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Learn how to use the Side-Side-Side (SSS) and Side-Angle-Side (SAS) congruence postulates to prove triangles congruent. Practice examples and proofs included.
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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 andH Corresponding angles ANSWER DB and HK 2. neither ANSWER
Complete the statement with the corresponding congruent part. D ANSWER ? J _____ 3. ? KH 4. ANSWER CB _____ 5. The triangles are congruent. Identify all pairs of corresponding congruent parts. Then write a congruence statement. B H, D J, C K,BD HJ, BC HK, CD KJ;∆BCD ∆HKJ ANSWER
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 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
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.
a. 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. angle ABD and angle CBD are (coz both are 90) Yes, we 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, it does not follow the SAS congruence postulate. So, No, we cannot use the SAS Congruence Postulate.
SOLUTION To set up the two column proof, start with the given: Example 3 Write a Proof Write a two-column proof that shows ∆JKL ∆NML. JL NL Lis the midpoint of KM. ∆JKL ∆NML
Example 3 Write a Proof Statements Reasons JL NL 1. 1. Given (Side) (Side) 2. 2. Lis the midpoint of KM. Given 3. Definition of midpoint 3. KL ML 4. JLK NLM 4. Vertical Angles Theorem (An included angle!)
Example 3 Write a Proof Statements Reasons JL NL 1. 1. Given Side side 2. 2. Lis the midpoint of KM. Given 5. SAS Congruence Postulate 5. ∆JKL ∆NML 3. Definition of midpoint 3. KL ML 4. JKL NML 4. Vertical Angles Theorem An included angle
From the figure, 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 DR AG RA RG
Checkpoint AC Statements Reasons _____ _____ _____ _____ ? ? ? ? CE CB 1. 1. 2. 2. Given DC 3. BCA ECD 3. 4. ∆BCA ∆ECD 4. Prove Triangles are Congruent Example. , ∆BCA ∆ECD CB CE ANSWER ANSWER ANSWER ANSWER
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 side 5. DR DR 6. ∆DRA ∆DRG angle RA RG Given 1. 1. side DRAG 2. 2. Given 3. DRAand DRG are right angles. 3. lines form right angles.
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 Fill in. , ∆BCA ∆ECD CB CE