Podcast
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?
- All three sides are proportional to corresponding sides of the other triangle.
- Two sides are proportional and the non-included angles are congruent.
- Two angles are congruent and the third angle is equal.
- Two sides are proportional and the included angle is congruent. (correct)
Which of the following is an example of proving similarity by AAA?
Which of the following is an example of proving similarity by AAA?
- Angle HGF is congruent to angle HTS and angle GHF is congruent to angle TSH. (correct)
- Side TU is proportional to side CD and side UV is proportional to side DE.
- Side VTU is proportional to side SQ and angle SQR is congruent to angle VTU.
- All sides of triangle VTU are proportional to triangle SQR.
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?
- The angles of only one triangle must add up to 180 degrees.
- At least one pair of sides must be proportional.
- All three pairs of corresponding sides must have equal ratios. (correct)
- Two angles must be equal in both triangles.
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?
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'?
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?
What is a key characteristic of similar triangles noted in the content?
What is a key characteristic of similar triangles noted in the content?
Flashcards
Similar Triangles
Similar Triangles
Two triangles are similar if their corresponding sides are proportional and their corresponding angles are congruent.
SAS Similarity
SAS Similarity
If two sides of one triangle are proportional to two sides of another triangle and the included angles are congruent, the triangles are similar.
AAA Similarity
AAA Similarity
If two angles of one triangle are congruent to two angles of another triangle, the triangles are similar.
SSS Similarity
SSS Similarity
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Similarity Statement
Similarity Statement
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Proportionality of Sides
Proportionality of Sides
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Congruence of Angles
Congruence of Angles
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Angle Sum Property
Angle Sum Property
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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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