Bell Ringer

Using Congruent Triangles

Example 1 Use Corresponding Parts In the diagram, AB and CD bisect each other at M. Prove that  A   B. Because  A and  B are corresponding angles in ∆ADM and ∆BCM, show that ∆ADM  ∆BCM to prove that  A   B. 2. SOLUTION First sketch the diagram and label any congruent segments and congruent angles. 1.

Example 1 Use Corresponding Parts Statements Reasons 1. AB and CD bisect each other at M. Given1. Definition of segment bisector2. MA  MB  AMD   BMC 3.Vertical Angles Theorem3. ∆ADM  ∆BCM 5.SAS Congruence Postulate5.  A   B 6.Corresponding parts of congruent triangles are congruent. 6. Definition of segment bisector4. MD  MC

Example 2 Visualize Overlapping Triangles SOLUTION Sketch the triangles separately and mark any given information. Think of ∆JGH moving to the left and ∆KHG moving to the right. 1. Sketch the overlapping triangles separately. Mark all congruent angles and sides. Then tell what theorem or postulate you can use to show ∆JGH  ∆KHG. Mark  GJH   HKG and  JHG   KGH.

Example 2 Visualize Overlapping Triangles Look at the original diagram for shared sides, shared angles, or any other information you can conclude. 2. Add congruence marks to GH in each triangle. In the original diagram, GH and HG are the same side, so GH  HG. You can use the AAS Congruence Theorem to show that ∆JGH  ∆KHG. 3.

Example 3 Use Overlapping Triangles In the original diagram,  C is the same in both triangles (  BCA   ECD). SOLUTION Sketch the triangles separately. Then label the given information and any other information you can conclude from the diagram. 1. Write a proof that shows AB  DE.  ABC   DEC AB  DE CB  CE

Show ∆ABC  ∆DEC to prove that AB  DE. Statements Reasons 1.  ABC   DEC Given1. Example 3 Use Overlapping Triangles Given2. CB  CE  C   C 3. Reflexive Prop. of Congruence 3. ASA Congruence Postulate4. ∆ABC  ∆DEC 4. Corresponding parts of congruent triangles are congruent. 5. AB  DE

Now You Try Use Overlapping Triangles ANSWER SAS. 1. Tell which triangle congruence theorem or postulate you would use to show that AB  CD.

Checkpoint Use Overlapping Triangles ANSWER Statements Reasons 1.Given1. Given2.  J   L ASA Congruence Postulate4. ∆KJN  ∆KLM 4. Corresponding parts of  triangles are . 5. NJ  ML KJ  KL Reflexive Prop. of Congruence3.  K   K 2. Given KJ  KL and  J   L, show NJ  ML. Redraw the triangles separately and label all congruences. Explain how to show that the triangles or corresponding parts are congruent. Now You Try

Checkpoint Use Overlapping Triangles Given  SPR   QRP and  Q   S, show ∆PQR  ∆RSP. 3. ANSWER Statements Reasons 1.Given1. Given2.  Q   S AAS Congruence Theorem4. ∆PQR  ∆RSP 4.  SPR   QRP Reflexive Prop. of Congruence3. PR  RP Now You Try

Checkpoint Now You Try

Checkpoint Now You Try

Checkpoint Now You Try

Checkpoint Now You Try

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Complete Pages #s 6-20 Even Only