Triangles 4 3 Congruent Triangles Warm Up Lesson

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Triangles 4 -3 Congruent Triangles Warm Up Lesson Presentation Lesson Quiz Holt Geometry

4 -3 Congruent Triangles 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. Holt Geometry

4 -3 Congruent Triangles Objectives Use properties of congruent triangles. Prove triangles congruent by using the definition of congruence. Holt Geometry

4 -3 Congruent Triangles Vocabulary corresponding angles corresponding sides congruent polygons Holt Geometry

4 -3 Congruent Triangles 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. Holt Geometry

4 -3 Congruent Triangles Holt Geometry

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

4 -3 Congruent Triangles 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. Holt Geometry

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

4 -3 Congruent Triangles 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 Holt Geometry

4 -3 Congruent Triangles 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 Holt Geometry

4 -3 Congruent Triangles 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 Holt Geometry Substitute values for m BCA and m BCD. Add 16 to both sides. Divide both sides by 2.

4 -3 Congruent Triangles 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° Holt Geometry Trans. Prop. of =

4 -3 Congruent Triangles 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 Holt Geometry Substitute values for AB and DE. Add 2 to both sides. Divide both sides by 2.

4 -3 Congruent Triangles Check It Out! Example 2 b Given: ∆ABC ∆DEF Find m F. m EFD + m DEF + m FDE = 180° ABC DEF 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° Holt Geometry ∆ Sum Thm. Substitute values for m DEF and m FDE. Simplify. Subtract 143 from both sides.

4 -3 Congruent Triangles 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 Holt Geometry

4 -3 Congruent Triangles 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 ∆ Holt Geometry

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

4 -3 Congruent Triangles 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 Holt Geometry

4 -3 Congruent Triangles 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 Holt Geometry

4 -3 Congruent Triangles 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. Holt Geometry Reasons Given Def. of bisector Vert. s Thm. Third s Thm. Def. of ∆s

4 -3 Congruent Triangles 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 Holt Geometry

4 -3 Congruent Triangles 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 Holt Geometry 5. 6. 7. 8. Def. of bisector Vert. s Thm. Third s Thm. Def. of ∆s

4 -3 Congruent Triangles 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 Holt Geometry

4 -3 Congruent Triangles 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 Holt Geometry