Exploring the World of Geometry: The Intricacies of Similarity

Step-by-step Guide: Similarity

1. Understanding Congruence vs. Similarity:
While congruent figures are identical in size and shape, similar figures only need to have the same shape. Their sizes may differ.

2. Identifying Similar Figures:
Two polygons are similar if:

• Their corresponding angles are congruent.
• Their corresponding sides are proportional.

3. Using the Similarity Ratio:
The ratio of the lengths of the corresponding sides of similar polygons is called the similarity ratio or scale factor. Given two similar triangles \(ABC\) and \(DEF\), if side \(AB\) is twice as long as side \(DE\), the similarity ratio is \(2:1\).

Examples

Example 1:
Given two triangles \(ABC\) and \(XYZ\) with \(\angle A = \angle X = 60^\circ\), \(\angle B = \angle Y = 50^\circ\), and \(\frac{AB}{XY} = \frac{2}{1}\), are the two triangles similar?

Solution:
First, compare angles:
\(\angle A = \angle X\)
\(\angle B = \angle Y\)
Thus, the third angles, \(\angle C\) and \(\angle Z\), will also be congruent because the angles in a triangle sum up to \(180^\circ\).
Next, compare the side ratio:
Given \(\frac{AB}{XY} = \frac{2}{1}\), if the other corresponding sides of the triangles are also in the ratio \(2:1\), the triangles are similar.
Given the congruent angles and the side ratio, triangles \(ABC\) and \(XYZ\) are indeed similar.

Example 2:
Rectangles \(PQRS\) and \(MNOA\) have the same angles (since all angles in a rectangle are \(90^\circ\)). If \(\frac{PQ}{MN} = \frac{3}{2}\), are the two rectangles similar?

Solution:
All angles in a rectangle are \(90^\circ\), so \(PQRS\) and \(MNOA\) have congruent angles.
Given \(\frac{PQ}{MN} = \frac{3}{2}\), if \(\frac{RS}{OA}\) is also in the ratio \(3:2\), the rectangles are similar.
Given the congruent angles and the side ratio, rectangles \(PQRS\) and \(MNOA\) are similar.

Practice Questions:

1. Two triangles \(EFG\) and \(LMN\) have sides in the ratio \(3:4\). If \(\angle E = \angle L\) and \(\angle F = \angle M\), are the two triangles similar?
2. Squares \(UVWX\) and \(QRST\) have side lengths in the ratio \(5:7\). Are the two squares similar?

Answers:

1. Yes, the triangles are similar.
2. Yes, the squares are similar.