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Triangle Definitions Help (page 2)

By — McGraw-Hill Professional
Updated on Oct 3, 2011

Congruent Triangles

There is disagreement in the literature about the exact meaning of the terms congruence and congruent when describing geometric figures in a plane. Some texts say two objects in a plane are congruent if and only if one can be placed exactly over the other after a rigid transformation (rotating it or moving it around, but not flipping it over). Other texts define congruence to allow flipping over, as well as rotation and motion. Let’s stay away from that confusion, and make two definitions.

Direct Congruence

Two triangles exhibit direct congruence (they are directly congruent ) if and only if they are directly similar, and the corresponding sides have the same lengths. Some examples are shown in Fig. 2-3. If you take one of the triangles and rotate it clockwise or counterclockwise to the correct extent, you can “paste” it precisely over any of the other triangles. Rotation and motion are allowed, but flipping over, also called mirroring , is forbidden. In general, triangles are not directly congruent if you must flip one of them over, in addition to rotating it, in order to be able to place it over the other.

Triangles Triangle Definitions Congruent Triangles

Fig. 2-3 . Directly congruent triangles.

Inverse Congruence

Two triangles exhibit inverse congruence (they are inversely congruent ) if and only if they are inversely similar, and they are also the same size. Rotation and motion are allowed, and mirroring is actually required.

If there are two triangles Δ ABC and Δ DEF that are directly congruent, we can symbolize this by writing Δ ABC ≅ Δ DEF . The direct congruence symbol is an equals sign with a direct similarity symbol on top. If the triangles Δ ABC and Δ DEF are inversely congruent, the same situation arises as is the case with inverse similarity. Three possibilities exist:

  • Points D and E are transposed, so Δ ABC ≅ Δ EDF
  • Points E and F are transposed, so Δ ABC ≅ Δ DFE
  • Points D and F are transposed, so Δ ABC ≅ Δ FED

Facts About Congruent Triangles

Direct Congruence Facts

Here are two important things you should remember about triangles that are directly congruent.

Direct Congruence Fact 1: Corresponding Sides

  • If two triangles are directly congruent, then their corresponding sides have equal lengths as you proceed around both triangles in the same direction. The converse of this is also true. If two triangles have corresponding sides with equal lengths as you proceed around them both in the same direction, then the two triangles are directly congruent.

Direct Congruence Fact 2: Corresponding Interior Angles

  • If two triangles are directly congruent, then their corresponding interior angles (that is, the interior angles opposite the corresponding sides) have equal measures as you proceed around both triangles in the same direction. The converse of this is not necessarily true. It is possible for two triangles to have corresponding interior angles with equal measures when you proceed around them both in the same direction, and yet the two triangles are not directly congruent.
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