite angle W, so the Cyclic Quadrilateral Theorem gives 180° − 80° = 100°.
  10. 90 · 5x plus 30 is 180, so x is 30 and C is 90.
Worksheet · LightMySky