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2.3 Biconditionals and Definitions

2.3 Biconditionals and Definitions. Geometry Chapter 2: Reasoning and Proof. Biconditional. Single true statement that combines a true conditional and its true converse Conditional : If the sum of the measures of two angles is 180, then the two angles are supplementary.

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2.3 Biconditionals and Definitions

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  1. 2.3 Biconditionals and Definitions Geometry Chapter 2: Reasoning and Proof

  2. Biconditional • Single true statement that combines a true conditional and its true converse • Conditional: If the sum of the measures of two angles is 180, then the two angles are supplementary. • Converse: If two angles are supplementary, then the sum of the measures of the two angles is 180. • Biconditional (Put it together): Two angles are supplementary ifandonlyif the sum of the measures of the two angles is 180.

  3. Represent! • Conditional (p → q): If the sum of the measures of two angles is 180, then the two angles are supplementary. • Converse (q→ p): If two angles are supplementary, then the sum of the measures of the two angles is 180. • Biconditional (p ↔ q): Two angles are supplementary ifandonlyif the sum of the measures of the two angles is 180.

  4. Let’s Work Together • Conditional: If two angles have equal measures, then the angles are congruent. • Converse: If two angles are congruent, then the angles have equal measures. • Biconditional: Two angles have equal measures if and only if the angles are congruent.

  5. Try It! • Conditional: If two numbers are reciprocals, then their product is 1. • Converse: If the product of two numbers is 1, then the numbers are reciprocals. • Biconditional: Two numbers are reciprocals if and only if their product is 1.

  6. Undo It A ray is an angle bisector ifandonlyif it divides an angle into two congruent angles. p: A ray is an angle bisector q: A ray divides an angle into two congruent angles. p → q: If a ray is an angle bisector, then it divides an angle into two congruent angles. q→ p: If a ray divides an angle into two congruent angles, then it is an angle bisector.

  7. Your Turn! An integer is divisible by 100 ifandonlyif its last two digits are zeros. p: An integer is divisible by 100 q: The last two digits of an integer are zeros. p → q: If an integer is divisible by 100, then its last two digits are zeros. q→ p: If the last two digits of an integer are zeros, then it is divisible by 100.

  8. DEFINITIONS • A statement that can help you identify or classify an object. • Good definition: • Uses clearly understood terms • Is precise (avoid words such as large, sort of, almost, etc.) • Is reversible (true biconditional)

  9. A straight angle is an angle that measures 180. • Is it reversible? • Write as a biconditional. • An angle is a straight angle if and only if its measure is 180.

  10. A quadrilateral is a polygon with four sides. • Reversible? • Biconditional: • A figure is a quadrilateral if and only if it is a polygon with four sides.

  11. Is it Good? • Which of the following is a good definition? • A fish is an animal that swims. • Rectangles have four corners. • Giraffes are animals with very long necks • A penny is a coin worth one cent.

  12. Practice Problems 22 problem Worksheet Conditionals Converse Inverse Contrapositive Biconditionals Definitions

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