Which Pair Of Triangles Can Be Proven Congruent By Sas

Which Pair of Triangles Can Be Proven Congruent by SAS?
Understanding how to determine whether two triangles are congruent using the Side‑Angle‑Side (SAS) postulate is a foundational skill in geometry. This article walks you through the concept, shows you exactly what to look for, and provides plenty of examples and practice so you can confidently answer the question “which pair of triangles can be proven congruent by SAS?”

Introduction

The SAS postulate states that if two sides and the included angle of one triangle are respectively equal to two sides and the included angle of another triangle, then the two triangles are congruent. In other words, the angle must be the one formed by the two given sides. Recognizing this pattern quickly lets you decide which pair of triangles satisfies SAS and therefore can be declared congruent without needing to measure all three sides or all three angles.

Understanding SAS Congruence

What Does SAS Mean? - S – Side

• A – Angle (the included angle)
• S – Side

The included angle is the angle that lies between the two sides you are comparing. If you have side AB, angle ∠BAC, and side AC in one triangle, and you find the same measurements in another triangle (side DE, angle ∠EDF, side DF), then the triangles are congruent by SAS.

Why the Included Angle Matters

If the angle is not between the two sides, the condition falls under SSA (Side‑Side‑Angle), which does not guarantee congruence (except in the special case of right triangles, known as HL). Therefore, always verify that the angle you are using is indeed the one formed by the two sides you have measured.

Identifying SAS Pairs: Step‑by‑Step Guide

Follow these steps whenever you are presented with two triangles and asked whether SAS applies:

1. Label the triangles (e.g., ΔABC and ΔDEF).
2. List the given side lengths for each triangle.
3. List the given angle measures for each triangle.
4. Check for a pair of sides that are equal in length in both triangles. 5. Locate the angle that sits between those two sides in each triangle.
5. Verify that the included angles are equal in measure.
6. If steps 4–6 are satisfied, conclude: the triangles are congruent by SAS.

If any step fails, SAS cannot be used; you may need to consider SSS, ASA, AAS, or HL instead.

Examples of Triangles Proven Congruent by SAS

Example 1: Simple Numeric Values

• ΔABC: AB = 5 cm, AC = 7 cm, ∠BAC = 60°
• ΔDEF: DE = 5 cm, DF = 7 cm, ∠EDF = 60°

Analysis

• Sides AB ↔ DE (both 5 cm)
• Sides AC ↔ DF (both 7 cm)
• Included angle ∠BAC ↔ ∠EDF (both 60°)

Since two sides and the angle between them match, ΔABC ≅ ΔDEF by SAS.

Example 2: Variable Expressions

• ΔPQR: PQ = 3x + 2, PR = 4x − 1, ∠QPR = 45°
• ΔSTU: ST = 3x + 2, SU = 4x − 1, ∠TSU = 45°

Analysis

• PQ = ST (identical expression)
• PR = SU (identical expression)
• ∠QPR = ∠TSU = 45°

Because the side lengths are expressed identically and the included angles are equal, the triangles are congruent by SAS for any value of x that keeps the side lengths positive.

Example 3: Diagram‑Based Reasoning

Imagine two triangles sharing a common side BC, with point A on one side of BC and point D on the other. You are given:

• AB = DC
• AC = DB
• ∠BAC = ∠CDB Even though the triangles overlap, the two sides AB & AC correspond to DC & DB, and the angle between each pair (∠BAC and ∠CDB) is equal. Hence, ΔABC ≅ ΔDCB by SAS. ---

Common Pitfalls When Using SAS

Practice Problems

Problem 1

ΔXYZ has XY = 9 cm, XZ = 12 cm, and ∠YXZ = 50°.
ΔUVW has UV = 9 cm, UW = 12 cm, and ∠VUW = 50°.

Question: Are the triangles congruent by SAS?

Solution: Yes. Two sides (XY ↔ UV, XZ ↔ UW) and the included angle (∠YXZ ↔ ∠VUW) are equal, so ΔXYZ ≅ ΔUVW by SAS.

Problem 2

In the figure below (imagine two triangles sharing side AD), you are given:

• AB = DC

• BD = EA

• ∠BAD = ∠EAD

Question: Are the triangles congruent by SAS?

Solution: Yes. We are given two sides (AB = DC, BD = EA) and the included angle (∠BAD = ∠EAD) are equal. Therefore, ΔABD ≅ ΔCDE by SAS.

Conclusion

The Side-Angle-Side (SAS) congruence postulate is a powerful tool for demonstrating that two triangles are identical. It relies on the fundamental principle that if two sides of a triangle and the included angle are congruent to the corresponding sides and included angle of another triangle, then the triangles must be congruent. Understanding the nuances of SAS, particularly the importance of the included angle and the correct order of sides, is crucial to applying this postulate accurately. By recognizing potential pitfalls and carefully analyzing given information, one can confidently use SAS to solve a wide range of geometric problems and solidify their understanding of triangle congruence. Mastering congruence postulates like SAS is a cornerstone of geometric reasoning and essential for success in further mathematical studies.

To further solidify understanding, consider additional practice problems and scenarios where SAS can be applied or misapplied.

Problem 3

ΔPQR has PQ = 7 cm, PR = 10 cm, and ∠QPR = 60°. ΔSTU has ST = 7 cm, SU = 10 cm, and ∠TSU = 60°.

Question: Are the triangles congruent by SAS?

Solution: Yes. Two sides (PQ ↔ ST, PR ↔ SU) and the included angle (∠QPR ↔ ∠TSU) are equal, so ΔPQR ≅ ΔSTU by SAS.

Problem 4

In the figure below (imagine two triangles sharing side EF), you are given:

• FG = EH
• FH = EG
• ∠FGE = ∠HEG

Question: Are the triangles congruent by SAS?

Solution: No. Here, while the sides FG = EH and FH = EG are given, the angle ∠FGE and ∠HEG are not the included angles between these sides. Therefore, SAS cannot be applied to conclude ΔFGE ≅ ΔHEG.

Problem 5

ΔABC has AB = 5 cm, AC = 8 cm, and ∠BAC = 45°. ΔDEF has DE = 5 cm, DF = 8 cm, and ∠EDF = 45°.

Question: Are the triangles congruent by SAS?

Solution: Yes. Two sides (AB ↔ DE, AC ↔ DF) and the included angle (∠BAC ↔ ∠EDF) are equal, so ΔABC ≅ ΔDEF by SAS.

Conclusion

The Side-Angle-Side (SAS) congruence postulate provides a robust method for determining the congruence of triangles. By ensuring that the correct sides and the included angle are compared, one can effectively apply this postulate. Recognizing the pitfalls, such as confusing the order of sides or overlooking given markings, is crucial for accurate application. Through careful analysis and practice, students can master the SAS postulate, enhancing their geometric reasoning skills and laying a strong foundation for more advanced mathematical studies. By internalizing these principles, one can tackle a variety of geometric problems with confidence and precision.