What are the Similarity Criteria? Exploring the Fundamentals of Shape Proportions in Geometry

Step-by-step Guide: Similarity Criteria

Definition of Similar Figures:
Two figures are said to be similar if their corresponding angles are congruent and their corresponding sides are proportional. Essentially, similar figures look alike but may be of different sizes.

• For triangles, if two triangles are similar, then their corresponding angles are equal, and their corresponding sides are in proportion.

Criteria for Similarity in Triangles:
a. Angle-Angle-Angle (AAA): If in two triangles, corresponding angles are equal, then their corresponding sides are in proportion and thus the triangles are similar.
b. Side-Angle-Side (SAS): If one angle of a triangle is equal to one angle of another triangle and the sides including these angles are proportional, then the triangles are similar.
c. Side-Side-Side (SSS): If the corresponding sides of two triangles are in proportion, then the triangles are similar.

Scale Factor:
The ratio of the lengths of corresponding sides of two similar figures is called the scale factor. For instance, if two similar triangles have corresponding sides of lengths \(AB\) and \(DE\), the scale factor is given by the ratio \( \frac{AB}{DE} \).

Examples

Example 1:
Triangle \(ABC\) has sides of lengths \(4 \text{ cm}\), \(5 \text{ cm}\), and \(6 \text{ cm}\). Triangle \(DEF\) has sides of lengths \(8 \text{ cm}\), \(10 \text{ cm}\), and \(12 \text{ cm}\). Are the triangles similar?

Solution:
Comparing the sides of triangles \(ABC\) and \(DEF\):
\( \frac{AB}{DE} = \frac{4 \text{ cm}}{8 \text{ cm}} = 0.5 \), \( \frac{BC}{EF} = \frac{5 \text{ cm}}{10 \text{ cm}} = 0.5 \), \( \frac{CA}{FD} = \frac{6 \text{ cm}}{12 \text{ cm}} = 0.5 \)

Since the ratios are the same, according to the SSS criterion, triangles \( ABC\) and \( DEF\) are similar with a scale factor of \( 0.5\).

Example 2:
Triangle \( PQR\) and triangle \( STU\) have angles measuring \(50^\circ\), \(60^\circ\), and \(70^\circ\). Are these triangles similar?

Solution:
Both triangles have corresponding angles that are equal. According to the AAA criterion, the triangles are similar.

Practice Questions

1. Triangle GHI has sides of lengths \(7 \text{ cm}\), \(24 \text{ cm}\), and \(25 \text{ cm}\). Triangle \(JKL\) has sides of lengths \(14 \text{ cm}\), \(48 \text{ cm}\), and \(50 \text{ cm}\). Are the triangles similar?
2. Triangle \(MNO\) has angles measuring \(35^\circ\), \(55^\circ\), and \(90^\circ\). Triangle \(PQR\) has angles measuring \(55^\circ\), \(90^\circ\), and \(35^\circ\). Are these triangles similar?

Answers:

1. Yes, triangles \(GHI\) and \(JKL\) are similar according to the SSS criterion with a scale factor of \(0.5\).
2. Yes, triangles \(MNO\) and \(PQR\) are similar according to the AAA criterion.