Proving Triangles Similar (AA~, SSS~, SAS~)

Similar Triangles Two triangles are similar if they are the same shape. That means the vertices can be paired up so the angles are congruent. Size changes.

AA Similarity (Angle-Angle or AA~) If 2 angles of one triangle are congruent to 2 angles of another triangle, then the triangles are similar. Given: and Conclusion: by AA~

SSS Similarity (Side-Side-Side or SSS~) If the lengths of the corresponding sides of 2 triangles are proportional, then the triangles are similar. Given: Conclusion: by SSS~

Example: SSS Similarity (Side-Side-Side) 5 11 22 8 16 10 Given: Conclusion: By SSS ~

SAS Similarity (Side-Angle-Side or SAS~) If the lengths of 2 sides of a triangle are proportional to the lengths of 2 corresponding sides of another triangle and the included angles are congruent, then the triangles are similar. Given: Conclusion: by SAS~

Example: SAS Similarity (Side-Angle-Side) 5 11 22 10 Conclusion: Given: By SAS ~

#1 A 80 D E 80 B C ABC ~ ADE by AA ~ Postulate 11111111#11# 1 11111111#11# 1  A #1 80 D E 80 B C ABC ~ ADE by AA ~ Postulate Slide from MVHS

  C #2 6 10 D E 5 3 A B CDE~ CAB by SAS ~ Theorem Slide from MVHS

  L #3 5 3 M 6 6 K N 6 10 O KLM~ KON by SSS ~ Theorem Slide from MVHS

  A 20 #4 D 30 24 16 B C 36 ACB~ DCA by SSS ~ Theorem Slide from MVHS

  L #5 15 P A 25 9 N LNP~ ANL by SAS ~ Theorem Slide from MVHS

Similarity is reflexive, symmetric, and transitive. Proving Triangles Similar Similarity is reflexive, symmetric, and transitive. Steps for proving triangles similar: 1. Mark the Given. 2. Mark … Reflexive (shared) Angles or Vertical Angles 3. Choose a Method. (AA~, SSS~, SAS~) Think about what you need for the chosen method and be sure to include those parts in the proof.

AA Problem #1 Step 1: Mark the given … and what it implies   Step 1: Mark the given … and what it implies Step 2: Mark the vertical angles AA Step 3: Choose a method: (AA,SSS,SAS) Step 4: List the Parts in the order of the method with reasons Statements Reasons 4. C D E G F Given Alternate Interior <s Alternate Interior <s   AA Similarity

SSS Problem #2 Step 1: Mark the given … and what it implies   Step 1: Mark the given … and what it implies SSS Step 2: Choose a method: (AA,SSS,SAS) Step 4: List the Parts in the order of the method with reasons Statements Reasons Given Substitution SSS Similarity

SAS Problem #3 Step 1: Mark the given … and what it implies   Step 1: Mark the given … and what it implies Step 2: Mark the reflexive angles SAS Step 3: Choose a method: (AA,SSS,SAS) Step 4: List the Parts in the order of the method with reasons Next Slide………….

Statements Reasons G is the Midpoint of H is the Midpoint of Given 2. EG = DG and EH = HF Def. of Midpoint 3. ED = EG + GD and EF = EH + HF Segment Addition Post. 4. ED = 2 EG and EF = 2 EH Substitution Division Property Reflexive Property SAS Postulate

The End 1. Mark the Given. 2. Mark … Shared Angles or Vertical Angles 3. Choose a Method. (AA, SSS , SAS) **Think about what you need for the chosen method and be sure to include those parts in the proof. The End