## Proving Triangle Congruence

#### Aligned To Common Core Standard:

**High School Congruence** - HSG-CO.B.8

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, six facts are always true:
1) Corresponding sides are identical : AB ≈ DE, BC ≈ EF, CA ≈ FD.
2) Corresponding angles are equivalent: < 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. Here is how you prove congruent triangles by following the ordered combinations.
**SSS (Side-Side-Side)** - If the three sides of a triangle are congruent to another, then the two triangles are equal.
**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.
**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.
**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.
**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 selection of lessons and worksheets help students learn how to prove that two triangles are congruent.

### Printable Worksheets And Lessons

- 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.

#### 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?