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IGCSEFM-ProofExercises

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IGCSE Further Maths Geometric Proofs Exercise
[Test Your Understanding]
Triangle 𝐴𝐡𝐢 is isosceles with 𝐴𝐢 = 𝐡𝐢.
Triangle 𝐢𝐷𝐸 is isosceles with 𝐢𝐷 = 𝐢𝐸.
𝐴𝐢𝐷 and 𝐷𝐸𝐹 are straight lines.
a) Prove that angle 𝐷𝐢𝐸 = 2π‘₯
b) Prove that 𝐷𝐹 is perpendicular to 𝐴𝐡.
Question 1 [Set 4 Paper 1 Q4]
𝐴𝐡𝐢 is a right-angled triangle. Angle 𝐴𝐢𝐡 = π‘₯. Angle
𝐡𝐴𝐷 = 90 − 2π‘₯.
Prove that 𝐴𝐢𝐷 is an isosceles triangle.
Question 2 [Set 4 Paper 1 Q9]
𝐴𝐡𝐢𝐷 is a quadrilateral.
Prove that π‘₯ = 𝑦.
Question 3 [Set 4 Paper 2 Q3]
𝐴𝐡 is parallel to 𝐢𝐷.
Is 𝑃𝑄 parallel to 𝑆𝑅? You must show your working.
IGCSE Further Maths Geometric Proofs Exercise
[Test Your Understanding]
Triangle 𝐴𝐡𝐢 is isosceles with 𝐴𝐢 = 𝐡𝐢.
Triangle 𝐢𝐷𝐸 is isosceles with 𝐢𝐷 = 𝐢𝐸.
𝐴𝐢𝐷 and 𝐷𝐸𝐹 are straight lines.
a) Prove that angle 𝐷𝐢𝐸 = 2π‘₯
b) Prove that 𝐷𝐹 is perpendicular to 𝐴𝐡.
Question 1 [Set 4 Paper 1 Q4]
𝐴𝐡𝐢 is a right-angled triangle. Angle 𝐴𝐢𝐡 = π‘₯. Angle
𝐡𝐴𝐷 = 90 − 2π‘₯.
Prove that 𝐴𝐢𝐷 is an isosceles triangle.
Question 2 [Set 4 Paper 1 Q9]
𝐴𝐡𝐢𝐷 is a quadrilateral.
Prove that π‘₯ = 𝑦.
Question 3 [Set 4 Paper 2 Q3]
𝐴𝐡 is parallel to 𝐢𝐷.
Is 𝑃𝑄 parallel to 𝑆𝑅? You must show your working.
Question 4 [Set 4 Paper 2 Q13]
𝑃𝑄𝑅𝑆 is a cyclic quadrilateral. 𝑄𝑆 = 𝑄𝑅. 𝑉𝑆𝑇 is a
tangent to the circle.
Work out the value of π‘₯. You must show your
working.
Question 5 [Specimen Paper 1 Q15]
𝐴, 𝐡, 𝐢 and 𝐷 are points on the circumference of a circle
such that 𝐡𝐷 is parallel to the tangent to the circle at 𝐴.
Prove that 𝐴𝐢 bisects angle 𝐡𝐢𝐷. Give reasons at each
stage of your working.
Question 6 [Specimen Paper 2 Q7]
Prove that 𝐴𝐡 is parallel to 𝐷𝐢.
Question 7 [June 2012 Paper 2 Q5]
𝐴𝐡𝐢 is a triangle. 𝑃 is a point on 𝐴𝐡 such that 𝐴𝑃 = 𝑃𝐢 = 𝐡𝐢. Angle
𝐡𝐴𝐢 = π‘₯.
a) Prove that angle 𝐴𝐡𝐢 = 2π‘₯.
b) You are also given that 𝐴𝐡 = 𝐴𝐢. Work out the value of π‘₯.
Question 4 [Set 4 Paper 2 Q13]
𝑃𝑄𝑅𝑆 is a cyclic quadrilateral. 𝑄𝑆 = 𝑄𝑅. 𝑉𝑆𝑇 is a
tangent to the circle.
Work out the value of π‘₯. You must show your
working.
Question 5 [Specimen Paper 1 Q15]
𝐴, 𝐡, 𝐢 and 𝐷 are points on the circumference of a circle
such that 𝐡𝐷 is parallel to the tangent to the circle at 𝐴.
Prove that 𝐴𝐢 bisects angle 𝐡𝐢𝐷. Give reasons at each
stage of your working.
Question 6 [Specimen Paper 2 Q7]
Prove that 𝐴𝐡 is parallel to 𝐷𝐢.
Question 7 [June 2012 Paper 2 Q5]
𝐴𝐡𝐢 is a triangle. 𝑃 is a point on 𝐴𝐡 such that 𝐴𝑃 = 𝑃𝐢 = 𝐡𝐢. Angle
𝐡𝐴𝐢 = π‘₯.
a) Prove that angle 𝐴𝐡𝐢 = 2π‘₯.
b) You are also given that 𝐴𝐡 = 𝐴𝐢. Work out the value of π‘₯.
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