SAS Congruence

SAS Congruence

Side-angle-side congruence. When two triangles have corresponding angles and sides that are congruent as shown below, the triangles are congruent.

Two triangles: △ABC ≅ △DEF, with tick marks showing AC=FD and BC=ED, and equal angle marks at C and D.

Key Formula

\text{If } \; AB = DE, \quad \angle B = \angle E, \quad BC = EF, \quad \text{then } \; \triangle ABC \cong \triangle DEF

Where:

$AB, DE$ = One pair of corresponding sides that are equal in length
$\angle B, \angle E$ = The included angles — each formed between the two known sides in its respective triangle
$BC, EF$ = The second pair of corresponding sides that are equal in length
$\cong$ = The congruence symbol, meaning the triangles are identical in shape and size

Worked Example

Problem: In triangle ABC, AB = 7 cm, BC = 10 cm, and the included angle B = 50°. In triangle DEF, DE = 7 cm, EF = 10 cm, and the included angle E = 50°. Prove that the two triangles are congruent.

Step 1: Identify the two sides and the included angle in each triangle. In △ABC, the two sides are AB and BC with angle B between them. In △DEF, the two sides are DE and EF with angle E between them.

Step 2: Compare the first pair of corresponding sides.

AB = DE = 7 \text{ cm}

Step 3: Compare the included angles.

\angle B = \angle E = 50°

Step 4: Compare the second pair of corresponding sides.

BC = EF = 10 \text{ cm}

Step 5: Since two sides and the included angle of △ABC match two sides and the included angle of △DEF, apply SAS Congruence.

\triangle ABC \cong \triangle DEF \quad (\text{by SAS})

Answer: △ABC ≅ △DEF by the SAS Congruence postulate.

Another Example

This example tests whether students can correctly identify corresponding vertices across two triangles with different labeling, and verify that the angle truly is the included angle between the two given sides.

Problem: In triangle PQR, PQ = 5 cm, QR = 8 cm, and angle Q = 60°. In triangle XYZ, XY = 5 cm, XZ = 8 cm, and angle X = 60°. A student claims the triangles are congruent by SAS. Is the student correct?

Step 1: In △PQR, the included angle Q sits between sides PQ and QR. So the SAS pairing is: PQ = 5, angle Q = 60°, QR = 8.

Step 2: In △XYZ, the angle X sits between sides XY and XZ. So the SAS pairing is: XY = 5, angle X = 60°, XZ = 8.

Step 3: Match corresponding parts: PQ ↔ XY = 5 cm ✓, angle Q ↔ angle X = 60° ✓, QR ↔ XZ = 8 cm ✓. All three corresponding parts are equal.

PQ = XY = 5, \quad \angle Q = \angle X = 60°, \quad QR = XZ = 8

Step 4: The angle in each triangle is between the two given sides, so it is indeed the included angle. SAS applies.

\triangle PQR \cong \triangle XYZ \quad (\text{by SAS})

Answer: Yes, the student is correct. △PQR ≅ △XYZ by SAS Congruence.

Frequently Asked Questions

What is the difference between SAS Congruence and SAS Similarity?

SAS Congruence requires two sides and the included angle to be exactly equal between triangles, proving the triangles are identical in shape and size. SAS Similarity requires only that two sides are in the same ratio (proportional) with the included angle equal, proving the triangles have the same shape but possibly different sizes.

Why does the angle have to be the included angle in SAS?

The angle must be between the two known sides because a non-included angle does not fix the triangle's shape uniquely. If you know two sides and an angle that is not between them (SSA), two different triangles can sometimes be formed — this is known as the ambiguous case. The included angle locks the two sides into a single configuration.

Can SAS Congruence be used for shapes other than triangles?

No. SAS Congruence is specific to triangles. Quadrilaterals and other polygons with more than three sides require additional information to prove congruence because knowing two sides and an included angle is not enough to determine their shape uniquely.

SAS Congruence vs. SAS Similarity

	SAS Congruence	SAS Similarity
What you compare	Two sides and the included angle are equal	Two sides are proportional and the included angle is equal
Conclusion	Triangles are congruent (same shape and size)	Triangles are similar (same shape, possibly different size)
Side condition	AB = DE and BC = EF	AB/DE = BC/EF (same ratio)
Symbol used	≅ (congruent)	~ (similar)
When to use	When you need to prove triangles are identical	When you need to prove triangles have the same shape

Why It Matters

SAS Congruence is one of the most frequently used triangle congruence postulates in geometry courses and standardized tests. You will rely on it in two-column proofs, coordinate geometry problems, and constructions where you need to show two triangles are identical. It also underpins many real-world applications such as engineering and surveying, where confirming exact measurements between structural components is essential.

Common Mistakes

Mistake: Using a non-included angle (SSA) and calling it SAS.

Correction: The angle must be the one formed between the two given sides. If the angle is opposite one of the sides rather than between them, you have an SSA arrangement, which does not guarantee congruence. Always check that the angle sits at the vertex where the two known sides meet.

Mistake: Mismatching corresponding vertices when writing the congruence statement.

Correction: The order of letters matters. If AB = DE and BC = EF with angle B = angle E, you must write △ABC ≅ △DEF, not △ABC ≅ △FED. Each letter position indicates which vertices correspond to each other.

Related Terms

Congruence Tests for Triangles — Overview of all triangle congruence postulates
SAS Similarity — Proves same shape using proportional sides
Congruent — Defines when figures are identical in shape and size
Corresponding — Matching parts between congruent or similar figures
Triangle — The polygon to which SAS Congruence applies
Angle — The included angle is central to the SAS test
Side of a Polygon — The two sides compared in the SAS condition