Proving Triangles are Congruent 4 3 by SSS

Proving Triangles are Congruent 4 -3 by SSS and SAS Warm Up Lesson Presentation Lesson Quiz Holt Geometry

4. 3 Congruent Triangles by SSS - SAS Warm Up 1. Name all sides and angles of ∆FGH. FG, GH, FH, F, G, H 2. What is true about K and L? Why? ; Third s Thm. 3. What does it mean for two segments to be congruent? They have the same length.

4. 3 Congruent Triangles by SSS - SAS Objectives Use properties of congruent triangles. Prove triangles congruent by using SSS and SAS.

4. 3 Congruent Triangles by SSS - SAS Vocabulary corresponding angles corresponding sides congruent polygons

4. 3 Congruent Triangles by SSS - SAS Geometric figures are congruent if they are the same size and shape. Corresponding angles and corresponding sides are in the same position in polygons with an equal number of sides. Two polygons are congruent polygons if and only if their corresponding sides are congruent. Thus triangles that are the same size and shape are congruent.

4. 3 Congruent Triangles by SSS - SAS

4. 3 Congruent Triangles by SSS - SAS Helpful Hint Two vertices that are the endpoints of a side are called consecutive vertices. For example, P and Q are consecutive vertices.

4. 3 Congruent Triangles by SSS - SAS To name a polygon, write the vertices in consecutive order. For example, you can name polygon PQRS as QRSP or SRQP, but not as PRQS. In a congruence statement, the order of the vertices indicates the corresponding parts.

4. 3 Congruent Triangles by SSS - SAS Helpful Hint When you write a statement such as ABC DEF, you are also stating which parts are congruent.

4. 3 Congruent Triangles by SSS - SAS Example 1: Naming Congruent Corresponding Parts Given: ∆PQR ∆STW Identify all pairs of corresponding congruent parts. Angles: P S, Q T, R W Sides: PQ ST, QR TW, PR SW

4. 3 Congruent Triangles by SSS - SAS Check It Out! Example 1 If polygon LMNP polygon EFGH, identify all pairs of corresponding congruent parts. Angles: L E, M F, N G, P H Sides: LM EF, MN FG, NP GH, LP EH

4. 3 Congruent Triangles by SSS - SAS Example 2 A: Using Corresponding Parts of Congruent Triangles Given: ∆ABC ∆DBC. Find the value of x. BCA and BCD are rt. s. Def. of lines. BCA BCD Rt. Thm. m BCA = m BCD Def. of s (2 x – 16)° = 90° 2 x = 106 x = 53 Substitute values for m BCA and m BCD. Add 16 to both sides. Divide both sides by 2.

4. 3 Congruent Triangles by SSS - SAS Example 2 B: Using Corresponding Parts of Congruent Triangles Given: ∆ABC ∆DBC. Find m DBC. m ABC + m BCA + m A = 180° ∆ Sum Thm. Substitute values for m BCA and m ABC + 139. 3 = 180 Simplify. m ABC + 90 + 49. 3 = 180 m ABC = 40. 7 DBC ABC Subtract 139. 3 from both sides. Corr. s of ∆s are . m DBC = m ABC Def. of s. m DBC 40. 7° Trans. Prop. of =

4. 3 Congruent Triangles by SSS - SAS Check It Out! Example 2 a Given: ∆ABC ∆DEF Find the value of x. AB DE Corr. sides of ∆s are . AB = DE Def. of parts. 2 x – 2 = 6 2 x = 8 x=4 Substitute values for AB and DE. Add 2 to both sides. Divide both sides by 2.

4. 3 Congruent Triangles by SSS - SAS Check It Out! Example 2 b Given: ∆ABC ∆DEF Find m F. m EFD + m DEF + m FDE = 180° ABC DEF ∆ Sum Thm. Corr. s of ∆ are . m ABC = m DEF Def. of s. m DEF = 53° Transitive Prop. of =. m EFD + 53 + 90 = 180 m F + 143 = 180 m F = 37° Substitute values for m DEF and m FDE. Simplify. Subtract 143 from both sides.

