Similar Triangles Notes and Worksheets - Lindsay Bowden - Free Printable
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Step-by-step solution for: Similar Triangles Notes and Worksheets - Lindsay Bowden
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Show Answer Key & Explanations
Step-by-step solution for: Similar Triangles Notes and Worksheets - Lindsay Bowden
To determine if the triangles in each pair are similar, we need to use the criteria for triangle similarity: SSS (Side-Side-Side), SAS (Side-Angle-Side), or AA (Angle-Angle). Let's analyze each pair step by step.
---
- Triangles: Two triangles with angles \(87^\circ\) and \(87^\circ\).
- Analysis: Both triangles have two equal angles (\(87^\circ\) and \(87^\circ\)). Since the sum of angles in a triangle is \(180^\circ\), the third angle in both triangles must also be equal.
- Conclusion: The triangles are similar by AA.
---
- Triangles: A right triangle with sides 3 and 8, and another right triangle with sides 6 and 16.
- Analysis: Both triangles are right triangles. We can check the ratios of the corresponding sides:
\[
\frac{3}{6} = \frac{1}{2}, \quad \frac{8}{16} = \frac{1}{2}
\]
The ratios of the corresponding sides are equal, and they share a right angle.
- Conclusion: The triangles are similar by SAS (or SSS, since all sides are proportional).
---
- Triangles: Two triangles with sides 3, 6, 10 and 5, 10, 20.
- Analysis: Check the ratios of the corresponding sides:
\[
\frac{3}{5}, \quad \frac{6}{10} = \frac{3}{5}, \quad \frac{10}{20} = \frac{1}{2}
\]
The ratios are not consistent.
- Conclusion: The triangles are not similar.
---
- Triangles: A right triangle with angles \(42^\circ\) and \(48^\circ\), and another right triangle with angles \(42^\circ\) and \(48^\circ\).
- Analysis: Both triangles have the same angles (\(42^\circ\), \(48^\circ\), and \(90^\circ\)).
- Conclusion: The triangles are similar by AA.
---
- Triangles: Two triangles with sides 4.7, 3, 4 and 4, 2, 3.
- Analysis: Check the ratios of the corresponding sides:
\[
\frac{4.7}{4}, \quad \frac{3}{2}, \quad \frac{4}{3}
\]
The ratios are not consistent.
- Conclusion: The triangles are not similar.
---
- Triangles: Two triangles with sides 9, 13.5, 4.5 and 3, 4.5, 1.5.
- Analysis: Check the ratios of the corresponding sides:
\[
\frac{9}{3} = 3, \quad \frac{13.5}{4.5} = 3, \quad \frac{4.5}{1.5} = 3
\]
The ratios of the corresponding sides are equal.
- Conclusion: The triangles are similar by SSS.
---
- Triangles: Two triangles with angles \(67^\circ\), \(35^\circ\), and \(77^\circ\).
- Analysis: Both triangles have the same angles.
- Conclusion: The triangles are similar by AA.
---
- Triangles: Two triangles with sides 5.7, 7.3, 3.2 and 10.95, 14.6, 6.4.
- Analysis: Check the ratios of the corresponding sides:
\[
\frac{5.7}{10.95} = \frac{1}{2}, \quad \frac{7.3}{14.6} = \frac{1}{2}, \quad \frac{3.2}{6.4} = \frac{1}{2}
\]
The ratios of the corresponding sides are equal.
- Conclusion: The triangles are similar by SSS.
---
- Triangles: Two triangles with sides 8, 7, 17.5 and 7, 19.5, 17.5.
- Analysis: Check the ratios of the corresponding sides:
\[
\frac{8}{7}, \quad \frac{7}{19.5}, \quad \frac{17.5}{17.5} = 1
\]
The ratios are not consistent.
- Conclusion: The triangles are not similar.
---
- Triangles: Two triangles with angles \(54^\circ\) and \(48^\circ\).
- Analysis: Both triangles have two equal angles (\(54^\circ\) and \(48^\circ\)). The third angle in both triangles must also be equal.
- Conclusion: The triangles are similar by AA.
---
- Triangles: Two triangles that appear to be the same shape but different sizes.
- Analysis: The triangles are clearly scaled versions of each other, indicating proportional sides.
- Conclusion: The triangles are similar by SSS.
---
- Triangles: Two triangles with sides 5, 4, 6 and 13, 10.4, 15.6.
- Analysis: Check the ratios of the corresponding sides:
\[
\frac{5}{13}, \quad \frac{4}{10.4} = \frac{5}{13}, \quad \frac{6}{15.6} = \frac{5}{13}
\]
The ratios of the corresponding sides are equal.
- Conclusion: The triangles are similar by SSS.
