Triangle ABC and Triangle LMN

Oct 13, 2025 by windailys.com 121 views

Select the Correct Answer: Which Triangles are Congruent According to the SAS Criterion?

Understanding the SAS Criterion

The SAS (Side-Angle-Side) criterion is a fundamental concept in geometry that helps determine whether two triangles are congruent or not. According to this criterion, if two triangles have three corresponding parts (two sides and the included angle) that are congruent, then the triangles are congruent. In other words, if we have two triangles, and we can show that two sides and the included angle of one triangle are congruent to two sides and the included angle of the other triangle, then the triangles are congruent.

Analyzing the Given Triangles

Let's analyze the given set of triangles ABC, LMN, EFG, and PQR. We are given that sides AB, MN, GF, and PQ are marked congruent, and sides AC, EF, and PR are marked congruent. We need to determine which of these triangles are congruent according to the SAS criterion.

Step 1: Identify the Congruent Sides

From the given information, we can see that the following sides are congruent:

AB ≅ MN
AB ≅ GF
AB ≅ PQ
AC ≅ EF
AC ≅ PR

Step 2: Identify the Included Angles

We need to identify the included angles for the congruent sides. The included angle is the angle between the two congruent sides.

For sides AB and AC, the included angle is angle BAC.
For sides MN and AC, the included angle is angle LAC.
For sides GF and AC, the included angle is angle EAC.
For sides PQ and AC, the included angle is angle PAC.

Step 3: Apply the SAS Criterion

Now, let's apply the SAS criterion to each pair of triangles.

Triangle ABC and Triangle LMN

Sides AB and MN are congruent.
Sides AC and LM are congruent.
The included angle between sides AB and AC is angle BAC.
The included angle between sides MN and LM is angle LAM.

Since the included angles are not marked congruent, we cannot conclude that triangles ABC and LMN are congruent.

Triangle ABC and Triangle EFG

Sides AB and GF are congruent.
Sides AC and EF are congruent.
The included angle between sides AB and AC is angle BAC.
The included angle between sides GF and EF is angle GEF.

Since the included angles are not marked congruent, we cannot conclude that triangles ABC and EFG are congruent.

Triangle ABC and Triangle PQR

Sides AB and PQ are congruent.
Sides AC and PR are congruent.
The included angle between sides AB and AC is angle BAC.
The included angle between sides PQ and PR is angle PQR.

Since the included angles are not marked congruent, we cannot conclude that triangles ABC and PQR are congruent.

Triangle LMN and Triangle EFG

Sides MN and GF are congruent.
Sides LM and EF are congruent.
The included angle between sides MN and LM is angle LAM.
The included angle between sides GF and EF is angle GEF.

Since the included angles are not marked congruent, we cannot conclude that triangles LMN and EFG are congruent.

Triangle LMN and Triangle PQR

Sides MN and PQ are congruent.
Sides LM and PR are congruent.
The included angle between sides MN and LM is angle LAM.
The included angle between sides PQ and PR is angle PQR.

Since the included angles are not marked congruent, we cannot conclude that triangles LMN and PQR are congruent.

Triangle EFG and Triangle PQR

Sides GF and PQ are congruent.
Sides EF and PR are congruent.
The included angle between sides GF and EF is angle GEF.
The included angle between sides PQ and PR is angle PQR.

Since the included angles are not marked congruent, we cannot conclude that triangles EFG and PQR are congruent.

Conclusion

Based on the given information, we cannot conclude that any of the triangles are congruent according to the SAS criterion. However, we can see that triangles LMN and EFG have two pairs of congruent sides, but the included angles are not marked congruent. Similarly, triangles LMN and PQR, and triangles EFG and PQR also have two pairs of congruent sides, but the included angles are not marked congruent.

Key Takeaways

The SAS criterion states that if two triangles have three corresponding parts (two sides and the included angle) that are congruent, then the triangles are congruent.
We need to identify the congruent sides and the included angles for each pair of triangles.
We cannot conclude that any of the triangles are congruent according to the SAS criterion based on the given information.

Final Answer

The final answer is: $\boxed{None}$
Q&A: Understanding Congruent Triangles According to the SAS Criterion

Frequently Asked Questions

In this article, we will address some of the most common questions related to congruent triangles according to the SAS criterion.

Q1: What is the SAS criterion?

A1: The SAS (Side-Angle-Side) criterion is a fundamental concept in geometry that helps determine whether two triangles are congruent or not. According to this criterion, if two triangles have three corresponding parts (two sides and the included angle) that are congruent, then the triangles are congruent.

Q2: How do I apply the SAS criterion to determine congruent triangles?

A2: To apply the SAS criterion, you need to identify the congruent sides and the included angles for each pair of triangles. Then, you need to check if the included angles are congruent. If they are, then the triangles are congruent.

Q3: What are the three corresponding parts that need to be congruent according to the SAS criterion?

A3: The three corresponding parts that need to be congruent according to the SAS criterion are:

Two sides
The included angle between the two sides

Q4: Can I use the SAS criterion to determine congruent triangles if the included angles are not marked congruent?

A4: No, you cannot use the SAS criterion to determine congruent triangles if the included angles are not marked congruent. The SAS criterion requires that the included angles be congruent.

Q5: What are some common mistakes to avoid when applying the SAS criterion?

A5: Some common mistakes to avoid when applying the SAS criterion include:

Failing to identify the congruent sides and the included angles
Assuming that the included angles are congruent without checking
Not checking if the three corresponding parts are congruent

Q6: Can I use the SAS criterion to determine congruent triangles if the triangles are not on the same plane?

A6: No, you cannot use the SAS criterion to determine congruent triangles if the triangles are not on the same plane. The SAS criterion only applies to triangles that are on the same plane.

Q7: What are some real-world applications of the SAS criterion?

A7: The SAS criterion has many real-world applications, including:

Architecture: Architects use the SAS criterion to determine if two triangles are congruent when designing buildings.
Engineering: Engineers use the SAS criterion to determine if two triangles are congruent when designing bridges and other structures.
Computer Graphics: Computer graphics artists use the SAS criterion to determine if two triangles are congruent when creating 3D models.

Q8: Can I use the SAS criterion to determine congruent triangles if the triangles have different side lengths?

A8: No, you cannot use the SAS criterion to determine congruent triangles if the triangles have different side lengths. The SAS criterion requires that the three corresponding parts be congruent.

Q9: What are some common misconceptions about the SAS criterion?

A9: Some common misconceptions about the SAS criterion include:

Assuming that the SAS criterion only applies to right triangles
Assuming that the SAS criterion only applies to triangles with equal side lengths
Assuming that the SAS criterion is only used in geometry

Q10: Can I use the SAS criterion to determine congruent triangles if the triangles are not similar?

A10: No, you cannot use the SAS criterion to determine congruent triangles if the triangles are not similar. The SAS criterion only applies to similar triangles.

Conclusion

In this article, we have addressed some of the most common questions related to congruent triangles according to the SAS criterion. We hope that this article has provided you with a better understanding of the SAS criterion and how to apply it to determine congruent triangles.