G.7 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 does not matter.

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 Given: Conclusion: By SAS ~

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

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

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

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

L 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 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 Step 5: Is there more? Statements Reasons 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 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? Statements Reasons 1. IJ = 3LN ; JK = 3NP ; IK = 3LP Given Division Property Substitution SSS Similarity

SAS Problem #3 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…………. Step 5: Is there more?

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

Similarity is reflexive, symmetric, and transitive.

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

Problem #1 Given: DE || FG Prove: DEC ~ FGC

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

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

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

Given: DE || FG Prove: DEC ~ FGC STATEMENTS REASONS 3. DEC ~ FGC

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

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

1. Given 2. Division Prop. 3. Substitution 4. SSS Similarity STATEMENTS REASONS 1. Given 2. Division Prop. 3. Substitution 4. SSS Similarity

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

Problem #3 Given: G is the midpoint of ED H is the midpoint of EF Prove: EGH~ EDF

Midpoint implies =/ @ segments Step 1: Mark the Given Given: … and what it implies Step 1: Mark the Given Given: G is the midpoint of ED H is the midpoint of EF Prove: EGH~ EDF Midpoint implies =/ @ segments

Reflexive Angles Vertical Angles Step 2: Mark . . . Given: G is the midpoint of ED H is the midpoint of EF Prove: EGH~ EDF

AA SSS SAS Step 3: Choose a Method Given: G is the midpoint of ED H is the midpoint of EF Prove: EGH~ EDF AA SSS SAS

1. Given 2. Def. of Midpoint 3. Seg. Add. Post. 4. Substitution Given: G is the midpoint of ED H is the midpoint of EF Prove: EGH~ EDF STATEMENTS REASONS 1. G is the midpoint of ED H is the midpoint of EF 1. Given 2. Def. of Midpoint 3. Seg. Add. Post. 4. Substitution

4. Substitution 5. Division Prop. = 6. Substitution 7. Reflexive Prop STATEMENTS REASONS 4. Substitution 5. Division Prop. = 6. Substitution 7. Reflexive Prop 8. SAS Similarity

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.