4. 3 Congruent Triangles by SSS - SAS Example 3: Proving Triangles Congruent Given: YWX and YWZ are right angles. YW bisects XYZ. W is the midpoint of XZ. XY YZ. Prove: ∆XYW ∆ZYW

4. 3 Congruent Triangles by SSS - SAS Statements Reasons 1. YWX and YWZ are rt. s. 1. Given 2. YWX YWZ 2. Rt. Thm. 3. YW bisects XYZ 3. Given 4. XYW ZYW 4. Def. of bisector 5. W is mdpt. of XZ 5. Given 6. XW ZW 6. Def. of mdpt. 7. YW 7. Reflex. Prop. of 8. X Z 8. Third s Thm. 9. XY YZ 9. Given 10. ∆XYW ∆ZYW 10. Def. of ∆

4. 3 Congruent Triangles by SSS - SAS Check It Out! Example 3 Given: AD bisects BE. BE bisects AD. AB DE, A D Prove: ∆ABC ∆DEC

4. 3 Congruent Triangles by SSS - SAS Statements Reasons 1. A D 1. Given 2. BCA DCE 2. Vertical s are . 3. ABC DEC 3. Third s Thm. 4. AB DE 4. Given 5. AD bisects BE, 5. Given BE bisects AD 6. BC EC, AC DC 6. Def. of bisector 7. ∆ABC ∆DEC 7. Def. of ∆s

4. 3 Congruent Triangles by SSS - SAS Example 4: Engineering Application The diagonal bars across a gate give it support. Since the angle measures and the lengths of the corresponding sides are the same, the triangles are congruent. Given: PR and QT bisect each other. PQS RTS, QP RT Prove: ∆QPS ∆TRS

4. 3 Congruent Triangles by SSS - SAS Example 4 Continued Statements 1. QP RT 2. PQS RTS 3. 4. 5. 6. 7. 1. 2. PR and QT bisect each other. 3. QS TS, PS RS 4. QSP TSR 5. QSP TRS 6. ∆QPS ∆TRS 7. Reasons Given Def. of bisector Vert. s Thm. Third s Thm. Def. of ∆s

4. 3 Congruent Triangles by SSS - SAS Check It Out! Example 4 Use the diagram to prove the following. Given: MK bisects JL. JL bisects MK. JK ML. JK || ML. Prove: ∆JKN ∆LMN

4. 3 Congruent Triangles by SSS - SAS Check It Out! Example 4 Continued Statements 1. JK ML 2. JK || ML 3. JKN NML Reasons 1. Given 2. Given 3. Alt int. s are . 4. JL and MK bisect each other. 4. Given 5. 6. 7. 8. JN LN, MN KN KNJ MNL KJN MLN ∆JKN ∆LMN 5. 6. 7. 8. Def. of bisector Vert. s Thm. Third s Thm. Def. of ∆s

4. 3 Congruent Triangles by SSS - SAS Lesson Quiz 1. ∆ABC ∆JKL and AB = 2 x + 12. JK = 4 x – 50. Find x and AB. 31, 74 Given that polygon MNOP polygon QRST, identify the congruent corresponding part. RS P 2. NO ____ 3. T ____ 4. Given: C is the midpoint of BD and AE. A E, AB ED Prove: ∆ABC ∆EDC

4. 3 Congruent Triangles by SSS - SAS Lesson Quiz 4. Statements Reasons 1. A E 1. Given 2. C is mdpt. of BD and AE 2. Given 3. AC EC; BC DC 3. Def. of mdpt. 4. AB ED 4. Given 5. ACB ECD 5. Vert. s Thm. 6. B D 6. Third s Thm. 7. ABC EDC 7. Def. of ∆s