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Math Study Guide: What is Congruence?

written by: Keren Perles • edited by: SForsyth • updated: 1/5/2012

You may want to translate the meaning of congruent in math as “equal." In truth, the definition is slightly more complex than that. This article will clarify the meaning of congruent, as well as give several examples and rules related to congruence.

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    The Basic Meaning of Congruence in Math

    If two geometric objects are congruent to each other, they have the same measurements. For example, a circle with a diameter of 3 units will be congruent with any other circle that has a diameter of 3 units. All of the other measurements of the circles will be identical. Congruence is notated by a symbol that looks like an equal sign (=) with a tilde (~) on top of it.

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    Examples

    • If two line segments each measure 10 units, they are congruent segments.
    • If two angles each measure 70 degrees, they are congruent angles, even if they are in different locations and facing different directions.
    • If two triangles have sides measuring 8, 6, and 4 units, and their corresponding angles are congruent, then the triangles are congruent too. This would be true even if one of the triangles is “flipped over" relative to the other triangle, so that they appear to be mirror images.
    • If two polygons have 15 sides, and the corresponding sides and angles are congruent, then the polygons are congruent too.
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    Congruent Triangles

    Possibly the most commonly used example of congruence is the congruence of two triangles. Congruence in triangles can be proven using one of several rules: side-angle-side, angle-side-angle, side-side-side, or angle-angle-side.

    • Side-angle-side – If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, the triangles are congruent.
    • Angle-side-angle – If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, the triangles are congruent.
    • Side-side-side – If three sides of one triangle are congruent to three sides of another triangle, the triangles are congruent.
    • Angle-angle-side – If two sides and a non-included angle of one triangle are congruent to two sides and a non-included angle of another triangle, the triangles are congruent.

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