What is SAS Similarity Criterion?

S3-SA2-0055

Grade Level:

Class 6

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

The SAS (Side-Angle-Side) Similarity Criterion helps us check if two triangles are similar. It says that if two sides of one triangle are proportional to two sides of another triangle, AND the angle between those proportional sides is equal in both triangles, then the two triangles are similar.

Simple Example

Quick Example

Imagine you have two photographs of the same building. One is a regular print, and the other is a smaller, zoomed-out version. If the ratio of the height and width of the building in the first photo is the same as in the second photo, and the corner angle of the building (between height and width) looks the same in both, then the photos show similar shapes of the building.

Worked Example

Step-by-Step

Let's check if Triangle ABC and Triangle PQR are similar using SAS.

Triangle ABC has side AB = 6 cm, side BC = 8 cm, and angle B = 50 degrees.
Triangle PQR has side PQ = 3 cm, side QR = 4 cm, and angle Q = 50 degrees.

Step 1: Identify the two sides and the included angle for each triangle.
For Triangle ABC: Sides AB and BC, included Angle B.
For Triangle PQR: Sides PQ and QR, included Angle Q.

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Step 2: Check if the ratio of corresponding sides is equal.
Ratio of AB to PQ = AB/PQ = 6 cm / 3 cm = 2.
Ratio of BC to QR = BC/QR = 8 cm / 4 cm = 2.

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Step 3: Compare the ratios. Both ratios are 2. So, AB/PQ = BC/QR.

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Step 4: Check if the included angles are equal.
Angle B = 50 degrees.
Angle Q = 50 degrees.
So, Angle B = Angle Q.

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Step 5: Since two sides are proportional (ratio is 2) and the included angles are equal (50 degrees), the SAS Similarity Criterion is met.

Answer: Yes, Triangle ABC is similar to Triangle PQR.

Why It Matters

Understanding similarity is super important for many fields. Architects use it to make scaled models of buildings, and engineers use it to design parts that fit together perfectly. It's even used in computer graphics to resize images without distorting them, which is crucial for creating realistic video games and movies.

Common Mistakes

MISTAKE: Students often check if any two sides are proportional, not specifically the sides that 'include' the angle. | CORRECTION: Always ensure the equal angle is positioned BETWEEN the two proportional sides you are comparing.

MISTAKE: Students forget to check both conditions – proportionality of sides AND equality of the included angle. | CORRECTION: For SAS, you MUST verify both the side ratio AND the angle equality. One without the other is not enough.

MISTAKE: Confusing SAS Similarity with SAS Congruence. | CORRECTION: For SAS Similarity, sides are PROPORTIONAL. For SAS Congruence, sides are EQUAL in length.

Practice Questions

Try It Yourself

QUESTION: Triangle XYZ has XY = 10 cm, YZ = 12 cm, Angle Y = 70 degrees. Triangle RST has RS = 5 cm, ST = 6 cm, Angle S = 70 degrees. Are these triangles similar by SAS? | ANSWER: Yes, they are similar.

QUESTION: Triangle LMN has LM = 4 cm, MN = 6 cm, Angle M = 60 degrees. Triangle OPQ has OP = 8 cm, PQ = 10 cm, Angle P = 60 degrees. Are these triangles similar by SAS? | ANSWER: No, they are not similar because MN/PQ (6/10 = 0.6) is not equal to LM/OP (4/8 = 0.5).

QUESTION: Two triangles have side lengths (3, 5) with an included angle of 45 degrees, and (6, 10) with an included angle of 45 degrees. What is the ratio of their corresponding sides? Are they similar? | ANSWER: The ratio of corresponding sides is 1:2 or 2:1. Yes, they are similar.

MCQ

Quick Quiz

Which condition is NOT required for SAS Similarity Criterion?

Two pairs of corresponding sides are proportional

The included angles between those sides are equal

All three angles are equal

The triangles are of different sizes

The Correct Answer Is:

SAS Similarity only requires two proportional sides and the included angle to be equal. All three angles being equal is for AAA Similarity. Triangles can be of different sizes and still be similar.

Real World Connection

In the Real World

When you use Google Maps or any navigation app to find your way to a friend's house or a new restaurant, the map itself is a scaled-down, similar version of the real world. The angles between roads on the map are the same as in reality, and the distances are proportional. This concept helps GPS systems accurately represent the real world on your phone screen.

Key Vocabulary

Key Terms

SIMILAR: Shapes that have the same shape but possibly different sizes. | PROPORTIONAL: Having a constant ratio between corresponding parts. | INCLUDED ANGLE: The angle formed by two specific sides of a triangle. | CRITERION: A rule or test used to judge something.

What's Next

What to Learn Next

Now that you understand SAS Similarity, you should explore the other similarity criteria: AAA (Angle-Angle-Angle) and SSS (Side-Side-Side). These criteria will give you more tools to identify similar triangles and solve geometry problems!