Use the SSS Congruence Postulate

1. Does the diagram give enough information to show that the triangles are congruent? Explain. Example 1 Use the SSS Congruence Postulate SOLUTION From the diagram you know that HJ LJ and HK  LK. By the Reflexive Property, you know that JK JK. Yes, enough information is given. Because corresponding sides are congruent, you can use the SSS Congruence Postulate to conclude that ∆HJK  ∆LJK. ANSWER

2. a. b. Example 2 Use the SAS Congruence Postulate Does the diagram give enough information to use the SAS Congruence Postulate? Explain your reasoning. SOLUTION From the diagram you know that AB CB and DB DB. a. The angle included between AB and DB is ABD. The angle included between CB and DB is CBD. Because the included angles are congruent, you can use the SAS Congruence Postulate to conclude that ∆ABD ∆CBD.

3. Example 2 Use the SAS Congruence Postulate b. You know that GF GH and GE GE. However, the congruent angles are not included between the congruent sides, so you cannot use the SAS Congruence Postulate.

4. SOLUTION The proof can be set up in two columns. The proof begins with the given information and ends with the statement you are trying to prove. Example 3 Write a Proof Write a two-column proof that shows ∆JKL  ∆NML. JL  NL Lis the midpoint of KM. ∆JKL  ∆NML

5. Example 3 Write a Proof Statements Reasons JL  NL 1. 1. Given These are the given statements. 2. 2. Lis the midpoint of KM. Given 3. JKL  NML 3. Vertical Angles Theorem This information is from the diagram.

6. Example 3 Write a Proof Statements Reasons 4. 4. Definition of midpoint KL  ML Statement 4 follows from Statement 2. 5. SAS Congruence Postulate Statement 5 follows from the congruences of Statements 1, 3, and 4. 5. ∆JKL  ∆NML

7. You are making a model of the window shown in the figure. You know that and  . Write a proof to show that ∆DRA ∆DRG. D A G R SOLUTION 1. Make a diagram and label it with the given information. Example 4 Prove Triangles are Congruent  DR AG RA RG

8. Example 4 Prove Triangles are Congruent 2. Write the given information and the statement you need to prove. DRAG,RA  RG  ∆DRA  ∆DRG 3. Write a two-column proof. List the given statements first.

9. Statements Reasons 4. Right angles are congruent. 5. Reflexive Property of Congruence 6. SAS Congruence Postulate Example 4 Prove Triangles are Congruent 4. DRA  DRG 5. DR  DR 6. ∆DRA  ∆DRG Given RA  RG 1. 1. DRAG 2. 2. Given  3. DRAand DRG are right angles. 3. lines form right angles. 

10. Checkpoint AC  AC  Statements Reasons _____ _____ _____ _____ ? ? ? ? DC CE CB  1. 1. Given ANSWER 2. 2. Given ANSWER DC Vertical Angles Theorem 3. BCA  ECD 3. ANSWER SAS Congruence Postulate 4. ∆BCA  ∆ECD 4. ANSWER Prove Triangles are Congruent 1. Fill in the missing statements and reasons. , ∆BCA  ∆ECD CB  CE