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Questions and Answers
What is required for two triangles to be proven similar by SAS similarity?
What is required for two triangles to be proven similar by SAS similarity?
Which of the following is an example of proving similarity by AAA?
Which of the following is an example of proving similarity by AAA?
What is the ratio of corresponding sides required for two triangles to be similar by SSS?
What is the ratio of corresponding sides required for two triangles to be similar by SSS?
If triangle ABC is similar to triangle DEF, which of the following statements is accurate?
If triangle ABC is similar to triangle DEF, which of the following statements is accurate?
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What conclusion can be drawn from this statement: 'Triangle PQR is similar to triangle STU, written as ΔPQR ~ ΔSTU'?
What conclusion can be drawn from this statement: 'Triangle PQR is similar to triangle STU, written as ΔPQR ~ ΔSTU'?
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For two triangles, if the ratio of two pairs of sides is 3:1 and the included angle is congruent, what can be concluded?
For two triangles, if the ratio of two pairs of sides is 3:1 and the included angle is congruent, what can be concluded?
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What is a key characteristic of similar triangles noted in the content?
What is a key characteristic of similar triangles noted in the content?
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Study Notes
Similar Triangles
- To prove two triangles are similar, two conditions must be met:
- Sides are proportional: The ratios of corresponding sides must be equal.
- Angles are congruent: Corresponding angles must have the same measure.
SAS (Side-Angle-Side) Similarity
- Conditions: Two sides of one triangle are proportional to two sides of another triangle, and the included angles are congruent.
- Example: Triangles TUV and CDE
- Given: Angle TUV is congruent to angle CDE
- To Prove: Side TU is proportional to side CD, and side UV is proportional to side DE.
- Ratios: TU/CD = 27/9 = 3; UV/DE = 32/10 = 3.2
- Result: Since the ratios are not equal (3 ≠ 3.2), the sides are not proportional. Note, this example shows triangles that are not similar.
AAA (Angle-Angle-Angle) Similarity
- Conditions: Two angles of one triangle are congruent to two angles of another triangle.
- Example: Triangles HGF and HTS
- Given: Angle HGF is congruent to angle HTS, and angle GHF is congruent to angle TSH.
- Reasoning: The third angles in both triangles must also be congruent due to the Angle Sum Property of Triangles.
- Conclusion: Triangles HGF and HTS are similar by AAA.
SSS (Side-Side-Side) Similarity
- Conditions: All three sides of one triangle are proportional to all three sides of another triangle.
- Example: Triangles VTU and SQR
- Given: VT/SQ = 42/15 = 2.8; TU/QR = 72/25 = 2.8; VU/SR = 84/30 = 2.8.
- Result: Since the ratios of all corresponding sides are equal, they are proportional.
- Conclusion: Triangles VTU and SQR are similar by SSS.
Similarity Statement
- A similarity statement shows the corresponding vertices of two similar triangles.
- Example: Triangle TUV is similar to triangle CDE, written as: ΔTUV ~ ΔCDE.
Key Takeaways:
- When proving similarity, focus on the proportionality of sides and congruence of angles using one of the three methods: SAS, AAA, or SSS.
- Use a calculator to find the ratios of corresponding sides.
- Write a similarity statement to identify the corresponding vertices of similar triangles.
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Description
This quiz covers the concepts of similar triangles, including the conditions for triangle similarity such as proportional sides and congruent angles. It also explores the SAS and AAA similarity criteria with examples. Test your understanding of these fundamental geometric principles!
