Let's talk about the CPCTC Theorem. CPCTC is an acronym that stands for the words, Corresponding Parts of Congruent Triangles are Congruent. It is a known fact that if two triangles are congruent then all corresponding parts of the two triangles are also congruent. Do you remember from earlier in this lesson that any two shapes are said to be congruent when they are exactly the same shape AND size?

Here is a new definition that helps to explain the fact stated in our first point. Corresponding parts are parts of shapes, or other objects, that exist in the exact same position relative to each shape. When I say that parts exist in the exact same position relative to each shape, I mean that the angles at the top of each shape are in the same relative position, or the side lengths that are shown on the right side are in the same relative position. Our example below will help to clarify.

When it comes to triangles, it is important to note that every triangle has six parts: three sides and three angles, and each part should be labeled to avoid confusion. In Geometry, we use symbols to indicate when objects are congruent, and we can use these symbols in a diagram or in written form.

For this example, I must first label each triangle in the diagram. I will use the capital letters A, B, and C for the triangle on the left, and the capital letters X, Y, and Z for the triangle on the right. Notice the angle next to letter A is in the same position as the angle next to letter X – these are angles in the same relative position, the top of the triangle. Angle B is in the same relative position as Y, and C is in the same relative position as Z. It is easy to see that these triangles are the exact same shape and size, we could verify by measuring side lengths and angles. Congruence between objects is kind of like saying that two objects are physically equal to one another, so the symbol used to show congruence looks a lot like an equals sign, only with one small difference. This diagram now shows that triangle ABC is congruent to Triangle XYZ. When in the written form, the symbols might look like this: Triangle ABC is congruent to Triangle XYZ. Notice the order we write the name of each triangle, we intentionally write the letters in the same order of reference.

Now we can identify corresponding parts of each triangle in the diagram. The three angles must be marked with arcs, and the three sides must be indicated with hatch marks, or sometimes called hash marks or even tick marks. Remember, corresponding parts of congruent triangles are congruent. Because of this, we want to use the exact same mark for each pair of corresponding parts.

Angle A corresponds with angle X, so I will use a single arc. Angle B corresponds with angle Y, so I will use a double arc, and angle C corresponds with angle Z, so I will use a triple arc. Now, all three pairs of angles in the diagram have been marked with arcs. Side AB corresponds with side XY, so I will use a single hatch mark. Side BC corresponds with YZ, so I will use a double hatch mark, and lastly, side AC corresponds with XZ, so I will use a triple hatch mark. Now each pair of sides in the diagram is marked with hatch marks. Here is how we can show the congruencies in written form. Side AB is congruent to side XY, side BC is congruent to side YZ, side AC is congruent to side XZ. And angle A is congruent to angle X, angle B is congruent to angle Y, and angle C is congruent to angle Z.

As a bonus point, I would like to point out that one way to check for congruent triangles is to measure all six parts of two triangles to verify each measurement is congruent. In later pages in this lesson, we will learn how to determine if two triangles are congruent by examining only three of the six parts of each triangle, and using Congruence Properties. Be on the lookout for this technique.

Good luck!