Worksheets · Foundation and Higher

Congruence criteria and geometric deductions

8 exam-style questions, grades 2 to 5. Worked solutions and the marks are on the last page.

Question 1Non-calculator · 1 mark
(a) Each of two triangles has sides of length 7 cm and 11 cm. The angle between these two sides is $38^\circ$ in each triangle. Which congruence test proves that the triangles are congruent? (1)
1. SSS
2. ASA
3. RHS
4. SAS
Question 2Non-calculator · 3 marks
Each part describes two triangles. Choose the condition that shows they are congruent.
(a) Triangle 1 has sides 5 cm and 7 cm with an angle of $40^\circ$ between them. Triangle 2 also has sides 5 cm and 7 cm with an angle of $40^\circ$ between them. (1)
1. ASA
2. RHS
3. SAS
4. SSS
(b) Both triangles have a right angle, a hypotenuse of 13 cm and another side of 5 cm. (1)
1. SSS
2. ASA
3. RHS
4. SAS
(c) Explain why two triangles with the same three angles need not be congruent. (1)
Question 3Non-calculator · 5 marks
ABCD is a parallelogram. Diagonal AC is drawn. AB = 5 cm, BC = 7 cm and AC = 10 cm.
(a) Prove that triangles ABC and CDA are congruent. (3)
(b) Find the perimeter of triangle CDA. (2)
Question 4Non-calculator · 4 marks
Triangles PQR and XYZ are right-angled at Q and Y. Their hypotenuses PR and XZ are both 15 cm. PQ = XY = 9 cm.
(a) Explain why the triangles are congruent. (2)
(b) Find YZ. (2)
Question 5Non-calculator · 4 marks
AB = AC = 7 cm in triangle ABC. M is the midpoint of BC, and BC = 8 cm.
(a) Prove that triangles ABM and ACM are congruent. (3)
(b) Deduce why AM is perpendicular to BC. (1)
Question 6Non-calculator · 3 marks
Triangle ABC has AB = 6 cm, BC = 8 cm and angle ABC = $50^\circ$ . Triangle PQR has PQ = 8 cm, QR = 6 cm and angle PQR = $50^\circ$ .
(a) Are the triangles congruent? Give the condition. (1)
(b) Which side of triangle PQR corresponds to AC? Select one answer. (1)
1. PR
2. PQ
3. QR
(c) AC = 6.2 cm. Write down the length of PR. (1)
Question 7Non-calculator · 4 marks
Two triangles each have angles of $40^\circ$ and $75^\circ$ . Each triangle has one side of length 9 cm.
(a) Find the third angle of each triangle. (2)
(b) Explain why the triangles need not be congruent. (1)
(c) In both triangles the 9 cm side is opposite the 40° angle. Are the triangles now congruent? Give the condition. (1)
Question 8Non-calculator · 3 marks
Triangle $PQR$ has $PQ = 7$ cm, angle $P = 50^\circ$ and angle $Q = 60^\circ$ . Triangle $XYZ$ has $XY = 7$ cm, angle $X = 50^\circ$ and angle $Z = 70^\circ$ .
(a) Are the two triangles congruent? You must show how you get your answer. (3)
1. Congruent
2. Not congruent

Worked solutions and marks

Question 1

(a) SAS

The $38^\circ$ angle lies between the given 7 cm and 11 cm sides.
Two sides and the included angle establish SAS congruence.

B1 Correct answer: SAS

Question 2

(a) SAS

Two sides and the angle between them (the included angle) match: SAS.

B1 Correct answer: SAS.

(b) RHS

Right angle, hypotenuse and one other side: RHS.

B1 Correct answer: RHS.

(c) They can be different sizes: equal angles only make them similar.

An enlargement keeps every angle but changes the side lengths.

C1 Stating that one triangle can be an enlargement of the other (similar, not congruent), so the sides can differ.

Question 3

(a) AB = CD and BC = DA because opposite sides of a parallelogram are equal. AC is common, so the triangles are congruent by SSS.

Identify the first corresponding side pair.
Identify the second pair and the common side.
Therefore AB = CD and BC = DA because opposite sides of a parallelogram are equal. AC is common, so the triangles are congruent by SSS.

