IGCSE Further Maths Geometric Proofs Exercise [Test Your Understanding] Triangle π΄π΅πΆ is isosceles with π΄πΆ = π΅πΆ. Triangle πΆπ·πΈ is isosceles with πΆπ· = πΆπΈ. π΄πΆπ· and π·πΈπΉ are straight lines. a) Prove that angle π·πΆπΈ = 2π₯ b) Prove that π·πΉ is perpendicular to π΄π΅. Question 1 [Set 4 Paper 1 Q4] π΄π΅πΆ is a right-angled triangle. Angle π΄πΆπ΅ = π₯. Angle π΅π΄π· = 90 − 2π₯. Prove that π΄πΆπ· is an isosceles triangle. Question 2 [Set 4 Paper 1 Q9] π΄π΅πΆπ· is a quadrilateral. Prove that π₯ = π¦. Question 3 [Set 4 Paper 2 Q3] π΄π΅ is parallel to πΆπ·. Is ππ parallel to ππ ? You must show your working. IGCSE Further Maths Geometric Proofs Exercise [Test Your Understanding] Triangle π΄π΅πΆ is isosceles with π΄πΆ = π΅πΆ. Triangle πΆπ·πΈ is isosceles with πΆπ· = πΆπΈ. π΄πΆπ· and π·πΈπΉ are straight lines. a) Prove that angle π·πΆπΈ = 2π₯ b) Prove that π·πΉ is perpendicular to π΄π΅. Question 1 [Set 4 Paper 1 Q4] π΄π΅πΆ is a right-angled triangle. Angle π΄πΆπ΅ = π₯. Angle π΅π΄π· = 90 − 2π₯. Prove that π΄πΆπ· is an isosceles triangle. Question 2 [Set 4 Paper 1 Q9] π΄π΅πΆπ· is a quadrilateral. Prove that π₯ = π¦. Question 3 [Set 4 Paper 2 Q3] π΄π΅ is parallel to πΆπ·. Is ππ parallel to ππ ? You must show your working. Question 4 [Set 4 Paper 2 Q13] πππ π is a cyclic quadrilateral. ππ = ππ . πππ is a tangent to the circle. Work out the value of π₯. You must show your working. Question 5 [Specimen Paper 1 Q15] π΄, π΅, πΆ and π· are points on the circumference of a circle such that π΅π· is parallel to the tangent to the circle at π΄. Prove that π΄πΆ bisects angle π΅πΆπ·. Give reasons at each stage of your working. Question 6 [Specimen Paper 2 Q7] Prove that π΄π΅ is parallel to π·πΆ. Question 7 [June 2012 Paper 2 Q5] π΄π΅πΆ is a triangle. π is a point on π΄π΅ such that π΄π = ππΆ = π΅πΆ. Angle π΅π΄πΆ = π₯. a) Prove that angle π΄π΅πΆ = 2π₯. b) You are also given that π΄π΅ = π΄πΆ. Work out the value of π₯. Question 4 [Set 4 Paper 2 Q13] πππ π is a cyclic quadrilateral. ππ = ππ . πππ is a tangent to the circle. Work out the value of π₯. You must show your working. Question 5 [Specimen Paper 1 Q15] π΄, π΅, πΆ and π· are points on the circumference of a circle such that π΅π· is parallel to the tangent to the circle at π΄. Prove that π΄πΆ bisects angle π΅πΆπ·. Give reasons at each stage of your working. Question 6 [Specimen Paper 2 Q7] Prove that π΄π΅ is parallel to π·πΆ. Question 7 [June 2012 Paper 2 Q5] π΄π΅πΆ is a triangle. π is a point on π΄π΅ such that π΄π = ππΆ = π΅πΆ. Angle π΅π΄πΆ = π₯. a) Prove that angle π΄π΅πΆ = 2π₯. b) You are also given that π΄π΅ = π΄πΆ. Work out the value of π₯.
