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G.7 Proving Triangles Similar (AA~, SSS ~, SAS ~ )

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1 G.7 Proving Triangles Similar (AA~, SSS ~, SAS ~ )

2 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 does not matter.

3 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. Conclusion: andGiven: by AA~

4 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~

5 Example: SSS Similarity (Side-Side-Side) Given : Conclusion : By SSS ~

6 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~

7 Example: SAS Similarity (Side-Angle-Side) Given : Conclusion : By SAS ~

8 A BC DE 80   ABC ~  ADE by AA ~ Postulate Slide from MVHS

9 AB C DE  CDE~  CAB by SAS ~ Theorem Slide from MVHS

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

11 C B A D  ACB~  DCA by SSS ~ Theorem Slide from MVHS

12 N L A P  LNP~  ANL by SAS ~ Theorem Slide from MVHS

13 Similarity is reflexive, symmetric, and transitive. 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. Steps for proving triangles similar: Proving Triangles Similar

14 Problem #1 C D E G F Step 1: Mark the given … and what it implies Step 2: Mark the vertical angles Step 3: Choose a method: (AA,SSS,SAS) Step 4: List the Parts in the order of the method with reasons Step 5: Is there more? StatementsReasons Given Alternate Interior

15 Problem #2 Step 1: Mark the given … and what it implies Step 2: Choose a method: (AA,SSS,SAS) Step 4: List the Parts in the order of the method with reasons Step 5: Is there more? StatementsReasons Given Division Property SSS Similarity Substitution SSS 1. IJ = 3LN ; JK = 3NP ; IK = 3LP

16 Problem #3 Step 1: Mark the given … and what it implies Step 3: Choose a method: (AA,SSS,SAS) Step 4: List the Parts in the order of the method with reasons Next Slide…………. Step 5: Is there more? SAS Step 2: Mark the reflexive angles

17 StatementsReasons 1. G is the Midpoint of H is the Midpoint of Given 2. EG = DG and EH = HFDef. of Midpoint 3. ED = EG + GD and EF = EH + HFSegment Addition Post. 4. ED = 2 EG and EF = 2 EHSubstitution Division Property Substitution Reflexive Property SAS Postulate

18 Similarity is reflexive, symmetric, and transitive.

19 Choose a Problem. Problem #1 Problem #2 Problem #3 End Slide Show SSS SAS AA

20 Problem #1 Given:DE||FG Prove:DEC  FGC

21 Step 1: Mark the Given … and what it implies Given:DE||FG Prove:DEC  FGC

22 Step 2: Mark... Reflexive Angles Vertical Angles … if they exist. Given:DE||FG Prove:DEC  FGC

23 Step 3: Choose a Method AA SSS SAS Given:DE||FG Prove:DEC  FGC

24 STATEMENTSREASONS 3. DEC  FGC Given:DE||FG Prove:DEC  FGC

25 Choose a Problem. Problem #1 Problem #2 Problem #3 End Slide Show SSS SAS AA

26 Problem #2 Choose a Method Based on the given info AA SSS SAS

27 STATEMENTSREASONS 1. Given 2. Division Prop. 3. Substitution 4. SSS Similarity

28 Choose a Problem. Problem #1 Problem #2 Problem #3 End Slide Show SSS SAS AA

29 Problem #3 Given: G is the midpoint ofED H is the midpoint ofEF Prove:EGH~EDF

30 Step 1: Mark the Given … and what it implies Midpoint implies =/  segments Given: G is the midpoint ofED H is the midpoint ofEF Prove:EGH~EDF

31 Reflexive Angles Vertical Angles Step 2: Mark... Given: G is the midpoint ofED H is the midpoint ofEF Prove:EGH~EDF

32 Step 3: Choose a Method AA SSS SAS Given: G is the midpoint ofED H is the midpoint ofEF Prove:EGH~EDF

33 STATEMENTSREASONS 1. G is the midpoint ofED H is the midpoint ofEF 1. Given 2. Def. of Midpoint 3. Seg. Add. Post. 4. Substitution Given: G is the midpoint ofED H is the midpoint ofEF Prove:EGH~EDF

34 STATEMENTSREASONS 4. Substitution 5. Division Prop. = 6. Substitution 7. Reflexive Prop 8. SAS Similarity

35 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.


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