Math 3
Postulates, Theorems and Converses Pg 38 #7
Name________________
Definitions:
A Postulate is a statement that is accepted without further justification.
A Theorem is a conjecture that is proven.
A Conditional is another name for an if-then statement.
A Converse is the statement formed by ___________________ the hypothesis and the conclusion of
an if-then statement.
Is a conditional statement (if-then statement) always true?
If the conditional statement is true, does the converse have to be true?
Example: If you live in Rapid City, then you live in South Dakota? True or False?
Converse: If you live South Dakota, then you live in Rapid City. True or False?
We have proven the following conjectures, so now we can call them Theorems. For each of the
following theorems, write the converse and determine whether or not it is true.
Alternate Interior Angles Theorem
Converse:
Alternate Exterior Angles Theorem
Converse:
Same Sided Interior Angles Theorem
Converse:
Same Sided Exterior Angles Theorem
Converse:
Corresponding Angles Assumption: If two parallel lines are cut by a transversal, then
corresponding angles have equal measure. We will now refer to this as the Corresponding Angles
Postulate.
Write the converse of this postulate and make a decision on whether or not it is true.
For the remainder of this unit and in future units, we can assume that both the Corresponding
Angles Postulate and its Converse are true. As a result, we can combine these two statements into
a true biconditional statement known as the Parallel Lines Postulate.
This type of statement is true forward and backwards.
Parallel Lines Postulate: In a plane, two lines cut by a transversal are parallel if and only if
corresponding angles have equal measure.
We will use this postulate in our next proof to prove that 2 lines are parallel.
7. Write a proof for the Same Sided Interior Angles Converse Theorem.
If two lines are cut by a transversal so that same sided interior angles are supplementary, then the
lines are parallel.
t
Given: Transversal t cutting l and m; ∠4 𝑎𝑛𝑑 ∠5 𝑎𝑟𝑒 𝑠𝑢𝑝𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦
1
Prove: 𝑙 // 𝑚 (l is parallel to m)
Statements
4
Reasons
5
8
2
3
6
1. Transversal t cutting l and m;
∠4 𝑎𝑛𝑑 ∠5 𝑎𝑟𝑒 𝑠𝑢𝑝𝑝𝑙𝑒𝑚𝑒𝑛𝑡𝑎𝑟𝑦
1.
7
2. 𝑚∠5 + 𝑚∠4 = 180°
2. Definition of __________________
3.
3.
4.
4.
5. 𝑚∠1 = 𝑚∠5
5.
6.
6. Parallel Lines Postulate using corresponding
angles ∠1 and ∠5