The SAS Congruence Theorem

Oct 13, 2025 by windailys.com 140 views

Introduction

In geometry, two triangles are said to be congruent if they have the same size and shape. This means that the corresponding sides and angles of the two triangles are equal. In this article, we will examine whether two triangles are congruent based on the given measurements of their sides. We will use the SAS (Side-Angle-Side) congruence theorem to determine if the triangles are congruent.

The SAS Congruence Theorem

The SAS congruence theorem states that if two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, then the two triangles are congruent. Mathematically, this can be expressed as:

If ∠A = ∠D, AB = CD, and AC = BD, then ΔABC ≅ ΔDCB.

Given Measurements

Let's examine the given measurements of the two triangles:

Triangle A: AB = 6.5 cm, BC = 4.5 cm
Triangle B: AB = 6.5 cm, BC = 4.5 cm, ∠B = 75°

Step 1: Determine the Included Angle

To apply the SAS congruence theorem, we need to determine the included angle of the two sides that are equal. In this case, the two sides that are equal are AB and BC. The included angle is ∠B.

Step 2: Check if the Included Angle is Equal

We are given that ∠B = 75°. However, we are not given the measure of ∠B in Triangle A. To determine if the triangles are congruent, we need to find the measure of ∠B in Triangle A.

Step 3: Use the Angle-Side-Angle (ASA) Congruence Theorem

Since we are given the measure of ∠B in Triangle B, we can use the ASA congruence theorem to find the measure of ∠B in Triangle A. The ASA congruence theorem states that if two angles and the included side of one triangle are equal to two angles and the included side of another triangle, then the two triangles are congruent.

Step 4: Apply the ASA Congruence Theorem

Let's apply the ASA congruence theorem to find the measure of ∠B in Triangle A:

If ∠A = ∠D, AB = CD, and ∠B = ∠E, then ΔABC ≅ ΔDCB.

We are given that ∠B = 75° in Triangle B. We can assume that ∠B = 75° in Triangle A as well.

Step 5: Check if the Triangles are Congruent

Now that we have found the measure of ∠B in Triangle A, we can check if the triangles are congruent using the SAS congruence theorem:

If ∠A = ∠D, AB = CD, and AC = BD, then ΔABC ≅ ΔDCB.

We are given that AB = 6.5 cm, BC = 4.5 cm, and ∠B = 75° in both triangles. We can assume that AC = BD as well.

Conclusion

Based on the given measurements, we can conclude that the two triangles are congruent using the SAS congruence theorem. The corresponding sides and angles of the two triangles are equal, which means that the triangles have the same size and shape.

Sibling Form

The two congruent triangles can be represented in sibling form as follows:

ΔABC ≅ ΔDCB

Key Takeaways

The SAS congruence theorem states that if two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, then the two triangles are congruent.
The ASA congruence theorem states that if two angles and the included side of one triangle are equal to two angles and the included side of another triangle, then the two triangles are congruent.
The two triangles are congruent based on the given measurements of their sides.

References

"Geometry: A Comprehensive Introduction" by Dan Pedoe
"Mathematics for Elementary Teachers" by Gary L. Musser and Christopher J. Musser

Glossary

Congruent: Two triangles are said to be congruent if they have the same size and shape.
SAS Congruence Theorem: A theorem that states that if two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, then the two triangles are congruent.
ASA Congruence Theorem: A theorem that states that if two angles and the included side of one triangle are equal to two angles and the included side of another triangle, then the two triangles are congruent.
Q&A: Congruent Triangles ==========================

Frequently Asked Questions

Q: What is a congruent triangle?

A: A congruent triangle is a triangle that has the same size and shape as another triangle. This means that the corresponding sides and angles of the two triangles are equal.

Q: What is the SAS congruence theorem?

A: The SAS congruence theorem states that if two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, then the two triangles are congruent.

Q: What is the ASA congruence theorem?

A: The ASA congruence theorem states that if two angles and the included side of one triangle are equal to two angles and the included side of another triangle, then the two triangles are congruent.

Q: How do I determine if two triangles are congruent?

A: To determine if two triangles are congruent, you need to check if the corresponding sides and angles of the two triangles are equal. You can use the SAS or ASA congruence theorems to help you determine if the triangles are congruent.

Q: What are some common mistakes to avoid when working with congruent triangles?

A: Some common mistakes to avoid when working with congruent triangles include:

Assuming that two triangles are congruent just because they have the same shape, without checking if the corresponding sides and angles are equal.
Failing to check if the included angle is equal when using the SAS congruence theorem.
Failing to check if the included side is equal when using the ASA congruence theorem.

Q: How do I represent congruent triangles in sibling form?

A: To represent congruent triangles in sibling form, you can use the following notation:

ΔABC ≅ ΔDCB

This notation indicates that triangle ABC is congruent to triangle DBC.

Q: What are some real-world applications of congruent triangles?

A: Congruent triangles have many real-world applications, including:

Architecture: Congruent triangles are used to design buildings and other structures.
Engineering: Congruent triangles are used to design bridges and other infrastructure.
Art: Congruent triangles are used to create symmetrical and balanced compositions.

Q: Can congruent triangles be used to solve problems in geometry?

A: Yes, congruent triangles can be used to solve problems in geometry. By using the SAS or ASA congruence theorems, you can determine if two triangles are congruent and use this information to solve problems.

Q: How do I practice working with congruent triangles?

A: To practice working with congruent triangles, you can try the following:

Draw two triangles and check if they are congruent using the SAS or ASA congruence theorems.
Use online resources or geometry software to practice working with congruent triangles.
Work with a partner or teacher to practice solving problems involving congruent triangles.

Conclusion

In conclusion, congruent triangles are an important concept in geometry that can be used to solve problems and understand the properties of triangles. By understanding the SAS and ASA congruence theorems and how to represent congruent triangles in sibling form, you can develop a deeper understanding of geometry and improve your problem-solving skills.