Overview: What Are Congruent Figures?
Geometric figures are congruent if they have exactly the same size and shape. This means that their sides are all the same size and their angles are all the same measurements. They differ from similar figures, in that similar figures have angles with the same measurement, but the measurement of side segments are in the same ratio.
Transforming Similar Figures into Congruent Figures
Similar figures can be transformed into congruent figures by making the smaller figure the same size as the larger figure or the larger figure the same size as the smaller figure. For example, suppose in the similar triangles ∆ABC and ∆DEF, the corresponding angles are all equal and the corresponding sides for ∆DEF measure 4, 5, and 6. The corresponding sides for ∆ABC are 8, 10, 12. To make the two triangles congruent, multiply the measurement of each side of ∆DEF by 2, so that the new measurement of ∆D’E’F’ is now 8, 10, 12 and the triangles are congruent.
Special Properties of Triangles
When two triangles are congruent, there are six congruent relationships: three equal sides and three equal angles. However, triangles already have some special properties that make it possible to simplify those relationships, so that one does not have to look at all six relationships. The angles of all triangles in normal space add up to 1800, so that if one has two measurements of angles in a triangle, the third can be deduced by subtracting the other two from 180. The length of sides also have special relationships with the size of angles in normal space.
Congruence Properties
There are three properties of congruent triangles that one can use to determine if triangles are congruent, known as Side-Angle-Side (SAS), Side-Side-Side (SSS), and Angle-Side-Angle (ASA). In Side-Angle-Side, if two sides and the angle between them are congruent in both triangles, then the angles are congruent. In Side-Side-Side, if all three sides of one triangle are the same length as all three sides of another triangle, then the triangles are congruent. In Angle-Side-Angle, if two angles of both triangles, as well as the common side between the angles they both share, are congruent , then the triangles are congruent.
Example of Congruence in Action
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For example, in the above figure, triangles ∆PQR and ∆LMN are congruent. According to Side-Angle-Side, this means that the segments PQ and QR are equal to LM and MN and angles Q and M are equal, as well as the segments QR and RP to MN and NL and angles R and N are equal. In addition RP and PQ are equal to LN and LM and angles P and L are equal. According to Side-Side-Side, PQ, QR, and RP are equal to LM, MN, and NL. According to Angle-Side-Angle, angles R and P and segment RP are equal to angles N and L and segment NL, angles P and Q and segment PQ are equal to angles L and M and segment LM, and angles Q and R and segment QR are equal to angles M and N and segment MN.
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