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Congruence

Identify the congruence tests (SSS, SAS, AAS, RHS) for pairs of triangles and use congruence to find unknown sides and angles.

Worked examples

Identifying a congruence test

Straightforward

Problem

Triangles ABCABC and DEFDEF have AB=DE=8AB = DE = 8 cm, BC=EF=6BC = EF = 6 cm, and AC=DF=10AC = DF = 10 cm. Which congruence test proves △ABC≅△DEF\triangle ABC \cong \triangle DEF?

Finding a missing angle using congruence

Moderate

Problem

Triangle PQR≅PQR \cong triangle XYZXYZ. ∠P=50°\angle P = 50° and ∠Q=80°\angle Q = 80°. Find ∠Z\angle Z.

Proving congruence in a rectangle and finding an angle

Challenging

Problem

In rectangle ABCDABCD, diagonal ACAC is drawn. Prove that △ABC≅△CDA\triangle ABC \cong \triangle CDA, then find ∠DCA\angle DCA given that ∠BAC=28°\angle BAC = 28°.

Practise

Q1·Straightforward
Triangles ABCABC and DEFDEF have AB=DEAB = DE, BC=EFBC = EF, and AC=DFAC = DF. Which congruence test proves △ABC≅△DEF\triangle ABC \cong \triangle DEF?
Q2·Straightforward
In triangles PQRPQR and XYZXYZ, PQ=XYPQ = XY, QR=YZQR = YZ, and ∠PQR=∠XYZ\angle PQR = \angle XYZ. Which congruence test applies?
Q3·Straightforward
In triangles ABCABC and DEFDEF, ∠A=∠D\angle A = \angle D, ∠B=∠E\angle B = \angle E, and BC=EFBC = EF. Which congruence test applies?
Q4·Straightforward
Triangles LMNLMN and RSTRST have ∠M=∠S=90°\angle M = \angle S = 90°, LN=RTLN = RT (hypotenuses), and MN=STMN = ST. Which congruence test applies?
Q5·Moderate
Triangle ABC≅ABC \cong triangle DEFDEF. AB=7AB = 7 cm, BC=11BC = 11 cm, ∠B=52°\angle B = 52°, DE=7DE = 7 cm, and ∠E=52°\angle E = 52°. Find EFEF, in cm.
Q6·Moderate
Triangle PQR≅PQR \cong triangle XYZXYZ. ∠P=42°\angle P = 42° and ∠Q=85°\angle Q = 85°. Find ∠Z\angle Z, in degrees.
Q7·Moderate
Triangle ABC≅ABC \cong triangle PQRPQR. AB=9AB = 9 cm, BC=14BC = 14 cm, AC=11AC = 11 cm, and QR=14QR = 14 cm. Find PRPR, in cm.
Q8·Moderate
Triangle LMN≅LMN \cong triangle STUSTU. ∠L=34°\angle L = 34° and ∠N=61°\angle N = 61°. Find ∠T\angle T, in degrees.
Q9·Challenging
In triangle ABCABC, DD is the midpoint of BCBC and AD⊥BCAD \perp BC. So BD=DCBD = DC, ∠ADB=∠ADC=90°\angle ADB = \angle ADC = 90°, and ADAD is common. This gives △ABD≅△ACD\triangle ABD \cong \triangle ACD by SAS. If AC=13AC = 13 cm, find ABAB, in cm.
Q10·Challenging
Triangles ABDABD and CBDCBD share side BDBD. AB=CB=10AB = CB = 10 cm and AD=CD=8AD = CD = 8 cm. These triangles are congruent by SSS. If ∠ABD=38°\angle ABD = 38°, find ∠CBD\angle CBD, in degrees.
Q11·Challenging
In rectangle ABCDABCD, diagonal ACAC is drawn. Since AB=CDAB = CD, ∠ABC=∠CDA=90°\angle ABC = \angle CDA = 90°, and BC=DABC = DA, we have △ABC≅△CDA\triangle ABC \cong \triangle CDA by SAS. If ∠BAC=32°\angle BAC = 32°, find ∠DCA\angle DCA, in degrees.
Q12·Challenging
Triangles PQRPQR and PSRPSR share side PRPR. ∠PQR=∠PSR=90°\angle PQR = \angle PSR = 90° and QR=SR=5QR = SR = 5 cm. These triangles are congruent by RHS (PRPR is the common hypotenuse). If PQ=12PQ = 12 cm, find PSPS, in cm.