M1 Identify the first corresponding side pair.
M1 Identify the second pair and the common side.
C1 Correct conclusion with supporting reasoning: AB = CD and BC = DA because opposite sides of a parallelogram are equal. AC is common, so the triangles are congruent by SSS.

(b) $22$ cm

Use the three corresponding side lengths.
$5+7+10$
Therefore $22$ cm.

M1 Use the three corresponding side lengths.
A1 Correct answer: $22$ cm

Question 4

(a) They have equal hypotenuses, one equal corresponding shorter side and a right angle, so RHS applies.

Identify the equal hypotenuses and the right angles.
They have equal hypotenuses, one equal corresponding shorter side and a right angle, so RHS applies.

M1 Identify the equal hypotenuses and the right angles.
C1 Correct conclusion with supporting reasoning: They have equal hypotenuses, one equal corresponding shorter side and a right angle, so RHS applies.

(b) $12$ cm

Use Pythagoras in either right triangle.
$\sqrt{(15)^{2}-(9)^{2}}$
Therefore $12$ cm.

M1 Use Pythagoras in either right triangle.
A1 Correct answer: $12$ cm

Question 5

(a) AB = AC, BM = CM because M is the midpoint, and AM is common. SSS proves congruence.

Use the midpoint to establish an equal side pair.
Use the given equal sides and the common segment.
Therefore AB = AC, BM = CM because M is the midpoint, and AM is common. SSS proves congruence.

M1 Use the midpoint to establish an equal side pair.
M1 Use the given equal sides and the common segment.
C1 Correct conclusion with supporting reasoning: AB = AC, BM = CM because M is the midpoint, and AM is common. SSS proves congruence.

(b) Congruence gives equal angles AMB and AMC. They lie on a straight line, so each is 90 degrees.

Congruence gives equal angles AMB and AMC. They lie on a straight line, so each is 90 degrees.

C1 Correct conclusion with supporting reasoning: Congruence gives equal angles AMB and AMC. They lie on a straight line, so each is 90 degrees.

Question 6

(a) Yes, SAS: each triangle has sides of 6 cm and 8 cm with the included angle of 50° between them.

Yes, SAS: each triangle has sides of 6 cm and 8 cm with the included angle of 50° between them.

C1 Correct conclusion with supporting reasoning: Yes, SAS: each triangle has sides of 6 cm and 8 cm with the included angle of 50° between them.

(b) PR

Therefore PR.

B1 Correct answer: PR

(c) $6.2$ cm

Therefore $6.2$ cm.

B1 Correct answer: $6.2$ cm

Question 7

(a) $65$ °

Subtract the two given angles from 180°.
$180-40-75$
Therefore $65$ °.

M1 Subtract the two given angles from 180°.
A1 Correct answer: $65$ °

(b) The 9 cm sides may be opposite different angles, so the equal sides might not correspond.

The 9 cm sides may be opposite different angles, so the equal sides might not correspond.

C1 Correct conclusion with supporting reasoning: The 9 cm sides may be opposite different angles, so the equal sides might not correspond.

(c) Yes, AAS: two angles and a corresponding side are equal.

Yes, AAS: two angles and a corresponding side are equal.

C1 Correct conclusion with supporting reasoning: Yes, AAS: two angles and a corresponding side are equal.

Question 8

(a) Yes: angle $Y = 60^\circ$ , so each triangle has a 7 cm side between angles of $50^\circ$ and $60^\circ$ (ASA).

Angles in a triangle add up to $180^\circ$ .
$\angle Y = 180 - 50 - 70 = 60$
In triangle $PQR$ the 7 cm side $PQ$ lies between angles of $50^\circ$ and $60^\circ$ .
In triangle $XYZ$ the 7 cm side $XY$ lies between angles of $50^\circ$ and $60^\circ$ .
So the triangles are congruent (ASA).

P1 Finding angle $Y = 60^\circ$ (or angle $R = 70^\circ$ ).
P1 Matching the 7 cm side between the same two angles in each triangle.
C1 "Yes, congruent (ASA)" from the matched side and angles.

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Congruence criteria and geometric deductions

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