Two figures are congruent if they have the same shape and the same size. When one figure is placed on top of the other, they coincide exactly. Congruent figures are copies of each other — they may be mirror images or rotations.
Congruent Line Segments and Angles:
Two line segments are congruent if they have the same length. Two angles are congruent if they have the same measure.
Congruence of Triangles:
Two triangles are congruent if their corresponding sides and angles are equal. We write triangle ABC is congruent to triangle DEF using the symbol ( triangle symbol ) and maintain vertex correspondence.
Criteria for Congruence:
SSS (Side-Side-Side): If all three sides of one triangle are equal to the corresponding three sides of another triangle, the triangles are congruent.
SAS (Side-Angle-Side): If two sides and the included angle (the angle between those two sides) of one triangle equal the corresponding two sides and included angle of another, the triangles are congruent.
ASA (Angle-Side-Angle): If two angles and the included side (the side between those two angles) of one triangle equal the corresponding two angles and included side of another, the triangles are congruent.
AAS (Angle-Angle-Side): If two angles and a non-included side of one triangle equal the corresponding parts of another, the triangles are congruent. (AAS follows from ASA.)
RHS (Right angle-Hypotenuse-Side): If the hypotenuse and one side of a right triangle equal the hypotenuse and one side of another right triangle, they are congruent.
Important Note: AAA (three equal angles) does NOT guarantee congruence — it only guarantees similar triangles, which may be different sizes.
Triangle ABC has AB = 5 cm, BC = 7 cm, CA = 6 cm. Triangle PQR has PQ = 5 cm, QR = 7 cm, RP = 6 cm. Are they congruent?
All three sides match: SSS criterion. Triangle ABC is congruent to triangle PQR.
In triangles ABC and DEF, AB = DE = 4 cm, angle B = angle E = 60 degrees, BC = EF = 6 cm. Which criterion applies?
Two sides and the included angle match: SAS criterion.
Two right triangles have hypotenuse 10 cm and one leg 6 cm each. By RHS, they are congruent.
Triangles ABC and XYZ have angle A = angle X, angle B = angle Y, and AB = XY. Identify the criterion.
Two angles and the included side: ASA criterion.
Show why SSA is not a valid congruence criterion.
SSA means two sides and a non-included angle match. It is possible to construct two different triangles satisfying these conditions, so SSA does not guarantee congruence.
In triangles PQR and STU, angle Q = angle T = 90 degrees, PR = SU (hypotenuse), QR = TU. By RHS, the triangles are congruent.
ABCD is a rectangle. Prove triangle ABC is congruent to triangle CDA.
AB = CD, BC = DA (opposite sides of rectangle), AC = CA (common). By SSS, triangle ABC is congruent to triangle CDA.
Summary
- Order of vertices in the congruence statement matters (it shows correspondence).
- Five valid criteria: SSS, SAS, ASA, AAS, RHS.
- AAA and SSA are NOT valid criteria.
Common mistakes
When writing congruence, ensure the vertex correspondence is correct. Confusing corresponding parts leads to errors. Remember the included angle in SAS must be the angle BETWEEN the two given sides.
Summary
Congruent triangles are identical in shape and size. Five criteria (SSS, SAS, ASA, AAS, RHS) guarantee congruence. These criteria allow us to prove triangles congruent and thereby deduce the equality of corresponding parts.