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Objectives

4.1 Congruent Polygons. Objectives. Define congruent polygons . Solve problems by using congruent polygons. 4.1 Congruent Polygons. Theorems, Postulates, & Definitions. Polygon Congruence Postulate 4.1.1:

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Objectives

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  1. 4.1 Congruent Polygons Objectives • Define congruent polygons. • Solve problems by using congruent polygons.

  2. 4.1 Congruent Polygons Theorems, Postulates, & Definitions Polygon Congruence Postulate 4.1.1: Two polygons are congruent if and only if there is a correspondence between their sides and angles such that: 1. Each pair of corresponding angles is congruent. 2. Each pair of corresponding sides is congruent.

  3. Congruent sides: AB  WX, BC  XY, CD  YZ, DA  ZW TOC 4.1 Congruent Polygons Key Skills Identify corresponding parts of congruent polygons. Given: ABCD WXYZ. Identify all pairs of congruent sides and angles. Congruent angles: A W, B X, C Y, DZ

  4. 4.2 Triangle Congruence Objectives • Develop three congruence postulates for triangles—SSS, SAS, and ASA.

  5. 4.2 Triangle Congruence Theorems, Postulates, & Definitions SSS (Side-Side-Side) Postulate 4.2.1: If the sides of one triangle are congruent to the sides of another triangle, then the two triangles are congruent. SAS (Side-Angle-Side) Postulate 4.2.2: If two sides and their included angle in one triangle are congruent to two sides and their included angle in another triangle, then the two triangles are congruent.

  6. 4.2 Triangle Congruence Theorems, Postulates, & Definitions ASA (Angle-Side-Angle) Postulate 4.2.3: If two angles and their included side in one triangle are congruent to two angles and their included side in another triangle, then the two triangles are congruent.

  7. TOC 4.2 Triangle Congruence Key Skills Use SSS, SAS, and ASA postulates to determine whether triangles are congruent. Name the postulate that allows you to conclude the following: ? SSS Postulate a. ABC ADC ? SAS Postulate b. PRS XYZ

  8. 4.3 Analyzing Triangle Congruence Objectives • Identify and use the SSS, SAS, and ASA Congruence Postulates and the AAS and HL Congruence Theorems. • Use counterexamples to prove that other side and angle combinations cannot be used to prove triangle congruence.

  9. 4.3 Analyzing Triangle Congruence Theorems, Postulates, & Definitions AAS (Angle-Angle-Side) Congruence Theorem 4.3.1: If two angles and a nonincluded side of one triangle are congruent to the corresponding angles and nonincluded side of another triangle, then the triangles are congruent. HL (Hypotenuse-Leg) Congruence Theorem 4.3.2: If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of another right triangle, then the two triangles are congruent.

  10. TOC 4.3 Analyzing Triangle Congruence Key Skills Use AAS and HL Theorems to determine whether triangles are congruent. a. ABCDCB by HL Theorem. b. WXYWZY by AAS. c. PRS may not be congruent to ONM because triangles cannot be proven congruent by SSA.

  11. 4.4 Using Triangle Congruence Theorems, Postulates, & Definitions Isosceles Triangle Theorem 4.4.1: If two sides of a triangle are congruent, then the angles opposite those sides are congruent. Converse of the Isosceles Triangle Theorem 4.4.2: If two angles of a triangle are congruent, then the sides opposite those angles are congruent.

  12. 4.4 Using Triangle Congruence Theorems, Postulates, & Definitions Corollary 4.4.3: The measure of each angle of an equilateral triangle is 60°. Corollary 4.4.4: The bisector of the vertex angle of an isosceles triangle is the perpendicular bisector of the base.

  13. Given: AC  CD and BC  EC Prove: AB  DE AC  CD and BC  EC AB  DE 4.4 Using Triangle Congruence Key Skills Use triangle congruence in proofs. StatementsReasons Given 1 2 Vertical Angles Theorem ABC DEC SAS Theorem CPCTC

  14. Given: DC  EC and AB || DE Prove: AC  BC Given DC  EC and AB || DE AC  BC Isosceles Triangle Thm. TOC 4.4 Using Triangle Congruence Key Skills Use properties of isosceles triangles in proofs. StatementsReasons 1  2 Isosceles Triangle Thm. 1 3 and 2  4 Corresponding Angles Postulate 3 4 Substitution Property

  15. 4.5 Proving Quadrilateral Properties Theorems, Postulates, & Definitions Theorem 4.5.1: A diagonal of a parallelogram divides the parallelogram into two congruent triangles. Theorem 4.5.2: Opposite sides of a parallelogram are congruent. Theorem 4.5.3: Opposite angles of a parallelogram are congruent.

  16. 4.5 Proving Quadrilateral Properties Theorems, Postulates, & Definitions Theorem 4.5.4: Consecutive angles of a parallelogram are supplementary. Theorem 4.5.5: The diagonals of a parallelogram bisect each other. Theorem 4.5.6: A rhombus is a parallelogram.

  17. 4.5 Proving Quadrilateral Properties Theorems, Postulates, & Definitions Theorem 4.5.7: A rectangle is a parallelogram. Theorem 4.5.8: The diagonals of a rhombus are perpendicular. Theorem 4.5.9: The diagonals of a rectangle are congruent.

  18. 4.5 Proving Quadrilateral Properties Theorems, Postulates, & Definitions Theorem 4.5.10: The diagonals of a kite are perpendicular. Theorem 4.5.11: A square is a rectangle. Theorem 4.5.12: A square is a rhombus. Theorem 4.5.13: The diagonals of a square are congruent and are the perpendicular bisectors of each other.

  19. AB  BC and CD  DA. So ABC and CDA are isosceles triangles. TOC 4.5 Proving Quadrilateral Properties Key Skills Given: Rhombus ABCD Prove: 1 2 3 4 Prove properties of quadrilaterals. ABC CDA by SSS.  1  4 and  2  3 by CPCTC. By the Isosceles Triangle Theorem, 1 2 and  3 4. By the Substitution Principle,1 2 3 4.

  20. 4.6 Conditions for Special Quadrilaterals Objective • Develop conjectures for special quadrilaterals—parallelograms, rectangles, and rhombuses.

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