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Fig. 1.5

Definition: Two figures are said to be Congruent if they have exactly the same shape and size.

Introduction: When naming congruent triangles, it is important to give the letters in the correct order so that it is clear which sides of the triangles correspond to each other. When congruent triangles are properly named, it is possible to find pairs of equal sides or equal angles without looking at the figure.

Note that in geometry the symbol \( \scriptsize \equiv \) means identically equal to”, or “is Congruent to”.

Therefore, ∆ABC \( \scriptsize \equiv \) ∆XYZ is short for “triangle ABC is congruent to triangle XYZ. The following gives the four sets of conditions for the congruency of two triangles.

Two triangles are congruent if:

I. Two sides and the included angle of one are respectively equal to two sides and the included angle of the other. (i.e. SAS).

Screenshot 2022 05 16 at 20.57.08
Fig. 1.6
\( \scriptsize \Delta ABC \equiv \Delta XYZ \)

Note: If the given angle is not included between the given sides, this is called the ambiguous case.

II. Two angles and a side are equal to two angles and the corresponding side of the other, (i.e. ASA or AAS).

Screenshot 2022 05 16 at 21.05.54
Fig. 1.7

\(\scriptsize \Delta LMN \equiv \Delta PQR\)

Note that it is not necessary that the corresponding sides should be between the angles.

III. The three sides of one are respectively equal to the three sides of the other, (i.e. SSS).

Screenshot 2022 05 16 at 21.12.17
Fig. 1.8
\(\scriptsize \Delta LNM \equiv \Delta RTS \)

Note that the corresponding angles are also equal.

IV. They are right-angled and have the hypotenuse and another side equal, (i.e. RHS).

Screenshot 2022 05 16 at 21.20.46
Fig. 1.9

\( \scriptsize \Delta KLM \equiv \Delta STV\)

Recall the ambiguous case in (I), however, this is the only case where a situation of corresponding non-included angle will give rise to the two triangles being congruent, i.e. provided the non-included angles are right-angled.

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