Triangle Similarity - Pre Algebra

1. Why are triangles with sides 2, 3, 4 and 4, 6, 9 not similar?

   The side ratios are 2, 2 and 2.25. They are not all equal, so the sides are not proportional.

2. What is the difference between similar and congruent triangles?

   Similar triangles have the same shape with proportional sides; congruent triangles also have the same size (scale factor 1).

3. Triangle $ABC$ is similar to triangle $DEF$. $AB = 4$, $DE = 10$ and $BC = 6$. What is $EF$?

   $EF = 15$. The scale factor is $\frac{10}{4} = 2.5$, and $6 \times 2.5 = 15$.

4. A triangle with sides 9, 12 and 15 is similar to a triangle with sides 6, 8 and $k$. What is $k$?

   $k = 10$. The scale factor is $\frac{6}{9} = \frac{2}{3}$, and $15 \times \frac{2}{3} = 10$.

5. What does the AA similarity criterion say?

   If two angles of one triangle are congruent to two angles of another, the triangles are similar.

6. How are corresponding sides of similar triangles related?

   They are proportional: all pairs share the same ratio, the scale factor.

7. What does the SAS similarity criterion say?

   If two pairs of sides are proportional and the angles between them are congruent, the triangles are similar.

8. One triangle has angles of $30^\circ$ and $80^\circ$. Another has angles of $30^\circ$ and $70^\circ$. Why are they similar?

   Both have angles of $30^\circ$, $70^\circ$ and $80^\circ$ (the third angles are $70^\circ$ and $80^\circ$), so AA applies.

9. What does the SSS similarity criterion say?

   If all three pairs of corresponding sides are proportional, the triangles are similar.

10. A triangle has sides 8, 10 and 12. A similar triangle has shortest side 6. What is its longest side?

    9. The scale factor is $\frac{6}{8} = \frac{3}{4}$, and $12 \times \frac{3}{4} = 9$.

11. A triangle with sides 3, 4 and 5 is similar to one with sides 6, 8 and 10. What is the scale factor from the first to the second?

    2. Each side of the second is twice the matching side of the first.

12. Two similar triangles have corresponding sides in the ratio $5 : 3$. The larger has area 100 square units. What is the smaller area?

    36 square units. The area ratio is $25 : 9$, and $100 \times \frac{9}{25} = 36$.

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