Lesson 2: Similarity and Proportional Reasoning

Property

Two figures are called similar if they have the same shape but different sizes. In similar figures:

• The corresponding angles are equal.
• We can multiply each side of one figure by the same factor (the scale factor) to get the corresponding side of the other figure.

Examples

• Two triangles both have angles $45^\circ$, $45^\circ$, and $90^\circ$. Because their corresponding angles are equal, they are similar, regardless of their side lengths.
• A rectangle with sides 4 cm and 6 cm is not similar to a square with sides 4 cm. Although they share a side length, their overall shapes and side ratios are different.
• A circle with a radius of 5 units and a circle with a radius of 15 units are similar. All circles have the same shape.

Explanation

Think of similar figures as a photo and its enlargement. The shape is exactly the same, but the size is different. Every part of the figure is scaled up or down by the same amount, and all the angles remain identical.

Property

The symbol $\sim$ means "is similar to" and is used to write similarity statements between figures. When writing similarity statements like $\triangle ABC \sim \triangle DEF$, the order of vertices must match the correspondence between the figures.

Examples

Property

When two triangles are similar, corresponding sides are proportional: $\frac{a}{a'} = \frac{b}{b'} = \frac{c}{c'}$. For indirect measurement, set up the proportion $\frac{\text{known side 1}}{\text{corresponding side 1}} = \frac{\text{known side 2}}{\text{unknown side}}$ and solve for the unknown.

Examples