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Name the circle theorem that justifies a step

10th grade (Year 11 in England) · 16 questions · Common Core: HSG.C.A.2, HSG.C.A.3 · England: GCSE Higher Geometry

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  1. Question 1

    O is the centre of the circle. Which theorem justifies this step? ∠OTP = 90°, where PT touches the circle at T

    • A. A tangent is perpendicular to the radius
    • B. The angle at the centre is twice the angle at the circumference
    • C. Tangents from an outside point are equal in length
    • D. The angle in a semicircle is 90°

  2. Question 2

    O is the centre of the circle. Which theorem justifies this step? ∠OTP = 90°, where PT touches the circle at T

    • A. Angles in the same segment are equal
    • B. Tangents from an outside point are equal in length
    • C. The angle at the centre is twice the angle at the circumference
    • D. A tangent is perpendicular to the radius

  3. Question 3

    O is the centre of the circle. Which theorem justifies this step? AB is a diameter, so ∠ACB = 90°

    • A. Angles in the same segment are equal
    • B. The angle in a semicircle is 90°
    • C. The angle at the centre is twice the angle at the circumference
    • D. Opposite angles of a cyclic quadrilateral add up to 180°

  4. Question 4

    O is the centre of the circle. Which theorem justifies this step? PA = PB, where PA and PB touch the circle

    • A. Angles in the same segment are equal
    • B. Opposite angles of a cyclic quadrilateral add up to 180°
    • C. The angle at the centre is twice the angle at the circumference
    • D. Tangents from an outside point are equal in length

  5. Question 5

    O is the centre of the circle. Which theorem justifies this step? ∠AOB = 2 × ∠ACB

    • A. The angle at the centre is twice the angle at the circumference
    • B. Tangents from an outside point are equal in length
    • C. The angle in a semicircle is 90°
    • D. Angles in the same segment are equal

  6. Question 6

    O is the centre of the circle. Which theorem justifies this step? ∠AOB = 2 × ∠ACB

    • A. The angle in a semicircle is 90°
    • B. Angles in the same segment are equal
    • C. Opposite angles of a cyclic quadrilateral add up to 180°
    • D. The angle at the centre is twice the angle at the circumference

  7. Question 7

    O is the centre of the circle. Which theorem justifies this step? ∠ADB = ∠ACB, both standing on chord AB on the same side

    • A. Angles in the same segment are equal
    • B. The angle in a semicircle is 90°
    • C. Opposite angles of a cyclic quadrilateral add up to 180°
    • D. Tangents from an outside point are equal in length

  8. Question 8

    O is the centre of the circle. Which theorem justifies this step? AB is a diameter, so ∠ACB = 90°

    • A. Angles in the same segment are equal
    • B. The angle in a semicircle is 90°
    • C. Tangents from an outside point are equal in length
    • D. Opposite angles of a cyclic quadrilateral add up to 180°

  9. Question 9

    O is the centre of the circle. Which theorem justifies this step? AB is a diameter, so ∠ACB = 90°

    • A. The angle in a semicircle is 90°
    • B. Opposite angles of a cyclic quadrilateral add up to 180°
    • C. A tangent is perpendicular to the radius
    • D. Tangents from an outside point are equal in length

  10. Question 10

    O is the centre of the circle. Which theorem justifies this step? ∠ADB = ∠ACB, both standing on chord AB on the same side

    • A. A tangent is perpendicular to the radius
    • B. Tangents from an outside point are equal in length
    • C. Angles in the same segment are equal
    • D. The angle at the centre is twice the angle at the circumference

  11. Question 11

    O is the centre of the circle. Which theorem justifies this step? ∠AOB = 2 × ∠ACB

    • A. The angle in a semicircle is 90°
    • B. The angle at the centre is twice the angle at the circumference
    • C. Tangents from an outside point are equal in length
    • D. Opposite angles of a cyclic quadrilateral add up to 180°

  12. Question 12

    O is the centre of the circle. Which theorem justifies this step? ∠AOB = 2 × ∠ACB

    • A. Angles in the same segment are equal
    • B. The angle at the centre is twice the angle at the circumference
    • C. A tangent is perpendicular to the radius
    • D. Opposite angles of a cyclic quadrilateral add up to 180°

  13. Question 13

    O is the centre of the circle. Which theorem justifies this step? ∠ADB = ∠ACB, both standing on chord AB on the same side

    • A. A tangent is perpendicular to the radius
    • B. Opposite angles of a cyclic quadrilateral add up to 180°
    • C. Tangents from an outside point are equal in length
    • D. Angles in the same segment are equal

  14. Question 14

    O is the centre of the circle. Which theorem justifies this step? ∠ABC + ∠ADC = 180° in quadrilateral ABCD drawn inside the circle

    • A. Opposite angles of a cyclic quadrilateral add up to 180°
    • B. Angles in the same segment are equal
    • C. Tangents from an outside point are equal in length
    • D. A tangent is perpendicular to the radius

  15. Question 15

    O is the centre of the circle. Which theorem justifies this step? ∠AOB = 2 × ∠ACB

    • A. The angle in a semicircle is 90°
    • B. The angle at the centre is twice the angle at the circumference
    • C. A tangent is perpendicular to the radius
    • D. Opposite angles of a cyclic quadrilateral add up to 180°

  16. Question 16

    O is the centre of the circle. Which theorem justifies this step? ∠ABC + ∠ADC = 180° in quadrilateral ABCD drawn inside the circle

    • A. A tangent is perpendicular to the radius
    • B. Opposite angles of a cyclic quadrilateral add up to 180°
    • C. The angle at the centre is twice the angle at the circumference
    • D. The angle in a semicircle is 90°

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Answer key: Name the circle theorem that justifies a step

  1. 1. A. A tangent is perpendicular to the radius

  2. 2. D. A tangent is perpendicular to the radius

  3. 3. B. The angle in a semicircle is 90°

  4. 4. D. Tangents from an outside point are equal in length

  5. 5. A. The angle at the centre is twice the angle at the circumference

  6. 6. D. The angle at the centre is twice the angle at the circumference

  7. 7. A. Angles in the same segment are equal

  8. 8. B. The angle in a semicircle is 90°

  9. 9. A. The angle in a semicircle is 90°

  10. 10. C. Angles in the same segment are equal

  11. 11. B. The angle at the centre is twice the angle at the circumference

  12. 12. B. The angle at the centre is twice the angle at the circumference

  13. 13. D. Angles in the same segment are equal

  14. 14. A. Opposite angles of a cyclic quadrilateral add up to 180°

  15. 15. B. The angle at the centre is twice the angle at the circumference

  16. 16. B. Opposite angles of a cyclic quadrilateral add up to 180°