




Figure 1 Congruent triangles.


<a href="http://ad.doubleclick.net/jump/CNSite/;navArea=CLIFFSNOTES2_MATH;type=Review_Topic;cat=MATH;kword=geometry;contentItemId=18851;tile=3;sz=300x250;ord=123456789?" target="_blank"><img src="http://ad.doubleclick.net/ad/CNSite/;navArea=CLIFFSNOTES2_MATH;type=Review_Topic;cat=MATH;kword=geometry;contentItemId=18851;tile=3;sz=300x250;ord=123456789?" width="300" height="250" border="0" alt="" /></a> 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 1 , 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...