Unit 3 Lesson 2.2: Biconditionals Lesson Goals Rewrite a definition as a biconditional statement. ESLRs: Becoming Competent Learners, Complex Thinkers, and Effective Communicators ESLRs: Becoming Competent Learners, Complex Thinkers, and Effective Communicators

Definition Biconditional Statement The combination of a conditional statement and its converse. The phrase “if and only if” is used to indicate this combination. A biconditional is only true when both the conditional and converse are true : All geometric definitions are biconditional statements. .

Example (not in notes) Complementary Angles: Two angles with measures that have a sum of 90o. Conditional: If two angles are complementary, then their measures have a sum of 90o. Converse: If the measures of two angles have a sum of 90o, then the angles are complementary. Biconditional: Two angles are complementary if and only if their measures have a sum of 90o

Example Perpendicular lines: Two lines that intersect to form a right angle. Conditional: If two lines are perpendicular, then they intersect to form a right angle. Converse: If two lines intersect to form a right angle, then they are perpendicular. Biconditional: Two lines are perpendicular if and only if they intersect to form a right angle.

Example Rewrite the definition as a conditional and its converse A ray bisects an angle if and only if it divides the angle into two congruent angles. Conditional: If a ray bisects an angle, then it divides the Angle into two congruent angles. Converse: If ray divides the angle into two congruent angles, then the ray bisects the angle.

Example (not in your notes) Rewrite the postulate as a conditional and its converse. Two lines intersect if and only if their intersection is exactly one point. Conditional: If two lines intersect, then they have exactly one point in common. Converse: If two lines have exactly one point in common, then the lines intersect.

example Points R, S, and T are collinear. Decide whether each statement about the diagram is true and explain using definitions. R S T U Points R, S, and T are collinear. Definition collinear: points that lie on the same line R, S, and T are on the same line. Therefore, it is true that R, S, and T are collinear.

Example Decide whether each statement about the diagram is true and explain using definitions. R S T U Definition perpendicular: Two lines that intersect to form a right angle.

Example Give a counterexample that demonstrates that the converse of the statement is false. Converse: Counterexample:

Example Determine whether the statement can be combined with its converse to form a true biconditional. True Converse: False It cannot be a true biconditional since the converse is false.

Example Determine whether the statement can be combined with its converse to form a true biconditional. True Converse: True

Today’s Assignment p. 82 : 1, 7, 13 – 19, 22, 25, 41, 42

Example Rewrite the postulate as a biconditional statement if possible. If two planes intersect, then they contain the same line. Converse: If two planes contain the same line, then they intersect. True Biconditional: Two planes intersect if and only if they contain the same line.

Example Supplementary Angles: Two angles with measures that have a sum of 180o. Conditional: If two angles are supplementary, then their measures have a sum of 180o. Converse: If the measures of two angles have a sum of 180o, then the angles are supplementary. Biconditional: Two angles are complementary if and only if their measures have a sum of 180o

Example Rewrite the definition as a conditional and its converse. Two angles are vertical angles if and only if they are the two non-adjacent angles formed by two intersecting lines. Conditional: If two angles are vertical, then they are the non-adjacent angles formed by two intersecting lines Converse: If two lines intersect, then the non-adjacent angles formed are vertical angles.

Example Rewrite the definition as a conditional and its converse Two angles form a linear pair if and only if they are adjacent angles and their non-common sides are opposite rays. Conditional: If two angles from a linear pair, then they are adjacent angles and their non-common sides are opposite rays. Converse: If adjacent angles have non-common sides that are opposite rays, then the angles form a linear pair.