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 · Non-calculator · 15 questions

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1.Triangle ABC has angle A = 40°, angle B = 65° and AB = 10 cm. Triangle DEF has angle D = 40°, angle E = 65° and DE = 10 cm. Which condition proves the two triangles are congruent?
(a)RHS - right angle, hypotenuse and one side equal
(b)SAS - two sides and the included angle equal
(c)AAS - two angles and a non-included side equal
(d)ASA - two angles and the included side equal
2.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)
3.A carpenter measures two triangular braces. Brace 1 has a 55° angle, a 65° angle, and, opposite the 55° angle, a side of 8 cm. Brace 2 has matching 55° and 65° angles and, opposite its 55° angle, a side of 8 cm. The carpenter says that because the equal 8 cm side is not between the two equal angles in either brace, he cannot be sure the two braces are the same size. Is the carpenter correct?
(a)No — any two angles enough alone
(b)Yes — third angle may differ
(c)Yes — side must be included
(d)No — third angle is also fixed
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.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
6.Triangles PQS and RQS share the common side QS. PQ equals RQ, and PS equals RS. Which condition proves that triangle PQS is congruent to triangle RQS?
(a)Cannot prove — no angle given
(b)Only two pairs of sides — not enough
(c)SAS, using the angle at Q
(d)SSS, using shared side QS
7.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
8.Two triangles are congruent to each other. What does this tell you about their angles and sides?
(a)All angles equal, sides may differ
(b)All sides equal, angles may differ
(c)Same area, angles and sides may differ
(d)All angles equal and all sides equal
9.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
10.Triangle ABC has AB = 9 cm, BC = 6 cm and angle A = 35°. Triangle DEF has DE = 9 cm, EF = 6 cm and angle D = 35°. Which of the following correctly identifies the condition shown here?
(a)SSA - not sufficient to prove congruence
(b)ASA - two angles and the included side equal
(c)SAS - two sides and the included angle equal
(d)AAS - two angles and a non-included side equal
11.In the RHS congruence condition, which three facts must be true about two triangles for RHS to prove they are congruent?
(a)Two sides and the angle between them
(b)Right angle, equal hypotenuse, one more side
(c)Right angle and all three sides equal
(d)Right angle, hypotenuse, one more angle
12.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
13.Triangle ABC has AB = 5 cm, BC = 7 cm and CA = 9 cm. Triangle DEF has DE = 5 cm, EF = 7 cm and FD = 9 cm. Which condition proves the two triangles are congruent?
(a)ASA - two angles and the included side equal
(b)SAS - two sides and the included angle equal
(c)RHS - right angle, hypotenuse and one side equal
(d)SSS - three sides equal
14.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
15.Triangle PQR and triangle XYZ have angle P = angle X, angle Q = angle Y and angle R = angle Z, with no side lengths given for either triangle. Which of the following correctly describes the relationship between the two triangles?
(a)Similar, but AAA alone does not prove congruence
(b)Congruent, because AAA proves congruence
(c)No relationship can be determined at all
(d)Congruent, because equal angles force equal sides

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