# Proving Triangle Congruence Worksheets

When we are able to prove that two triangles are congruent, meaning that all there corresponding angles and sides are equal, we can use that information to learn a great deal of information about the coordinate system they may be in and the surrounding shapes. You will find this skill especially helpful when you begin writing geometric proofs. This selection of lessons and worksheets help students learn how to prove that two triangles are congruent. We will explore all the different theorems and show you how to use them in action based lessons.

### Aligned Standard: High School Geometry - HSG-CO.B.8

- Which Proof Does It? Step-by-step Lesson - Which of the theorems (ASA, SAS, or SSS) proves the triangles are congruent?
- Guided Lesson - This is great for a quick rapid fire lesson too. State the proof that needs to be used (ASA, SAS, or SSS).
- Guided Lesson Explanation - It is always a great idea to review what the markings on the triangles mean. Sometimes the message gets lost in the details.
- Practice Worksheet - I bent some of the triangles to make it a little more challenging for the kids.
- Matching Worksheet - Probably wouldn't have hurt the change the letters of the angles. Now I notice it!
- Geometric Constructions- Congruence Worksheet Five Pack - See what other information you can learn from just knowing that two triangles are similar.
- Congruence of Triangles Worksheet Five Pack - Find missing sides and angles by using common geometry concepts.
- Equal Triangles Worksheet Five Pack - This will start to lead us toward proofs in a round about way.

- Answer Keys - These are for all the unlocked materials above.

### Homework Sheets

Determine which proof helps you explain the given information.

- Homework 1 - Side-Side-Side Postulate (SSS) – If three sides of a triangle are congruent to three sides of another triangle, the triangles are congruent.
- Homework 2 - Side-Angle-Side Postulate (SAS) – If two sides and the included angle of a triangle are congruent to two sides and the angle of another triangle; the triangles are congruent.
- Homework 3 - Angle-Side-Angle Postulate (ASA) – If two angles and the included side of a triangle are congruent to two angles and the included side of another triangle, the triangles are congruent.

### Practice Worksheets

Obviously the triangles we present you with are congruent. I see the questions phrased in a way that leads to believe it is possible that they are congruent.

- Practice 1 - For the set below, determine if the triangles are congruent. State the proof needed (ASA, SAS, or SSS).
- Practice 2 - Look at all the marks to make your decision.
- Practice 3 - What side matches the other.

### Math Skill Quizzes

You will not find a better mix of problems on this anywhere else. At least that's what a geometry teacher told us.

- Quiz 1 - Triangle PQR ≈ Triangle GHI and the perimeter of Triangle PQR is 300 cm. If the sum of two sides of Triangle GHI is 150 cm, what is the length of the third side of Triangle PQR?
- Quiz 2 - For the set below, determine what postulate would be used to prove congruence.
- Quiz 3 - Which postulate would prove this?

### How Do You Prove Triangle Congruence?

When two triangles are congruent, one can be moved through more rigid motions to coincide with the other. All the corresponding angles and sides will be equal. When triangles are congruent, two facts are always true. Their corresponding sides are identical this is usually represented in a manner like this: AB ≈ DE, BC ≈ EF, CA ≈ FD. Congruent triangles have corresponding angles that are equivalent. This is often displayed in a fashion similar to this: ∠ A ≈ ∠ D, ∠ B ≈ ∠ E, ∠ C ≈ ∠ F.

The best part is when you are trying to prove the congruency of a triangle. It is not necessary to prove all the six points to show their congruency. Below are some of the methods of proving congruency. This is somewhat of a cheat sheet you can use to prove that two triangles are congruent. The names of the combinations indicate things that can prove and congruent on corresponding locations of that figure. If you can prove those locations to be equal on both figures, you have proven that they are congruent figures.

**SSS (Side-Side-Side)** - If the three sides of a triangle are congruent to another, then the two triangles are equal. To use this you will need to know the lengths of sides.

**SSS (Side-Angle-Side)** - If two sides and included angle of one of the triangles are congruent to corresponding parts of another, they are identical. Remember that the angle must be included mean that it needs between those two sides for this to be true.

**ASA (Angle-Side-Angle)** - If two angles and included side of one of the triangle are congruent to the corresponding parts of another triangle, they are equal. Once again, we need to realize that the side must be included or between those angles for this to work.

**AAS (Angle-Angle-Side)** - If two angles and a non-included side of one triangle are congruent to the corresponding sides of another triangle, they are congruent to each other. Not the need for a non-included side.

**HL (Hypotenuse-Leg)** - If the hypotenuse and leg of one triangle are congruent to the corresponding part of another right-angled triangle, they are the same. This is a lesser often used form, but definitely something to keep in your back pocket for more puzzling proofs.