Printable · GCSE Foundation · ages 14-16

Congruence criteria for triangles worksheet — GCSE Foundation

Fifteen questions on "congruence criteria for triangles" — DfE statement G5. Print it, or print three versions so neighbours cannot copy by letter; the key gives the letter for each version.

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Congruence criteria for triangles worksheet — GCSE Foundation

Maths · GCSE Foundation · G5 · Congruence criteria for triangles · 15 questions

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1.Triangle ABC has a right angle at B and hypotenuse AC = 17 cm, with AB = 8 cm. Triangle DEF has a right angle at E and hypotenuse DF = 17 cm. Which extra fact about DEF would complete the RHS condition for proving that ABC ≅ DEF, with the vertices matching in that order?
(a)Angle D = angle A
(b)BC = 8 cm
(c)EF = 8 cm
(d)DE = 8 cm
2.Triangle ABC has AB = 6 cm and BC = 9 cm, with angle A = 40°. Triangle DEF has DE = 6 cm and EF = 9 cm, with angle D = 40°. Meera says these facts prove the triangles are congruent by SAS. Is Meera correct?
(a)Yes — two sides and one angle match
(b)No — the angle given is not the included angle
(c)No — SSS needs three equal sides, not two
(d)Yes — SAS allows any equal angle, not just the included one
3.For two triangles to be proved congruent using SAS, which two facts must be true about both triangles?
(a)Two sides equal, with the angle between them equal
(b)Two angles equal, with the side between them equal
(c)All three pairs of sides equal
(d)Two sides equal, and any one angle equal
4.Triangle LMN has a right angle at M, with hypotenuse LN = 15 cm and LM = 9 cm. Triangle PQR has a right angle at Q, with hypotenuse PR = 15 cm and PQ = 9 cm. Which condition proves the two triangles are congruent?
(a)SSS
(b)SAS
(c)RHS
(d)ASA
5.Triangle ABC has AB = (2x + 3) cm, BC = 11 cm and angle B = 90°. Triangle DEF has DE = 13 cm, EF = 11 cm and angle E = 90°. Given that the two triangles are congruent by SAS, work out the value of x.
(a)4
(b)10
(c)8
(d)5
6.Two metal window frames are checked before installation. Frame A has edges of 40 cm and 65 cm, with a marked angle of 35° at the far end of the 65 cm edge, away from where the two given edges meet. Frame B has matching edges of 40 cm and 65 cm, with a matching 35° angle in the same position. The fitter wants to know if these measurements alone guarantee the two frames are congruent. What should the fitter be told?
(a)No — the angle is not between the two sides
(b)Yes — SSS, since three measurements are given
(c)Yes — SAS, since two sides and an angle match
(d)No — because only two sides were measured
7.A sailmaker cuts two triangular sail panels from a pattern. Panel X has a 10 m side, with angles of 50° and 75° at its two ends. Panel Y also has a 10 m side, with angles of 50° and 75° at its two ends, in the same arrangement. The sailmaker wants to check the panels will match exactly, without cutting a third measurement. Which condition proves the two panels are congruent?
(a)SAS
(b)RHS
(c)ASA
(d)SSS
8.Triangle ABC and triangle DEF both contain a right angle, at B and E respectively, and the hypotenuses AC and DF are equal in length. Which extra fact would prove the triangles are congruent by RHS?
(a)angle A = angle D
(b)AB = DE
(c)AC is parallel to DF
(d)the triangles are drawn the same way up
9.Triangle ABC has AB = 6 cm, BC = 8 cm and angle B = 90°. Triangle XYZ has XY = 6 cm, YZ = 8 cm and angle Y = 90°. Which congruence statement correctly shows the matching vertices?
(a)ABC ≅ ZXY
(b)ABC ≅ ZYX
(c)ABC ≅ XYZ
(d)ABC ≅ YXZ
10.Triangle ABC has a right angle at B. Triangle DEF has a right angle at E. The hypotenuse of triangle ABC equals the hypotenuse of triangle DEF, and side AB equals side DE. Which condition proves that the two triangles are congruent?
(a)ASA
(b)SAS
(c)SSS
(d)RHS
11.An isosceles triangle ABC has AB equal to AC. AM is a line from vertex A, perpendicular to BC, meeting BC at point M. Which condition proves that triangle ABM is congruent to triangle ACM?
(a)SAS, right angle as included angle
(b)ASA, AB as the included side
(c)SSS, using BM = CM as a fact
(d)RHS, using AM as common side
12.A roof has two triangular sections, P and Q. Section P has a sloping edge of 3.6 m and a base edge of 2.4 m, with an angle of 70° between them. Section Q has a sloping edge of 3.6 m and a base edge of 2.4 m, with an angle of 70° between them, arranged the same way round. A roofer wants to cut identical triangular felt panels for both sections without measuring Section Q separately. Which condition confirms that the two sections are congruent?
(a)ASA — angle between two sides
(b)SSS — but only two sides given
(c)Cannot prove — only one angle
(d)SAS — two sides, included angle
13.Triangles ABE and CDE share the vertex E, where lines AC and BD cross at E. AE equals CE, and BE equals DE. Angle AEB and angle CED are formed as vertically opposite angles where the lines cross. Which condition proves that triangle ABE is congruent to triangle CDE?
(a)ASA, vertically opposite angle at E
(b)Cannot prove — no angle measured
(c)SSS, three equal side pairs
(d)SAS, vertically opposite angle included
14.Triangle ABC has a right angle at B, hypotenuse AC = 13 cm and AB = 5 cm. Triangle DEF has a right angle at E, hypotenuse DF = 13 cm and DE = 5 cm. Which condition proves the two triangles are congruent?
(a)SSS - three sides equal
(b)RHS - right angle, hypotenuse and one side equal
(c)ASA - two angles and the included side equal
(d)SAS - two sides and the included angle equal
15.Triangle ABC has AB = 8 cm, BC = 10 cm and CA = 6 cm. Triangle LMN has LM = 10 cm, MN = 6 cm and NL = 8 cm. The two triangles are congruent by SSS. Which of the following correctly matches the corresponding vertices?
(a)A↔L, B↔M, C↔N (ABC≅LMN)
(b)A↔N, B↔M, C↔L (ABC≅NML)
(c)A↔N, B↔L, C↔M (ABC≅NLM)
(d)A↔M, B↔N, C↔L (ABC≅MNL)

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