Congruent Triangles worksheet

10 GCSE practice questions with answers, free to print. Every sheet is generated, so you can make a fresh one whenever you need it.

10 questions 26 marks Sheet 1

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3 marks
\(ABCD\) is a quadrilateral. The diagonal \(AC\) has been drawn.

\(AB = AD\), as the dashes show, and \(AC\) bisects angle \(BAD\).

(a)
Write down the condition (SSS, SAS, ASA or RHS) that shows triangle \(ABC\) and triangle \(ADC\) are congruent.

(b)
Hence write down the length that must be equal to \(CB\).
2 marks
Triangle \(ABC\) and triangle \(PQR\) are congruent, with \(A\) matching \(P\), \(B\) matching \(Q\) and \(C\) matching \(R\).

\(\angle ABC = \angle PQR\), \(BC = QR\) and \(\angle BCA = \angle QRP\).

In triangle \(ABC\), \(AB = 8\) cm, \(BC = 15\) cm and \(AC = 17\) cm.

(a)
State the condition (SSS, SAS, ASA or RHS) that the three facts above show.

(b)
State the length of \(PR\).
3 marks
\(ABC\) is a triangle with \(AB = AC = 10\) cm and \(BC = 14\) cm.

The dashed line runs from \(A\) to the point \(M\) on \(BC\), and \(AM\) is perpendicular to \(BC\).

(a)
Write down the condition (SSS, SAS, ASA or RHS) that shows triangle \(ABM\) and triangle \(ACM\) are congruent.

(b)
Hence write down the length of \(BM\).
2 marks
Triangle \(ABC\) has \(AB = 13\) cm, \(BC = 17\) cm and \(\angle ABC = 40^\circ\).

Triangle \(PQR\) has \(PQ = 13\) cm, \(QR = 17\) cm and \(\angle PQR = 40^\circ\).

Must the two triangles be congruent?

Write Yes or No, and give a reason for your answer.
3 marks
\(ABCD\) is a parallelogram with \(AB = 11\) cm and \(BC = 10\) cm. The diagonal \(AC\) has been drawn.

Angle \(ABC = 59^\circ\).

(a)
Write down the condition (SSS, SAS, ASA or RHS) that shows triangle \(ABC\) and triangle \(CDA\) are congruent.

(b)
Write down the size of angle \(CDA\).
3 marks
\(ABCD\) is a quadrilateral. The diagonal \(AC\) has been drawn.

\(AB = AD\), as the dashes show, and \(CB = CD\).

(a)
Write down the condition (SSS, SAS, ASA or RHS) that shows triangle \(ABC\) and triangle \(ADC\) are congruent.

(b)
Hence write down the angle that must be equal to angle \(BAC\).
2 marks
Triangle \(ABC\) and triangle \(PQR\) are congruent, with \(A\) matching \(P\), \(B\) matching \(Q\) and \(C\) matching \(R\).

\(AB = PQ\), \(AC = PR\) and \(\angle BAC = \angle QPR\).

In triangle \(ABC\), \(AB = 9\) cm, \(BC = 12\) cm and \(AC = 15\) cm.

(a)
Write down the condition (SSS, SAS, ASA or RHS) that the three facts above show.

(b)
Write down the length of \(PR\).
3 marks
\(ABC\) is a triangle with \(AB = AC = 14\) cm and \(BC = 12\) cm.

The dashed line runs from \(A\) to the point \(M\) on \(BC\), and \(AM\) is perpendicular to \(BC\).

(a)
Write down the condition (SSS, SAS, ASA or RHS) that shows triangle \(ABM\) and triangle \(ACM\) are congruent.

(b)
Hence write down the length of \(BM\).
2 marks
Triangle \(ABC\) has \(AB = 14\) cm, \(BC = 15\) cm and \(\angle BCA = 62^\circ\).

Triangle \(PQR\) has \(PQ = 14\) cm, \(QR = 15\) cm and \(\angle QRP = 62^\circ\).

Must the two triangles be congruent?

Write Yes or No, and give a reason for your answer.
3 marks
\(ABCD\) is a parallelogram with \(AB = 14\) cm and \(BC = 6\) cm. The diagonal \(AC\) has been drawn.

Angle \(ABC = 78^\circ\).

(a)
Write down the condition (SSS, SAS, ASA or RHS) that shows triangle \(ABC\) and triangle \(CDA\) are congruent.

(b)
State the size of angle \(CDA\).

Answers

(a) SAS
(b) \(CD\)
(a) ASA
(b) \(17\ \text{cm}\)
(a) RHS
(b) \(7\ \text{cm}\)
Yes, because the angle given lies between the two sides given, so this is SAS
(a) SSS
(b) \(59^\circ\)
(a) SSS
(b) \(\angle DAC\)
(a) SAS
(b) \(15\ \text{cm}\)
(a) RHS
(b) \(6\ \text{cm}\)
No, because the angle given does not lie between the two sides given, and two different triangles can be drawn from that information
(a) SSS
(b) \(78^\circ\)

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