1. Introduction
Congruent triangles have corresponding parts with angle measures that are the same and side lengths that are the same. If two triangles are congruent, they are also similar. Similar triangles have the same shape, but may be different in size. It is possible for two triangles to be similar but not congruent. Just like with determining congruency, it is possible to determine similarity based on the angle measures and lengths of the sides of the triangles.
1.6.1: Defining Similarity

2. Key Concepts
To determine whether two triangles are similar, observe the angle measures and the side lengths of the triangles.
When a triangle is transformed by a similarity transformation (a rigid motion [reflection, translation, or rotation] followed by a dilation), the result is a triangle with a different position and size, but the same shape.
If two triangles are similar, then their corresponding angles are congruent and the measures of their corresponding sides are proportional, or have a constant ratio.
1.6.1: Defining Similarity

3. Key Concepts, continued
The ratio of corresponding sides is known as the ratio of similitude.
The scale factor of the dilation is equal to the ratio of similitude.
Similar triangles with a scale factor of 1 are congruent triangles.
Like with congruent triangles, corresponding angles and sides can be determined by the order of the letters.
If is similar to , the vertices of the two triangles correspond in the same order as they are named.
1.6.1: Defining Similarity

4. Key Concepts, continued
The symbol shows that parts are corresponding.
; they are equivalent.
The corresponding angles are used to name the corresponding sides.
1.6.1: Defining Similarity

5. Key Concepts, continued
Observe the diagrams of and
The symbol for similarity ( ) is used to show that figures are similar.
1.6.1: Defining Similarity

6. Key Concepts, continued
1.6.1: Defining Similarity

7. Common Errors/Misconceptions
incorrectly identifying corresponding parts of triangles
assuming corresponding parts indicate congruent parts
assuming alphabetical order indicates congruence
changing the order of named triangles, causing parts to be incorrectly interpreted as congruent
1.6.1: Defining Similarity

8. Guided Practice Example 1
Use the definition of similarity in terms of similarity transformations to determine whether the two figures are similar. Explain your answer.
1.6.1: Defining Similarity

9. Guided Practice: Example 1, continued
Examine the orientation of the triangles.
The orientation of the triangles has remained the same, indicating translation, dilation, stretch, or compression.
ABC
XYZ
Side
Orientation
Top left side of triangle
Bottom left side of triangle
Right side of triangle
1.6.1: Defining Similarity

10. Guided Practice: Example 1, continued
Determine whether a dilation has taken place by calculating the scale factor.
First, identify the vertices of each triangle.
A (–4, 1), B (1, 4), and C (2, –2)
X (–8, 2), Y (2, 8), and Z (4, –4)
Then, find the length of each side of and using the distance formula,
1.6.1: Defining Similarity

11. Guided Practice: Example 1, continued
Calculate the distance of
Distance formula
Substitute (–4, 1) and (1, 4) for (x1, y1) and (x2, y2).
Simplify.
The distance of is units.
1.6.1: Defining Similarity

12. Guided Practice: Example 1, continued
Calculate the distance of
Distance formula
Substitute (1, 4) and (2, –2) for (x1, y1) and (x2, y2).
Simplify.
The distance of is units.
1.6.1: Defining Similarity

13. Guided Practice: Example 1, continued
Calculate the distance of
Distance formula
Substitute (–4, 1) and (2, – 2) for (x1, y1) and (x2, y2).
Simplify.
The distance of is units.
1.6.1: Defining Similarity

14. Guided Practice: Example 1, continued
Calculate the distance of
Distance formula
Substitute (–8, 2) and (2, 8) for (x1, y1) and (x2, y2).
Simplify.
The distance of is units.
1.6.1: Defining Similarity

15. Guided Practice: Example 1, continued
Calculate the distance of
Distance formula
Substitute (2, 8) and (4, –4) for (x1, y1) and (x2, y2).
Simplify.
The distance of is units.
1.6.1: Defining Similarity

16. Guided Practice: Example 1, continued
Calculate the distance of
Distance formula
Substitute (–8, 2) and (4, – 4) for (x1, y1) and (x2, y2).
Simplify.
The distance of is units.
1.6.1: Defining Similarity

17. Guided Practice: Example 1, continued
Calculate the scale factor of the changes in the side lengths.
Divide the side lengths of by the side lengths of
The scale factor is constant between each pair of corresponding sides.
1.6.1: Defining Similarity

18. Guided Practice: Example 1, continued
Determine if another transformation has taken place.
Multiply the coordinate of each vertex of the preimage by the scale factor, k.
Dk(x, y) = (kx, ky) Dilation by a scale factor of k
D2(–4, 1) = [2(–4), 2(1)] = (–8, 2)
D2(1, 4) = [2(1), 2(4)] = (2, 8)
D2(2, –2) = [2(2), 2(–2)] = (4, –4)
You can map onto by the dilation with a scale factor of 2.
1.6.1: Defining Similarity

19. ✔ Guided Practice: Example 1, continued State your conclusion.
A dilation is a similarity transformation; therefore, and are similar. The ratio of similitude is 2. ✔
1.6.1: Defining Similarity

20. Guided Practice: Example 1, continued
1.6.1: Defining Similarity

21. Guided Practice Example 2
Use the definition of similarity in terms of similarity transformations to determine whether the two figures are similar. Explain your answer.
1.6.1: Defining Similarity

22. Guided Practice: Example 2, continued
Examine the angle measures of the triangles.
Use a protractor or construction methods to determine if corresponding angles are congruent.
None of the angles of are congruent to the angles of
1.6.1: Defining Similarity

23. ✔ Guided Practice: Example 2, continued
Summarize your findings. Similarity transformations preserve angle measure.
The angles of and are not congruent.
There are no sequences of transformations that will map onto
and are not similar triangles. ✔
1.6.1: Defining Similarity

24. Guided Practice: Example 2, continued
1.6.1: Defining Similarity