Congruent Triangles

Triangles that have exactly the same size and shape are called congruent triangles. The symbol for congruent is ≅. Two triangles are congruent when the three sides and the three angles of one triangle have the same measurements as three sides and three angles of another triangle. The triangles in Figure 1 are congruent triangles.





Figure 1 Congruent triangles.

Corresponding parts

The parts of the two triangles that have the same measurements (congruent) are referred to as corresponding parts. This means that Corresponding Parts of Congruent Triangles are Congruent (CPCTC). Congruent triangles are named by listing their vertices in corresponding orders. In Figure , Δ BAT ≅ Δ ICE.

Example 1: If Δ PQR ≅ Δ STU which parts must have equal measurements?





These parts are equal because corresponding parts of congruent triangles are congruent.

Tests for congruence


To show that two triangles are congruent, it is not necessary to show that all six pairs of corresponding parts are equal. The following postulates and theorems are the most common methods for proving that triangles are congruent (or equal).

Postulate 13 (SSS Postulate): If each side of one triangle is congruent to the corresponding side of another triangle, then the triangles are congruent (Figure 2).



                                             

Figure 2 The corresponding sides (SSS) of the two triangles are all congruent.



Postulate 14 (SAS Postulate): If two sides and the angle between them in one triangle are congruent to the corresponding parts in another triangle, then the triangles are congruent (Figure 3).






Figure 3 Two sides and the included angle (SAS) of one triangle are congruent to the

corresponding parts of the other triangle.



Postulate 15 (ASA Postulate): If two angles and the side between them in one triangle are congruent to the corresponding parts in another triangle, then the triangles are congruent (Figure 4).






Figure 4 Two angles and their common side (ASA) in one triangle are congruent to the

corresponding parts of the other triangle.



Theorem 28 (AAS Theorem): If two angles and a side not between them in one triangle are congruent to the corresponding parts in another triangle, then the triangles are congruent (Figure 5).






Figure 5 Two angles and the side opposite one of these angles (AAS) in one triangle

are congruent to the corresponding parts of the other triangle.



Postulate 16 (HL Postulate): If the hypotenuse and leg of one right triangle are congruent to the corresponding parts of another right triangle, then the triangles are congruent (Figure 6).






Figure 6 The hypotenuse and one leg (HL) of the first right triangle are congruent to the

corresponding parts of the second right triangle.


Theorem 29 (HA Theorem): If the hypotenuse and an acute angle of one right triangle are congruent to the corresponding parts of another right triangle, then the triangles are congruent (Figure 7).






Figure 7 The hypotenuse and an acute angle (HA) of the first right triangle are congruent

to the corresponding parts of the second right triangle.


Theorem 30 (LL Theorem): If the legs of one right triangle are congruent to the corresponding parts of another right triangle, then the triangles are congruent (Figure 8).






Figure 8 The legs (LL) of the first right triangle are congruent to the corresponding parts

of the second right triangle.



Theorem 31 (LA Theorem): If one leg and an acute angle of one right triangle are congruent to the corresponding parts of another right triangle, then the triangles are congruent (Figure 9).





Figure 9 One leg and an acute angle (LA) of the first right triangle are congruent to the

corresponding parts of the second right triangle.


Example 2: Based on the markings in Figure 10, complete the congruence statement Δ ABC ≅Δ .






Figure 10 Congruent triangles.


Δ YXZ, because A corresponds to Y, B corresponds to X, and C corresponds, to Z.

Example 3: By what method would each of the triangles in Figures 11 (a) through 11 (i) be proven congruent?






Figure 11  Methods of proving pairs of triangles congruent.


  • (a) SAS.
  • (b) None. There is no AAA method.
  • (c) HL.
  • (d) AAS.
  • (e) SSS. The third pair of congruent sides is the side that is shared by the two triangles.
  • (f) SAS or LL.
  • (g) LL or SAS.
  • (h) HA or AAS.
  • (i) None. There is no SSA method.

Example 4: Name the additional equal corresponding part(s) needed to prove the triangles in Figures 12 (a) through 12 (f) congruent by the indicated postulate or theorem.






Figure 12 Additional information needed to prove pairs of triangles congruent.



  • (a) BC = EF or AB = DE ( but not AC = DF because these two sides lie between the equal angles).
  • (b) GI = JL.
  • (c) MO = PO and NO = RO.
  • (d) TU = WX and SU = VX.



  • (e) mT = mE and m ∠TOW = m ∠ EON.
  • (f) IX = EN or SX = TN (but not IS = ET because they are hypotenuses).