In this section we will be proving that given triangles are congruent. The first definition we will go over is CPCTC. It states that if two triangles are congruent, then there corresponding parts will also be congruent.

Notice in the diagram below, the corresponding angles are written in the same order in the congruency statement ∆ABC ≅ ∆DEF. The corresponding sides also match up in the congruency statement. This allows easy identification of corresponding parts with out a diagram.

Notice in the diagram below, the corresponding angles are written in the same order in the congruency statement ∆ABC ≅ ∆DEF. The corresponding sides also match up in the congruency statement. This allows easy identification of corresponding parts with out a diagram.

Another property you will use frequently when proving triangles congruent is pretty self explanatory. Often triangles will be joined by common sides or angles. Remember from algebra that the reflexive property states simply that a value is equal to itself (a = a). This is also used in geometry with sides and angles. Copy the diagram into your notes.

The theorems/postulates below will be what you will use in your proofs. YOU MUST MEMORIZE THESE THEOREMS! Copy them into your notes, then watch the corresponding videos for an explanation of each.

The theorems/postulates listed above work for all triangles. Notice there is no Angle-Side-Side Theorem because this scenario IS NOT enough information to prove congruence. However if the triangles are right triangles, it can prove congruency by the theorem Hypotenuse-Leg (HL). This is the most important of the right triangle congruence theorems. There are other right triangle congruency theorems (see below) however these are not commonly used because they can be proven with the theorems from above (SSS, SAS, ASA, AAS). The theorems below ONLY work for right triangles.

Before we begin with the proofs we will need to practice Identifying the theorems/postulates that prove given triangles are congruent. Watch the example video and take notes. Complete the worksheet and check your answers.

## Proving Triangle Congruence

When constructing proofs the same process is used from the worksheet above. The difference is that each conclusion or statement is written down with the evidence that backs up the statement. Below are several examples with videos that correspond. Copy the examples before you begin the practice.