GMAT Exam: A Simple Guide To Geometry - Congruence and Similarity

Congruence
Congruent geometric figures have the same size and same shape.

They will fit exactly on top of each other.

The symbol @ denotes congruence.


Corresponding parts of congruent figures are congruent.
Congruent parts (for example, angles and sides) of congruent figures have the same measure.
Congruent triangles are triangles for which corresponding sides and corresponding angles are congruent.
In the figure shown, triangles ABC and DEF are congruent.

Hash marks identify the congruent parts.



Two triangles are congruent in the following situations.
- Side-Side-Side (SSS): If three sides of one triangle are congruent, correspondingly, to three sides of the other triangle.
- Side-Angle-Side (SAS): If two sides and the included angle of one triangle are congruent, correspondingly, to two sides and the included angle of the other triangle.
- Angle-Side-Angle (ASA): If two angles and the included side of one triangle are congruent, correspondingly, to two angles and the included side of the other triangle.
- Angle-Angle-Side (AAS): If two angles and the nonincluded side of one triangle are congruent, correspondingly, to two angles and the nonincluded side of the other triangle.

Two situations that do NOT guarantee congruence are AAA (three corresponding angles congruent) and SSA (two corresponding sides and the nonincluded angle congruent).

Similarity
Similar geometric figures have the same shape, but not necessarily the same size. The symbol ~ denotes similarity.


Corresponding angles of similar shapes are congruent. Corresponding lengths of similar shapes are proportional.
That is, the ratios of the lengths are equal.
The ratio of the areas of two similar figures is the square of the ratio of the lengths of any two corresponding sides.
For instance, if corresponding sides of two squares are in the ratio 4 to 1, then the ratio of the area of the larger square to the area of the smaller square is 42 to 12, or 16 to 1.