When discussing shapes in Geometry, the idea of something being "equal" is often confused with it being "congruent". While these terms have similar meanings, they have distinct definitions in mathematics. In Geometry, "equal" refers to two sides or angles having the same measurement, while "congruent" refers to two shapes having the exact same size and shape, including equal sides and angles. This concept is crucial to understand when examining congruent triangles.
To determine if two triangles are congruent, we must compare them to each other. It is not enough to look at a single triangle, as it will always be congruent to itself. Let's further explore this idea using a simple example of two triangles.
Imagine you and a friend each have a right triangle that is identical in size and shape. However, when you both place your triangles on a table with the right angle on the left side, they may appear different. Yet, with a simple turn and drag, the triangles can be perfectly overlapped, proving that they are congruent. This applies to any number of triangles that can be overlapped precisely.
In essence, congruent triangles have equal sides and angles, but may be positioned differently in space. On the other hand, non-congruent triangles have different shapes and/or sizes.
In Geometry, it is essential to be able to identify which angles and sides from one triangle correspond to those of another. To do this, notation is used. Usually, equal sides are marked with dash-like lines and equal angles have curved markings above them. For example, in the image below, sides AB and DE are equal, as well as BC and EF, and AC and DF.
Additionally, the symbol "≅" is used to indicate that two triangles are congruent. In more complex cases, double and triple marks may be used to differentiate between multiple equal sides or angles.
It is important to note that congruent triangles can only be transformed through translation or rotation, while non-congruent triangles require a change in size or shape to overlap precisely. Understanding these distinctions is crucial in Geometry as it allows for precise comparisons of shapes and figures.
Non-congruent triangles, also known as non-identical triangles, are triangles that have different shapes and/or sizes in comparison to each other.
To illustrate, let's consider the two triangles shown in the image below. These triangles are not congruent, as they have distinct shapes and sizes regardless of how they are moved or rotated.
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