Ratios can be used to show a multiplicative relationship between two quantities. When two ratios are equivalent, you can use a proportion to relate the ratios.

Two figures are said to be similar if they are proportionally related. In this section, you will investigate the relationship between corresponding side lengths and corresponding angle measures of similar figures. From that investigation, you will make a formal definition of similar figures.

Use the interactive to begin your investigation of similar figures. In this interactive, triangle *ABC* and triangle *DEF* are similar triangles. Use the check boxes to reveal or hide ratios of the lengths of corresponding sides and to reveal or hide the measures of corresponding angles. Move the vertices of triangle *ABC* around, and pay special attention to patterns in the ratios of the lengths of corresponding sides and to patterns in the measures of corresponding angles.

**Investigating Similar Figures**

Click and drag on *A*, *B*, or *C* to adjust the size of triangle *ABC*.

Click on the Show/Hide Ratios box to show or hide the ratios of the lengths of corresponding sides.

Click on the Show/Hide Angle Measures box to show or hide the angle measures.

Need additional directions?

Use the interactive to answer the following questions.

1. Given that the two triangles are similar, what did you notice about the ratios of the lengths of the corresponding sides of triangle *ABC* and triangle *DEF*?

2. Given that the two triangles are similar, what did you notice about the measures of the corresponding angles of triangle *ABC *and triangle *DEF*?

3. Use what you noticed about the ratios of the lengths of the corresponding sides and the measures of the corresponding angles to write a definition of similar triangles.

4. Do you think the relationship you noticed for triangles would be true for quadrilaterals and other polygons?