---
1. AA
2. SAS
3. No
4. AA
5. No
6. SSS
7. AA
8. SSS
9. No
10. AA
11. SSS
12. SSS
\[
\boxed{\text{AA, SAS, No, AA, No, SSS, AA, SSS, No, AA, SSS, SSS}}
\]
---
1.
- Triangles: Two triangles with angles \(87^\circ\) and \(87^\circ\).
- Analysis: Both triangles have two equal angles (\(87^\circ\) and \(87^\circ\)). Since the sum of angles in a triangle is \(180^\circ\), the third angle in both triangles must also be equal.
- Conclusion: The triangles are similar by AA.
---
2.
- Triangles: A right triangle with sides 3 and 8, and another right triangle with sides 6 and 16.
- Analysis: Both triangles are right triangles. We can check the ratios of the corresponding sides:
\[
\frac{3}{6} = \frac{1}{2}, \quad \frac{8}{16} = \frac{1}{2}
\]
The ratios of the corresponding sides are equal, and they share a right angle.
- Conclusion: The triangles are similar by SAS (or SSS, since all sides are proportional).
---
3.
- Triangles: Two triangles with sides 3, 6, 10 and 5, 10, 20.
- Analysis: Check the ratios of the corresponding sides:
\[
\frac{3}{5}, \quad \frac{6}{10} = \frac{3}{5}, \quad \frac{10}{20} = \frac{1}{2}
\]
The ratios are not consistent.
- Conclusion: The triangles are not similar.
---
4.
- Triangles: A right triangle with angles \(42^\circ\) and \(48^\circ\), and another right triangle with angles \(42^\circ\) and \(48^\circ\).
- Analysis: Both triangles have the same angles (\(42^\circ\), \(48^\circ\), and \(90^\circ\)).
- Conclusion: The triangles are similar by AA.
---
5.
- Triangles: Two triangles with sides 4.7, 3, 4 and 4, 2, 3.
- Analysis: Check the ratios of the corresponding sides:
\[
\frac{4.7}{4}, \quad \frac{3}{2}, \quad \frac{4}{3}
\]
The ratios are not consistent.
- Conclusion: The triangles are not similar.
---
6.
- Triangles: Two triangles with sides 9, 13.5, 4.5 and 3, 4.5, 1.5.
- Analysis: Check the ratios of the corresponding sides:
\[
\frac{9}{3} = 3, \quad \frac{13.5}{4.5} = 3, \quad \frac{4.5}{1.5} = 3
\]
The ratios of the corresponding sides are equal.
- Conclusion: The triangles are similar by SSS.
---
7.
- Triangles: Two triangles with angles \(67^\circ\), \(35^\circ\), and \(77^\circ\).
- Analysis: Both triangles have the same angles.
- Conclusion: The triangles are similar by AA.
---
8.
- Triangles: Two triangles with sides 5.7, 7.3, 3.2 and 10.95, 14.6, 6.4.
- Analysis: Check the ratios of the corresponding sides:
\[
\frac{5.7}{10.95} = \frac{1}{2}, \quad \frac{7.3}{14.6} = \frac{1}{2}, \quad \frac{3.2}{6.4} = \frac{1}{2}
\]
The ratios of the corresponding sides are equal.
- Conclusion: The triangles are similar by SSS.
---
9.
- Triangles: Two triangles with sides 8, 7, 17.5 and 7, 19.5, 17.5.
- Analysis: Check the ratios of the corresponding sides:
\[
\frac{8}{7}, \quad \frac{7}{19.5}, \quad \frac{17.5}{17.5} = 1
\]
The ratios are not consistent.
- Conclusion: The triangles are not similar.
---
10.
- Triangles: Two triangles with angles \(54^\circ\) and \(48^\circ\).
- Analysis: Both triangles have two equal angles (\(54^\circ\) and \(48^\circ\)). The third angle in both triangles must also be equal.
- Conclusion: The triangles are similar by AA.
---
11.
- Triangles: Two triangles that appear to be the same shape but different sizes.
- Analysis: The triangles are clearly scaled versions of each other, indicating proportional sides.
- Conclusion: The triangles are similar by SSS.
---
12.
- Triangles: Two triangles with sides 5, 4, 6 and 13, 10.4, 15.6.
- Analysis: Check the ratios of the corresponding sides:
\[
\frac{5}{13}, \quad \frac{4}{10.4} = \frac{5}{13}, \quad \frac{6}{15.6} = \frac{5}{13}
\]
The ratios of the corresponding sides are equal.
- Conclusion: The triangles are similar by SSS.
---
Final Answers:
1. AA
2. SAS
3. No
4. AA
5. No
6. SSS
7. AA
8. SSS
9. No
10. AA
11. SSS
12. SSS
\[
\boxed{\text{AA, SAS, No, AA, No, SSS, AA, SSS, No, AA, SSS, SSS}}
\]
Parent Tip: Review the logic above to help your child master the concept of triangle similarity theorems worksheet.
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