Warm Up On Desk (5 min) Do Daily Quiz 5.1 (10 min)

Review -go over the Daily Quiz items in 5.1

5.2 ESSENTIAL OBJECTIVE Show triangles are congruent using SSS and SAS.

In Exercises 1–5, use the triangles below. Determine whether the given angles or sides represent corresponding angles, corresponding sides, or neither. 1. B and H ANSWER Corresponding angles 2. DB and HK ANSWER neither

Complete the statement with the corresponding congruent part. 3. J  _____ ? ANSWER D 4. CB  _____ ? ANSWER KH The triangles are congruent. Identify all pairs of corresponding congruent parts. Then write a congruence statement. 5. ANSWER B  H, D  J, C  K, BD  HJ, BC  HK, CD  KJ; ∆BCD  ∆HKJ

5.2

VOCABULARY A proof is a convincing argument that shows why a statement is true.

Side-Side-Side Congruence Postulate (SSS) If three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent.

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. 9

Side-Angle-Side Congruence Postulate (SAS) If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent.

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. SOLUTION a. From the diagram you know that AB  CB and DB  DB. angle ABD and angle CBD are  (coz both are 90) Yes, we can use the SAS Congruence Postulate to conclude that ∆ABD  ∆CBD.

No, we cannot use the SAS Congruence Postulate. Example 2 Use the SAS Congruence Postulate b. You know that GF  GH and GE  GE. However, it does not follow the SAS congruence postulate. So, No, we cannot use the SAS Congruence Postulate.

Introducing: Two Column Proof

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 To set up the two column proof, start with the given:

Example 3 Statements Reasons Given Given 4. Write a Proof (Side) Statements Reasons Given 1. JL  NL Given 2. L is the midpoint of KM. 3. Definition of midpoint 3. KL  ML (An included angle!) JLK  NLM 4. Vertical Angles Theorem

Example 3 Statements Reasons Given Given 4. 5. Write a Proof Side side Statements Reasons Given 1. JL  NL Given 2. L is the midpoint of KM. 3. Definition of midpoint 3. KL  ML An included angle JKL  NML 4. Vertical Angles Theorem ∆JKL  ∆NML 5. SAS Congruence Postulate

proof to show that ∆DRA  ∆DRG. Example 4 A D G R  RG RA AG DR From the figure, and  proof to show that ∆DRA  ∆DRG. . Write a SOLUTION 1. Make a diagram and label it with the given information.

Checkpoint Example. ∆BCA  ∆ECD DC AC  CB  CE , CB  CE Statements Prove Triangles are Congruent Example. ∆BCA  ∆ECD DC AC  CB  CE , CB  CE Statements Reasons 1. ? _____ ANSWER Given 2. ? _____ ANSWER BCA  ECD 3. ? _____ ANSWER ∆BCA  ∆ECD 4. ? _____ ANSWER

DRA and DRG are right angles. lines form right angles.  Example 4 Prove Triangles are Congruent Statements Reasons 1. RA  RG Given side 2. DR AG  Given 3. DRA and DRG are right angles. lines form right angles.  4. DRA  DRG Right angles are congruent. angle 5. DR  DR Reflexive Property of Congruence side 6. ∆DRA  ∆DRG SAS Congruence Postulate 22

Checkpoint Fill in. ∆BCA  ∆ECD DC AC  CB  CE , CB  CE Statements Prove Triangles are Congruent Fill in. ∆BCA  ∆ECD DC AC  CB  CE , CB  CE Statements Reasons 1. ? _____ Given ANSWER Given 2. ? _____ DC AC  ANSWER Vertical Angles Theorem ANSWER BCA  ECD 3. ? _____ SAS Congruence Postulate ANSWER ∆BCA  ∆ECD 4. ? _____ 23

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