SAS Similarity — Definition, Rules & Examples

SAS Similarity

Side-angle-side similarity. When two triangles have corresponding angles that are congruent and corresponding sides with identical ratios as shown below, the triangles are similar.

Two triangles ABC and DEF with marked congruent angles at A and D, and proportion AB/DE = AC/DF showing SAS Similarity.

Key Formula

\text{If } \frac{AB}{DE} = \frac{AC}{DF} \quad \text{and} \quad \angle A \cong \angle D, \quad \text{then } \triangle ABC \sim \triangle DEF

Where:

$AB, AC$ = The two sides of the first triangle that form the included angle A
$DE, DF$ = The two corresponding sides of the second triangle that form the included angle D
$\angle A, \angle D$ = The included angles between the two pairs of corresponding sides
$\sim$ = Denotes that the two triangles are similar (same shape, proportional sides)

Worked Example

Problem: In triangle ABC, AB = 6, AC = 9, and angle A = 50°. In triangle DEF, DE = 4, DF = 6, and angle D = 50°. Determine whether the triangles are similar by SAS Similarity.

Step 1: Identify the included angles. Angle A and angle D are both 50°, so the included angles are congruent.

\angle A = \angle D = 50°

Step 2: Compute the ratio of the first pair of corresponding sides (the sides that start at the vertex of the included angle).

\frac{AB}{DE} = \frac{6}{4} = \frac{3}{2}

Step 3: Compute the ratio of the second pair of corresponding sides.

\frac{AC}{DF} = \frac{9}{6} = \frac{3}{2}

Step 4: Compare the two ratios. Both equal 3/2, and the included angles are congruent, so the SAS Similarity condition is satisfied.

\frac{AB}{DE} = \frac{AC}{DF} = \frac{3}{2} \quad \text{and} \quad \angle A \cong \angle D

Answer: Yes, triangle ABC is similar to triangle DEF by SAS Similarity, with a scale factor of 3/2.

Another Example

This example shows a case where the test fails — the included angles match, but the side ratios are unequal. Students need to verify both conditions, not just one.

Problem: In triangle PQR, PQ = 10, PR = 15, and angle P = 40°. In triangle STU, ST = 6, SU = 8, and angle S = 40°. Are the triangles similar by SAS Similarity?

Step 1: Check the included angles. Angle P and angle S are both 40°, so the included angles are congruent.

\angle P = \angle S = 40°

Step 2: Compute the ratio of the first pair of corresponding sides.

\frac{PQ}{ST} = \frac{10}{6} = \frac{5}{3}

Step 3: Compute the ratio of the second pair of corresponding sides.

\frac{PR}{SU} = \frac{15}{8}

Step 4: Compare the ratios. 5/3 ≈ 1.667, while 15/8 = 1.875. The ratios are not equal, so the SAS Similarity condition is NOT satisfied.

\frac{5}{3} \neq \frac{15}{8}

Answer: No, triangles PQR and STU are not similar by SAS Similarity because the ratios of the sides that form the included angle are not equal.

Frequently Asked Questions

What is the difference between SAS Similarity and SAS Congruence?

SAS Congruence requires two pairs of corresponding sides to be equal in length with an equal included angle, proving the triangles are identical in size and shape. SAS Similarity only requires the two pairs of sides to be in the same ratio (not necessarily equal) with a congruent included angle, proving the triangles have the same shape but possibly different sizes.

Does the angle in SAS Similarity have to be between the two sides?

Yes. The angle must be the included angle — the angle formed by the two sides you are comparing. If the congruent angle is not between the proportional sides, you cannot conclude similarity using SAS. Using a non-included angle can lead to incorrect results, similar to the ambiguous case in triangle construction.

How many similarity tests are there for triangles?

There are three standard similarity tests: AA (Angle-Angle), SAS (Side-Angle-Side), and SSS (Side-Side-Side). AA requires two pairs of congruent angles. SAS requires two pairs of proportional sides with a congruent included angle. SSS requires all three pairs of corresponding sides to be in the same ratio.

SAS Similarity vs. SAS Congruence

	SAS Similarity	SAS Congruence
What it proves	Triangles are similar (same shape, possibly different size)	Triangles are congruent (same shape and same size)
Side condition	Two pairs of corresponding sides in the same ratio	Two pairs of corresponding sides equal in length
Angle condition	Included angles are congruent	Included angles are congruent
Scale factor	Can be any positive number	Must be exactly 1
Symbol used	∼ (similar)	≅ (congruent)

Why It Matters

SAS Similarity appears throughout geometry courses, especially in proofs involving parallel lines, midpoints, and triangle proportionality. It is also essential in trigonometry and real-world applications like indirect measurement, where you determine unknown distances by setting up similar triangles. Many standardized tests, including the SAT and ACT, include problems that require recognizing when SAS Similarity applies.

Common Mistakes

Mistake: Using a non-included angle instead of the included angle between the two proportional sides.

Correction: The angle must be the one formed by the two sides whose ratios you are comparing. Always check that the angle sits at the vertex where the two measured sides meet.

Mistake: Setting up the side ratios with mismatched corresponding sides (e.g., pairing the shorter side of one triangle with the longer side of the other).

Correction: Carefully match corresponding vertices first. If angle A corresponds to angle D, then sides AB and AC must be paired with DE and DF respectively, based on the vertex labeling and position in each triangle.

Related Terms

SAS Congruence — Congruence version requiring equal sides, not ratios
Similar — The property that SAS Similarity establishes
Similarity Tests for Triangles — Overview of all three similarity criteria
Triangle — The geometric figure being tested for similarity
Corresponding — Matching sides and angles between triangles
Ratio — Used to compare corresponding side lengths
Congruent — Describes the equal included angles required
Angle — The included angle is a key part of SAS Similarity