Triangle A: 5 cm, 7 cm, 9 cm Triangle B: 5 cm, 7 cm, 8 cm Triangle C: 9 cm, 7 cm, 5 cm Triangle D: 5 cm, 6 cm, 9 cm
Write down the letters of the two triangles that are congruent to each other.
1 mark
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A and C
2
In triangles PQR and XYZ,
PQ = XY = 6 cm angle PQR = angle XYZ = 40° QR = YZ = 8 cm
The two triangles are congruent.
Write down the congruence condition that shows the triangles are congruent.
1 mark
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SAS
3
For each pair of triangles, state whether they must be congruent. If they must be, give the condition (SSS, SAS, ASA or RHS).
Triangle ABC and triangle DEF: AB = DE = 6 cm, BC = EF = 8 cm, angle ABC = angle DEF = 55°.
Triangle GHJ and triangle KLM: angle G = angle K = 40°, angle H = angle L = 75°, GH = KL = 10 cm.
Triangle NPQ and triangle RST: the angles in each triangle are 50°, 60° and 70°.
Triangle UVW and triangle XYZ: angle V = angle Y = 90°, UW = XZ = 13 cm, VW = YZ = 5 cm.
4 marks
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a) SASb) ASAc) Not necessarily congruent (AAA is not a condition)d) RHS
4
ABCD is a kite in which AB = AD and CB = CD.
The diagonal AC is drawn.
Prove that triangle ABC is congruent to triangle ADC.
3 marks
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Triangle ABC is congruent to triangle ADC by SSS
5
In triangle ABC, AB = AC.
The point D lies on BC so that AD bisects angle BAC.
Prove that triangle ABD is congruent to triangle ACD.
3 marks
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Triangle ABD is congruent to triangle ACD by SAS
6
ABC is an isosceles triangle with AB = AC.
M is the midpoint of BC.
Prove that triangle ABM is congruent to triangle ACM.
Hence prove that angle ABC = angle ACB.
4 marks
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a) SSS: AB = AC, BM = CM, AM common b) corresponding angles of congruent triangles are equal
7
In quadrilateral ABCD the diagonal BD is drawn.
Angle ABD = angle CDB and angle ADB = angle CBD.
Prove that triangle ABD is congruent to triangle CDB.
3 marks
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Triangle ABD is congruent to triangle CDB by ASA
8
OA and OB are two straight lines that meet at O.
The point P lies between the two lines.
M is the point on OA such that PM is perpendicular to OA. N is the point on OB such that PN is perpendicular to OB.
PM = PN
Prove that triangle OMP is congruent to triangle ONP.
3 marks
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Triangle OMP is congruent to triangle ONP by RHS
9
A and B are points on a circle with centre O.
AB is not a diameter. M is the midpoint of the chord AB.
Prove that OM bisects angle AOB.
3 marks
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Triangles OAM and OBM are congruent (SSS), so angle AOM = angle BOM and OM bisects angle AOB
10
M is the midpoint of the line XY. The line PM is perpendicular to XY.
Prove that PX = PY.
3 marks
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Triangles PMX and PMY are congruent (SAS), so PX = PY
11
Neither chord is a diameter.
AB and CD are chords of a circle with centre O.
AB = CD
Prove that triangle OAB is congruent to triangle OCD.
Hence explain why angle AOB = angle COD.
4 marks
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a) OA = OC (radii), OB = OD (radii), AB = CD (given), so triangle OAB is congruent to triangle OCD (SSS)b) Angle AOB and angle COD are corresponding angles in congruent triangles, so they are equal
12
ABCD is a quadrilateral in which AB is parallel to DC and AB = DC.
The diagonal BD is drawn.
Prove that triangle ABD is congruent to triangle CDB.
Hence prove that AD = CB.
4 marks
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a) AB = CD (given), angle ABD = angle CDB (alternate angles, AB parallel to DC), BD common, so triangle ABD is congruent to triangle CDB (SAS)b) AD corresponds to CB in the congruent triangles, so AD = CB
13
ABCD is a parallelogram.
The diagonals AC and BD intersect at the point M.
Prove that the diagonals of the parallelogram bisect each other.
5 marks
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Triangles ABM and CDM are congruent (ASA), so AM = CM and BM = DM, which means M is the midpoint of both diagonals
Every question here is original and written by Robinson Tuition. The grade follows the Edexcel grade descriptors and suits AQA and OCR too; this is not an exam-board paper.