Edexcel GCSE Maths (1MA1), Higher tier · Geometry and measures
Practise Congruence and geometric proof. 15 exam-style questions plus unlimited generated ones on this subtopic, at up to four difficulty levels, with full mark schemes and a progress tracker. Free, no account needed.
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Write down the condition for congruence when two sides and the angle between them are equal.
SAS
Sample questions
Written for this site in the style of Edexcel exam questions. They are not taken from real past papers.
Question 1Easy2 marks
(a) Ahmed says, 'If two triangles have the same three angles, they must be congruent.' Explain why Ahmed is wrong.[1]
(b) Write down the condition for congruence when two sides and the angle between them are equal.[1]
Show the answer and mark scheme
(a)Answer: Triangles with the same angles can be different sizes (they are similar, not necessarily congruent), e.g. equilateral triangles with sides 2 cm and 5 cm.
C1 for explaining that the triangles could be different sizes (an enlargement), e.g. two equilateral triangles with different side lengths
Worked solution: An equilateral triangle of side 2 cm and one of side 5 cm both have angles 60°, 60°, 60°, but they are not the same size, so they are not congruent.
(b)Answer: SAS
B1 for SAS
Worked solution: SAS (side, angle, side).
Question 2Medium7 marks
The diagram shows a circle, centre O. AB is a chord of the circle. M is the midpoint of AB. Diagram NOT accurately drawn
[object Object]
(a) Prove that triangle OAM is congruent to triangle OBM.[3]
(b) Hence prove that OM is perpendicular to AB.[2]
(c) OA = 9 cm and AB = 14 cm. Work out the length of OM. Give your answer as a surd in its simplest form.[2]
Show the answer and mark scheme
(a)Answer: OA = OB (radii), AM = BM (M is the midpoint of AB), OM is common, so the triangles are congruent (SSS).
C1 for OA = OB, with the reason that they are radii (of the same circle)
C1 for AM = BM, with the reason that M is the midpoint of AB
C1 for OM common to both triangles, and the conclusion that the triangles are congruent (SSS)
Worked solution: In triangles OAM and OBM: OA = OB (radii of the same circle) AM = BM (M is the midpoint of AB) OM = OM (common side) So triangle OAM is congruent to triangle OBM (SSS).
(b)Answer: Angle OMA = angle OMB (corresponding angles of congruent triangles). AMB is a straight line, so angle OMA + angle OMB = 180°, so each angle is 90°.
C1 for angle OMA = angle OMB, with the reason that they are corresponding angles of congruent triangles
C1 for angle OMA + angle OMB = 180° (angles on a straight line), so angle OMA = 90° and OM is perpendicular to AB
Worked solution: From part (a), angle OMA = angle OMB (corresponding angles of congruent triangles). AMB is a straight line, so angle OMA + angle OMB = 180° (angles on a straight line add up to 180°). So 2 × angle OMA = 180° and angle OMA = 90°. Therefore OM is perpendicular to AB.
(c)Answer: \(OM = 4\sqrt{2}\) cm
M1 for AM = 7 and \(OM^2 = 9^2 - 7^2\) (= 32)
A1 for \(4\sqrt{2}\)
Worked solution: AM = ½ × 14 = 7 cm. Since OM is perpendicular to AB, triangle OAM is right-angled at M: \(OM = \sqrt{9^2 - 7^2} = \sqrt{32} = \sqrt{16 \times 2} = 4\sqrt{2}\) cm
Question 3Hard7 marks
The diagram shows a circle, centre O. AB and CD are chords of the circle, with AB = CD. M is the point on AB and K is the point on CD such that OM is perpendicular to AB and OK is perpendicular to CD. Diagram NOT accurately drawn
[object Object]
(a) Prove that OM = OK.[4]
(b) OA = 15 cm and AB = 24 cm. Work out the length of OM.[2]
(c) Write down the length of OK.[1]
Show the answer and mark scheme
(a)Answer: AM = ½AB and CK = ½CD (the perpendicular from the centre to a chord bisects the chord), so AM = CK; OA = OC (radii); angle OMA = angle OKC = 90°; so triangles OAM and OCK are congruent (RHS) and OM = OK.
C1 for AM = ½AB and CK = ½CD, with the reason that the perpendicular from the centre of a circle to a chord bisects the chord
C1 for AM = CK, since AB = CD
C1 for OA = OC, with the reason that they are radii (of the same circle)
C1 for angle OMA = angle OKC = 90°, triangles OAM and OCK congruent (RHS), so OM = OK (corresponding sides of congruent triangles)
Worked solution: OM is perpendicular to AB, so AM = ½AB (the perpendicular from the centre to a chord bisects the chord). Similarly CK = ½CD. AB = CD, so AM = CK. OA = OC (radii of the same circle). In triangles OAM and OCK: angle OMA = angle OKC = 90°, the hypotenuses OA and OC are equal, and AM = CK. So the triangles are congruent (RHS). Therefore OM = OK (corresponding sides of congruent triangles).
(b)Answer: 9 cm
M1 for AM = 12 and \(OM^2 = 15^2 - 12^2\) (= 81)
A1 for 9
Worked solution: AM = ½ × 24 = 12 cm. \(OM = \sqrt{15^2 - 12^2} = \sqrt{81} = 9\) cm