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

From the diagram you know that AB  CB and DB  DB. Example 2 Use the SAS Congruence Postulate Does the diagram give enough information to use the SAS Congruence Postulate? Explain your reasoning. a. b. SOLUTION a. From the diagram you know that AB  CB and DB  DB. 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.

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.

Write a two-column proof that shows ∆JKL  ∆NML. Example 3 Write a Proof Write a two-column proof that shows ∆JKL  ∆NML. ∆JKL  ∆NML JL  NL L is the midpoint of KM. 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 Statements Reasons Given Given 3. Write a Proof These are the given statements. Statements Reasons Given 1. JL  NL Given 2. L is the midpoint of KM. This information is from the diagram. JKL  NML 3. Vertical Angles Theorem

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

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

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

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

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