Worksheets · Foundation and Higher

Congruence criteria and geometric deductions

3 exam-style questions. Answers and the mark for each step 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 · 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 3Non-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

Answers and marks

(a) SAS

(a) 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.

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

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

Congruence criteria and geometric deductions

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