1. 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

2. 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. .

3. 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

4. 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.

5. 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.

6. 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.

7. 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.

8. 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.

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

10. 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.

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

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

16. 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.

17. 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

18. 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.

19. 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.