Quadrilaterals Worksheet With Answers Page 20
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50Let’s consider the following properties of a parallelogram stated in the “if-then” format.
a) If a quadrilateral is a parallelogram, then the opposite sides are parallel.
b) If a quadrilateral is a parallelogram, then the opposite sides are congruent.
c) If a quadrilateral is a parallelogram, then the opposite angles are
congruent.
d) If a quadrilateral is a parallelogram, then consecutive angles are
supplementary.
e) If a quadrilateral is a parallelogram, then the diagonals bisect each other.
Each of these statements is called a conditional statement. The “if” part of the statement
is called the hypothesis and the “then” part is called the conclusion. When the hypothesis
and conclusion are reversed, the converse of the statement is formed as follows.
a) If the opposite sides of a quadrilateral are parallel, then the quadrilateral is
a parallelogram.
b) If the opposite sides of a quadrilateral are congruent, then the quadrilateral
is a parallelogram.
c) If the opposite angles of a quadrilateral are congruent, then the
quadrilateral is a parallelogram.
d) If consecutive angles of a quadrilateral are supplementary, then the
quadrilateral is a parallelogram.
e) If the diagonals of a quadrilateral bisect each other, then the quadrilateral
is a parallelogram.
Each of these converse statements is a theorem that will be proven in the following
examples or exercises. While each of these converse statements is true, the converse of a
statement may not always be true. Such is the case in the following statement and its
converse.
°
Statement: If it is 23
outside, then we have cold weather.
°
Converse: If we have cold weather, then it is 23
outside.
A statement that is true could have a converse that is false. If there is one
counterexample for a given statement, then it is not always true and cannot be considered
a true statement.
In the following example, both the statement and its converse are true.
Statement: If a quadrilateral is a parallelogram, then one pair of sides is both
parallel and congruent.
Converse: If a quadrilateral has one pair of sides both parallel and
congruent, then the quadrilateral is a parallelogram.
We will prove the converse statement and establish another way to show that a
quadrilateral is a parallelogram in example 1.
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