Congruent Triangles Study Guide
Introduction to Congruent Triangles
This lesson will help you identify corresponding parts of congruent triangles and to name the postulate or theorem that shows that two triangles are congruent.
Congruent triangles are commonly used in the construction of quilts, buildings, and bridges. Congruent triangles are also used to estimate inaccessible distances, such as the width of a river or the distance across a canyon. In this lesson, you will learn simple ways to determine whether two triangles are congruent.
When you buy floor tiles, you get tiles that are all the same shape and size. One tile will fit right on top of another. In geometry, you would say one tile is congruent to another tile. Similarly, in the following figure, ΔABC and ΔXYZ are congruent. They have the same size and shape.
Imagine sliding one triangle over to fit on top of the other triangle. You would put point A on point X; point B on point Y; and point C on point Z. When the vertices are matched in this way, A and X are called corresponding angles, and and are called corresponding sides.
Corresponding angles and corresponding sides are often referred to as corresponding parts of the triangles. In other words, you could say Corresponding Parts of Congruent Triangles are Congruent. This statement is often referred to by the initials CPCTC.
When ΔABC is congruent to ΔXYZ, you write ΔABC ΔXYZ. This means that all of the following are true:
Suppose instead of writing ΔABC ΔXYZ, you started to write ΔCAB ______. Since you started with C to name the first triangle, you must start with the corresponding letter, Z, to name the second triangle. Corresponding parts are named in the same order. If you name the first triangle ΔCAB, then the second triangle must be named ΔZXY. In other words, ΔCAB ΔZXY.
Side-Side-Side (SSS) Postulate
If you have three sticks that make a triangle and a friend has identical sticks, would it be possible for each of you to make different-looking triangles? No, it is impossible to do this. A postulate of geometry states this same idea. It is called the side-side-side postulate.
Take a look at the following triangles to see this postulate in action:
The hatch marks on the triangles show which sides are congruent to which in the two triangles. For example, and both have one hatch mark, which shows that these two segments are congruent. is congruent to as shown by the two hatch marks, and are congruent as shown by the three hatch marks.
Since the markings indicate that the three pairs of sides are congruent, you can conclude that the three pairs of angles are also congruent. From the definition of congruent triangles, it follows that all six parts of ΔABC are congruent to the corresponding parts of ΔRST.
Side-Angle-Side (SAS) Postulate
If you put two sticks together at a certain angle, there is only one way to finish forming a triangle. Would it be possible for a friend to form a different-looking triangle if he or she started with the same two lengths and the same angle? No, it would be impossible. Another postulate of geometry states this same idea; it is called the side-angle-side postulate.
Look at the following two triangles to see an example of this postulate:
Angle-Side-Angle (ASA) Postulate
There is one more postulate that describes two congruent triangles. Angle-side-angle involves two angles and a side between them. The side is called an included side.
Take a look at the following two triangles:
Practice problems for these concepts can be found at: Congruent Triangles Practice Questions.
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