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**G.7 Proving Triangles Similar**

(AA~, SSS~, SAS~)

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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.

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**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~

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**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~

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**Example: SSS Similarity (Side-Side-Side)**

5 11 22 8 16 10 Given: Conclusion: By SSS ~

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**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~

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**Example: SAS Similarity (Side-Angle-Side)**

5 11 22 10 Given: Conclusion: By SAS ~

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

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

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

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

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

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**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.

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**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

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**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

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**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?

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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

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**Similarity is reflexive, symmetric, and transitive.**

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**Choose a Problem. Problem #1 AA Problem #2 SSS Problem #3 SAS**

End Slide Show Problem #1 AA Problem #2 SSS Problem #3 SAS

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

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**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

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**Step 2: Mark . . . Reflexive Angles Vertical Angles Given: DE || FG**

Prove: DEC ~ FGC … if they exist.

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

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

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**Choose a Problem. Problem #1 AA Problem #2 SSS Problem #3 SAS**

End Slide Show Problem #1 AA Problem #2 SSS Problem #3 SAS

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

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**1. Given 2. Division Prop. 3. Substitution 4. SSS Similarity**

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

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**Choose a Problem. Problem #1 AA Problem #2 SSS Problem #3 SAS**

End Slide Show Problem #1 AA Problem #2 SSS Problem #3 SAS

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**Problem #3 Given: G is the midpoint of ED H is the midpoint of EF**

Prove: EGH~ EDF

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**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

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**Reflexive Angles Vertical Angles Step 2: Mark . . . Given:**

G is the midpoint of ED H is the midpoint of EF Prove: EGH~ EDF

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**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

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**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

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**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

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**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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There are 3 ways to show two triangles are similar to each other. Those 3 ways are: 1. Angle-Angle Similarity Postulate. (AA~) 2. Side-Angle-Side Similarity.

There are 3 ways to show two triangles are similar to each other. Those 3 ways are: 1. Angle-Angle Similarity Postulate. (AA~) 2. Side-Angle-Side Similarity.

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