In geometry, congruent shapes have the same size and shape. This means that the sides and segments or two shapes have the same length. And, the angles possess the same measurements. When working with triangles, there are postulates and theorems that can be used to prove that the shapes are congruent. Here are ones you should know.
The ASA Postulate states that when two triangles have two equal angles and when the side between those angles are also the same length, the two triangles are congruent shapes. For example, check out these two triangles:
Here we have two equal acute angles, shown by the letter A. There are also two equal obtuse angles, shown by the letter B. The side between angles A and B are marked by two dashes, showing that these sides are also equal. Therefore, we know that these triangles follow the ASA Postulate. Therefore, they are congruent.
For the SAS Postulate, the two triangles need to have two identical sides and an identical angle. The identical angle should be positioned between the identical sides. Let’s take a look at the following example:
In this example, the triangles have listed measurements for comparison. The long side of each triangle has a measurement of 5, and the shorter side has a measurement of 3 in both triangles. The angle between these sides is 45-degrees. Because of this, we can surmise that the triangle follows the SAS Postulate.
The SSS Postulate simply states that all the sides of the two triangles are equal. So, if you can measure each side of the triangle and end up with three equal corresponding sides, you know that the two triangles are congruent. Check out the following example, and notice that all three sides have equal lengths:
With the measurements of the three sides in the first triangle corresponding to the measurements of the second triangle, we can use the SSS Postulate to determine that the triangles are congruent.
With the AAS Theorem, you can prove that two triangles are congruent by checking the measurement of two angles. If the two angles are equal on each triangle, and the side that is not between those angles is also equal, then the triangles are congruent.
In the following example, the obtuse angles in each triangle have the same measurement of 110-degrees. One of the acute angles has a measurement of 30-degrees. And, there is a side that isn’t between these angles with a measurement of 8. Since these measurements are the same on both triangles, we can use the AAS Theorem to say that these triangles are congruent.
The Hypotenuse-Leg Theorem only works for right triangles. If you have two right triangles, then check the measurement of the hypotenuse and one of the legs. If those lengths are the same on each triangle, then the two triangles are congruent. Remember that the hypotenuse is the largest side and is located opposite from the right angle. The sides touching the right angle are referred to as legs. Choose one of these legs when assessing the triangles.
For example, in the following example we have two right triangles. Therefore, you can look at the side lengths for the hypotenuses (20) and one of the legs (16). Because these lengths are the same on both of the triangles, it can be deduced through the Hypotenuse-Leg Theorem that the triangles are congruent.
Although two triangles might seem like congruent shapes, it’s important to use one of these postulates or theorems to make sure that they are actually congruent. Once you have some of the angle measurements and side lengths to back up your theory, you can truly determine whether or not the triangles are congruent.