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Chapter 2 Practice Paper: Proofs in Plane Geometry — question paper

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QQ/S4AM/CH2

Secondary 4 · Additional Mathematics (G3)

Chapter 2 Practice Paper

Proofs in Plane Geometry

Duration: 40 minutes
Total: 30 marks
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Class:
Register No.:
Date:

READ THESE INSTRUCTIONS FIRST

  • Write your name, class and register number in the spaces at the top of this page.
  • Answer all questions.
  • Show all necessary working. Omission of essential working will result in loss of marks.
  • Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless a different level of accuracy is specified in the question.
  • The number of marks is given in brackets [ ] at the end of each question or part question.
  • The total number of marks for each paper is shown on its cover.

Original practice material written for the MOE Additional Mathematics syllabus. It is not an official MOE or school examination paper. Chapter 2 (G3 Proofs in plane geometry).

For Examiner’s Use
Section A/ 30
Total/ 30

Section A: Structured questions[30 marks]

Answer all questions in the spaces provided. Show all necessary working and give a reason for every deductive step in a proof. Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless told otherwise.

  1. 1

    In triangle PQR, PQ = PR and the bisector of angle QPR meets QR at M. Angle QPR = 96°.

    1. (a)

      Prove that triangles PQM and PRM are congruent, and deduce that PM is perpendicular to QR.

      [3]
    2. (b)

      Find angle QPM and angle PMR.

      angles =[2]
  2. 2

    Two parallel lines l and m are cut by a transversal that meets l at S and m at T. The angle between l and the transversal at S, on the upper side, is 117°.

    1. (a)

      Write down the angle between m and the transversal at T on the same side of the transversal and the same side of m, giving a reason.

      [2]
    2. (b)

      Find the angle that is co-interior with the 117° angle at T.

      [2]
    3. (c)

      Explain why the transversal makes the same acute angle with both parallel lines.

      [2]
  3. 3

    A, B, C and D lie on a circle with centre O, and C and D both lie on the major arc AB. Angle AOB = 130°.

    1. (a)

      Find angle ACB.

      [2]
    2. (b)

      Write down angle ADB, giving a reason.

      [2]
    3. (c)

      The tangent to the circle at A is drawn. Given that angle OAB = 25°, find the angle between this tangent and the chord AB. Explain how your answer agrees with the alternate segment theorem.

      [3]
  4. 4

    In triangle ABC, D lies on AB with AD = 3 cm and DB = 6 cm, and E lies on AC with AE = 4 cm and EC = 8 cm.

    1. (a)

      Show that triangles ADE and ABC are similar.

      [3]
    2. (b)

      Given BC = 9.6 cm, find DE.

      DE =[1]
    3. (c)

      Given that the area of triangle ADE is 2.1 cm², find the area of triangle ABC.

      [2]
  5. 5

    TA and TB are tangents to a circle with centre O from an external point T, and angle ATB = 54°. C is a point on the major arc AB of the circle.

    1. (a)

      Find angle TAB.

      [2]
    2. (b)

      Find angle AOB.

      [2]
    3. (c)

      Find angle ACB, where C stands on the major arc AB.

      [2]

— End of paper —