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An included angle is an angle formed by two adjacent sides of a polygon.

An included angle is an angle formed by two adjacent sides of a polygon.  B is the included angle between sides AB and BC. Reflexive Property. In Algebra: In Geometry: a = a b = b 2 + 3 = 2 + 3. or. Sides: PQ  ST , QR  TW , PR  SW.

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An included angle is an angle formed by two adjacent sides of a polygon.

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  1. An included angle is an angle formed by two adjacent sides of a polygon. B is the included angle between sides AB and BC.

  2. Reflexive Property • In Algebra: In Geometry: • a = a • b = b • 2 + 3 = 2 + 3 or

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

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

  5. Given:YWXandYWZ are right angles. YW bisects XYZ. W is the midpoint of XZ. XY  YZ. Prove: ∆XYW  ∆ZYW Example 3: Proving Triangles Congruent

  6. 5.W is mdpt. of XZ 6.XW ZW 7.YW YW 9.XY YZ 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. Given 6. Def. of mdpt. 7. Reflex. Prop. of  8.X  Z 8. Third s Thm. 9. Given 10.∆XYW  ∆ZYW 10. Def. of  ∆

  7. For example, you only need to know that two triangles have three pairs of congruent corresponding sides. This can be expressed as the following postulate.

  8. It is given that AC DC and that AB  DB. By the Reflexive Property of Congruence, BC  BC. Therefore ∆ABC  ∆DBC by SSS. Example 1: Using SSS to Prove Triangle Congruence Use SSS to explain why ∆ABC  ∆DBC.

  9. It is given that AB CD and BC  DA. By the Reflexive Property of Congruence, AC  CA. So ∆ABC  ∆CDA by SSS. Check It Out! Example 1 Use SSS to explain why ∆ABC  ∆CDA.

  10. Caution The letters SAS are written in that order because the congruent angles must be between pairs of congruent corresponding sides.

  11. It is given that XZ VZ and that YZ  WZ. By the Vertical s Theorem. XZY  VZW. Therefore ∆XYZ  ∆VWZ by SAS. Example 2: Engineering Application The diagram shows part of the support structure for a tower. Use SAS to explain why ∆XYZ  ∆VWZ.

  12. It is given that BA BD and ABC  DBC. By the Reflexive Property of , BC  BC. So ∆ABC  ∆DBC by SAS. Check It Out! Example 2 Use SAS to explain why ∆ABC  ∆DBC.

  13. 2. QP bisects RQS 1. QR  QS 4. QP  QP Check It Out! Example 4 Given: QP bisects RQS. QR QS Prove: ∆RQP  ∆SQP Statements Reasons 1. Given 2. Given 3. RQP  SQP 3. Def. of bisector 4. Reflex. Prop. of  5. ∆RQP  ∆SQP 5. SAS Steps 1, 3, 4

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