Unit 6 Lesson 6.5: Triangle Similarity Theorems

Unit 6 Lesson 6.5:

Triangle Similarity Theorems

MT: Using Transformations to Prove Similarity

Angle-Angle (AA) Similarity Theorem

​Theorem: If two angles of one triangle are congruent to two angles of another triangle, then the two triangles are similar.

PRACTICE:

Find the length of BE, if possible.​

​First we have to determine whether or not these triangles are similar.

​

If the triangles ARE similar, you CAN find BE.

If the triangles ARE NOT similar, you CANNOT find BE.

​

PRACTICE:

PRACTICE:

Step 2: ​Use a theorem/postulate to determine if any are congruent.

• ​If you look at the indicators, we have PARALLEL LINES and two TRANSVERSALS.

• Parallel Lines & Transversals means we have:

  • SSIA (Same-Side Interior Angles)

  • AIA (Alternate Interior Angles)

  • AEA (Alternate Exterior Angles)

  • VA (Vertical Angles)

  • CA​ (Corresponding Angles)

​

PRACTICE:

​#1

​#2

​#3

PRACTICE:

​#1

​#2

​#3

PRACTICE:

PRACTICE:

Step 3: Solve the proportion. ​

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Multiple Choice

You are given that $$\angle RSV\cong\angle RTU$$  by the red indicators.

Which other pair of angles make these two triangle similar by the AA Similarity Theorem?

• $$\angle SRV\cong\angle TRU$$  

• $$\angle SVR\cong\angle TUR$$  

Multiple Choice

Find the length of RT, if possible.

• 6

• 5

• 15

• Not possible.

Multiple Select

Which two angle pairs result in $$\Delta ACB\sim\Delta CDA$$ by the AA Similarity Theorem.

Select all that apply.

• $$\angle ADC\cong\angle BCA$$  

• $$\angle DAC\cong\angle CBA$$  

• $$\angle DCA\cong\angle CAB$$  

Multiple Choice

Find the length of AC, if possible.

• 5.1

• 11

• 20.4

• Not possible.

Multiple Choice

Find the length of PQ, if possible.

• 11

• 20

• 12

• 21

Side-Side-Side (SSS) Similarity Theorem

​Theorem: If all 3 sides of one triangle are proportional to all 3 sides of another triangle, then the two triangles are similar.

Side-Angle-Side (SAS) Similarity Theorem

​Theorem: If 2 sides of one triangle are proportional to 2 sides of another triangle and their included angles are congruent, then the two triangles are similar.

PRACTICE:

Determine whether the given triangles are similar.  Justify your reasoning using a triangle similarity theorem.

PRACTICE:

Determine whether the given triangles are similar.  Justify your reasoning using a triangle similarity theorem.

Step 2: Determine which theorem you can use (process of elimination).

​AA (must have info for at least 2 angles)

SSS (must have info for all 3 sides)​

SAS (must have info for 2 sides and an angle in between)

​

PRACTICE:

Step 3: Check angle measures & ratios.

They aren't similar yet. Use those charts like we did in Unit 5!

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Multiple Choice

Determine whether the given triangles are similar.  Justify your reasoning.

• YES

• NO

• CANNOT BE DETERMINED

Multiple Choice

Determine whether the given triangles are similar.  Justify your reasoning.

• YES

• NO

• CANNOT BE DETERMINED

Multiple Choice

Determine whether the given triangles are similar.  Justify your reasoning.

• YES

• NO

• CANNOT BE DETERMINED

Multiple Choice

Given that $$\Delta ABC$$  is the PRE-IMAGE and $$\Delta BDC$$   is the IMAGE, what scale factor was used to prove these are similar by the SSS Similarity Theorem?

• 3.3

• 0.3

• 2.5

• 0.4