What is SAS Similarity Criterion? | Simple Explanation for Class 6

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What is the SAS Similarity Criterion?

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 states that if two sides of one triangle are proportional to two sides of another triangle, AND the included angle (the angle between those two sides) of the first triangle is equal to the included angle of the second triangle, then the two triangles are similar.

Simple Example

Quick Example

Imagine you have two photographs of the same monument, but one is a smaller print and the other is a larger print. If the ratio of the height of the monument in the small photo to the height in the large photo is the same as the ratio of the width in the small photo to the width in the large photo, and the angle at the corner of the monument is the same in both photos, then the photos show similar shapes of the monument.

Worked Example

Step-by-Step

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

Given:
Triangle ABC: Side AB = 4 cm, Side BC = 6 cm, Angle B = 50 degrees.
Triangle PQR: Side PQ = 8 cm, Side QR = 12 cm, Angle Q = 50 degrees.

Step 1: Check the ratio of corresponding sides.
Ratio of AB to PQ = AB/PQ = 4/8 = 1/2.

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Step 2: Check the ratio of the other pair of corresponding sides.
Ratio of BC to QR = BC/QR = 6/12 = 1/2.

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Step 3: Compare the ratios. Are they equal?
Yes, 1/2 = 1/2. The sides are proportional.

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

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Step 5: Apply the SAS Similarity Criterion.
Since two sides are proportional (AB/PQ = BC/QR) and the included angles are equal (Angle B = Angle Q), Triangle ABC is similar to Triangle PQR.

Answer: Yes, Triangle ABC is similar to Triangle PQR by SAS Similarity.

Why It Matters

Understanding similarity helps engineers design smaller models of bridges or airplanes that behave like the real thing, saving time and money. Architects use it to create miniature building models. Even in computer graphics, similarity helps scale objects correctly, making games and animations look realistic.

Common Mistakes

MISTAKE: Students compare any two sides and any angle. | CORRECTION: Remember, the angle MUST be the 'included angle' – the angle BETWEEN the two sides you are comparing.

MISTAKE: Students check if the sides are equal instead of proportional. | CORRECTION: For similarity, sides must be in the same ratio (proportional), not necessarily equal in length.

MISTAKE: Students compare the included angle with a non-included angle. | CORRECTION: Always ensure the angle you are comparing is the one formed by the two proportional sides in both triangles.

Practice Questions

Try It Yourself

QUESTION: Triangle XYZ has sides XY=5 cm, YZ=7 cm, and Angle Y=60 degrees. Triangle MNO has sides MN=10 cm, NO=14 cm, and Angle N=60 degrees. Are these triangles similar by SAS? | ANSWER: Yes, because XY/MN = 5/10 = 1/2, YZ/NO = 7/14 = 1/2, and Angle Y = Angle N = 60 degrees.

QUESTION: Triangle ABC has AB=3 cm, BC=4 cm, and Angle B=70 degrees. Triangle DEF has DE=6 cm, EF=9 cm, and Angle E=70 degrees. Are they similar? | ANSWER: No, because AB/DE = 3/6 = 1/2, but BC/EF = 4/9. The side ratios are not equal.

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 if they are similar? | ANSWER: The ratio is 1/2 (or 2/1), because 3/6 = 1/2 and 5/10 = 1/2, and the included angles are equal.

MCQ

Quick Quiz

Which condition is NOT required for SAS Similarity?

Two pairs of corresponding sides are proportional.

The included angles between those sides are equal.

All three angles of both triangles are equal.

The triangles have the same shape.

The Correct Answer Is:

SAS Similarity only requires two pairs of proportional sides and their included angles to be equal. Option C describes the AAA (Angle-Angle-Angle) similarity criterion, which is a different way to prove similarity, not a requirement for SAS.

Real World Connection

In the Real World

When you use your phone's camera to zoom in on a distant object, the image you see on your screen is a similar (scaled) version of the actual object. The camera lens and software use principles like SAS similarity to maintain the object's proportions while changing its size, ensuring the zoomed image looks correct.

Key Vocabulary

Key Terms

SIMILAR TRIANGLES: Triangles that have the same shape but possibly different sizes, with corresponding angles equal and corresponding sides proportional. | PROPORTIONAL: When two ratios are equal. For example, 1/2 and 2/4 are proportional. | INCLUDED ANGLE: The angle formed by two specific sides of a triangle. | CORRESPONDING SIDES: Sides that are in the same relative position in two different triangles. | CRITERION: A rule or test used to judge something.

What's Next

What to Learn Next

Now that you understand SAS Similarity, you can explore other similarity criteria like SSS (Side-Side-Side) and AAA (Angle-Angle-Angle). These will give you more tools to determine if triangles are similar, which is super useful in geometry and beyond!