Angles (Including Parallel Lines): WJEC GCSE Intermediate June 2018 Unit 1 Q18

Intermediate June 2018 Unit 1 Q18

18. In the triangle ABC shown below, \(B\hat{A}C = 40^\circ\) and \(A\hat{C}B = 80^\circ\).
X is a point on side AC such that BX = BC.

Diagram not drawn to scale

Prove that AX = BX.
Give reasons for each step of your proof.
You must show all your working. [5]

Mark scheme

Scheme	Marks	Comments
		Angles shown on the diagram take precedence. If any angle is not named then it must be unambiguously identified either on the diagram, from a given reason or in further work. (e.g. must be convincing that X = 80 is referring to BXC and not AXB.) If initial incorrect assumptions are made then allow correct FT methods to calculate other relevant angles.
∠BXC = 80(°) Reason: ‘BX = BC’ OR ‘Isosceles triangle’	B1
∠AXB (= 180 – 80 ) = 100(°) Reason: ‘Angles on a straight line’.	B1	FT 180 – ‘their ∠BXC’
∠ABX (= 180 – 40 – 100) = 40(°) Reason: ‘Angles in a triangle’.	B1	FT 180 – 40 − ‘their ∠AXB’.
Statement ‘So AX = BX’, Reason: ‘Two equal angles (in a triangle)’ OR ∠ ABX = ∠BAX OR ‘Isosceles triangle’.	B1	Only available if ∠ABX stated or shown to be 40(°)
Sight of at least TWO of the above reasons.	E1	Reasons must be appropriate AND are dependent on associated B1 gained.

Scheme	Marks	Comments
∠BXC = 80(°) Reason: ‘BX = BC’ OR ‘Isosceles triangle’.	B1
∠CBX (= 180 – 80 − 80) = 20(°) Reason: ‘Angles in a triangle’.	B1	FT 180 – 80 − ‘their ∠BXC’.
∠ABX (= 180 – 80 – 40 – 20) = 40(°) Reason: ‘Angles in a triangle’.	B1	FT 180 – 80 – 40 − ‘their ∠CBX’.
Statement ‘So AX = BX’. Reason: ‘Two equal angles (in a triangle)’ OR ∠ABX = ∠BAX OR ‘Isosceles triangle’.	B1	Only available if ∠ABX stated or shown* to be 40(°)*
Sight of at least TWO of the above reasons.	E1	Reasons must be appropriate AND are dependent on associated B1 gained.

Scheme	Marks	Comments
∠ABX = 40(°) Reason: ‘AX = BX’ OR ‘Isosceles triangle’.	B1
∠AXB (= 180 – 40 − 40) = 100(°) Reason: ‘Angles in a triangle’.	B1	FT 180 – 40 − ‘their ∠ABX’.
∠ BXC = 80(°) Reason: ‘Angles on a straight line’.	B1	FT 180 − ‘their ∠AXB’.
Statement ‘So BX = BC’ (as given) Reason: ‘Two equal angles (in a triangle)’ OR ‘∠BXC = ∠BCX’ OR ‘Isosceles triangle’.	B1	Only available if ∠BXC stated or shown* to be 80(°)*
Sight of at least TWO of the above reasons.	E1	Reasons must be appropriate AND are dependent on associated B1 gained.

Scheme	Marks	Comments
∠ABX = 40(°) Reason: ‘AX = BX’ OR ‘Isosceles triangle’.	B1
∠CBX (= 180 – 80 − 40− 40) = 20(°) Reason: ‘Angles in a triangle’.	B1	FT 180 – 80 – 40 − ‘their ∠ABX’.
∠ BXC (= 180 – 80 – 20) = 80(°) Reason: ‘Angles in a triangle’.	B1	FT 180 – 80 − ‘their ∠CBX’.
Statement ‘So BX = BC’ (as given) Reason: ‘Two equal angles (in a triangle)’ OR ‘∠BXC = ∠BCX’ OR ‘Isosceles triangle’.	B1	Only available if ∠BXC stated or shown* to be 80(°)*
Sight of at least TWO of the above reasons.	E1	Reasons must be appropriate AND are dependent on associated B1 gained.

(The alternative methods are printed in italics in the original, with their headings underlined.)