Rachel thinks that the price of the bench will be \($10\) off of listed price of \($89.99\). Then take \(20\%\) off of the new price of \($79.99\). Kathleen thinks that \(20\%\) will be taken off the list price of \($89.99\). Then take \($10\) off the new price of \($71.99\). Who do you agree with and why?
|
Many things in life have a cause and effect relationship. For example, think about most advertisements: “If you buy this item, you will get another one for \(50\%\) off!” The cause is buying an item and the effect is getting another one \(50\%\) off. Statements that have this cause-effect relationship are called conditional statements.
Conditional statements can be written in this form:
If ___________ , then _________.
(cause) (effect)
The cause in each conditional statement is called the hypothesis, where the effect is called the conclusion.
If ___________ , then _________.
(cause) (effect)
The cause in each conditional statement is called the hypothesis, where the effect is called the conclusion.
Definitions of new terms can often be written as conditional statements. Recall from Basics of Geometry Target D that complementary angles are angles whose measures add up to equal \(90\) degrees. To write that as a conditional statement, we will write:
Statement 1: If two angles are complementary, then their measures add up to \(90\) degrees.
In this example, “two angles are complementary” is the hypothesis whereas “their measures add up to \(90\) degrees” is the conclusion. Before we can conclude that the measures of two angles adds up to \(90^{\circ}\), we need to know that they are complementary. This cause-effect relationship actually allows us to discover and build onto more ideas in Geometry, which is extremely powerful!
Notice, if we switch the hypothesis and conclusion in Statement 1 from above, we create a new statement:
Statement 2: If the measures of two angles add up to \(90^{\circ}\), then they are complementary.
Statement 2 is called a converse statement.
Statement 1: If two angles are complementary, then their measures add up to \(90\) degrees.
In this example, “two angles are complementary” is the hypothesis whereas “their measures add up to \(90\) degrees” is the conclusion. Before we can conclude that the measures of two angles adds up to \(90^{\circ}\), we need to know that they are complementary. This cause-effect relationship actually allows us to discover and build onto more ideas in Geometry, which is extremely powerful!
Notice, if we switch the hypothesis and conclusion in Statement 1 from above, we create a new statement:
Statement 2: If the measures of two angles add up to \(90^{\circ}\), then they are complementary.
Statement 2 is called a converse statement.
Converse
Switching the hypothesis and conclusion of a conditional statement.
Switching the hypothesis and conclusion of a conditional statement.
Since the hypothesis and conclusion are switched around from its original conditional statement. This new statement still has a cause-effect relationship even though we changed the hypothesis and conclusion! Since our original statement was a definition, it lends itself to having a converse statement that is true.
Let's look at another example:
Statement 3: If an angle is acute angle, then its measure is between \(0^{\circ}\) and \(90^{\circ}\).
Converse of Statement 3:
If the measure of an angle is between \(0^{\circ}\) and \(90^{\circ}\) degrees, then it is an acute angle.
Conditional statements can either be true or false. A conditional statement that is true, must be true in every instance or example. A conditional statement that is false, only has to be false in at least one instance or example. An instance that is false is called a counterexample. A counterexample satisfies the hypothesis, but shows that the conclusion is false.
Example 1:
Consider the conditional statement: If I am a teenager, then I am \(13\). Is this statement true or false?
Solution:
If someone is teenager that is not \(13\), then that person is a counterexample. For example, a student that is \(15\) years old is a counterexample. So the statement is false. A person that is \(25\) would not be a counterexample. That person does not satisfy the hypothesis, "I am a teenager." The converse of the conditional statement is true.
Some statements, though not written in the traditional “if - then” format, can still be written that way if there is still a cause-effect relationship. Let’s take a look at the example below.
Example 2:
Rewrite the following statements as conditional statements.
a) Free delivery on all furniture purchases!
b) An apple a day keeps the doctor away.
Solution:
a) This statement talks about furniture purchases and free delivery. Though it isn’t written as a conditional statement, it still has a cause-effect relationship. The cause is buying furniture and the effect is that you get free delivery! So one way to write it as a conditional statement is: “If you purchase furniture, then you get free delivery.”
b) Again, there is a cause-effect relationship in this statement. You must first eat an apple, then the result is that you will keep the doctor away. To write it as a conditional statement, we can write it like: “If you eat an apple a day, then you will keep the doctor away.”
Example 3:
In each of the following statements, identify the hypothesis and the conclusion.
a) If I travel overseas, then I will need my passport.
b) If two angles are vertical, then they are congruent
c) I will want some milk if I eat a chocolate chip cookie.
d) If a point is a midpoint, then it splits the segment into two congruent segments.
Solution:
a) Usually, the hypothesis comes right after the “if” in a conditional statement. Or, you can think about which part is the cause in this statement. Because of this, the hypothesis in this conditional statement is: “I travel overseas”. The conclusion (or effect) is “I will need my passport.”
b) The hypothesis is “two angles are vertical” and the conclusion is “they are congruent.”
c) This statement is a bit tricky. If you reread it carefully again, you will notice that the cause is actually eating a chocolate chip cookie, whereas the effect is wanting some milk. So even though eating a chocolate chip cookie doesn’t come first in the statement, the hypothesis is “I eat a chocolate chip cookie” and the conclusion is “I will want some milk”.
d) The hypothesis is “a point is a midpoint” and the conclusion is “it splits the segment into two congruent segments.”
Example 4:
Write the converse of the following statement: If an angle is a right angle, then it measures \(90^{\circ}\).
Solution:
The hypothesis is “an angle is a right angle” and the conclusion is “it measures \(90^{\circ}\).” If we switch our hypothesis and conclusion, we will get our converse statement: If an angle measures \(90^{\circ}\), then it is a right angle.
Quick Check
1) Rewrite the statement as a conditional statement. Getting \(8\) hours of sleep at night makes me less cranky.
2) Identify the hypothesis and conclusion of the following statement: If two angles are congruent, then their measures are equal.
3) Write the converse of the following statement: If three points are collinear, then they lie on the same line.
4) Determine if the statement is true or false. If it is false, give a counterexample. If a triangle is acute, then one of the angles is \(60^{\circ}\).
Quick Check Solutions
1) Rewrite the statement as a conditional statement. Getting \(8\) hours of sleep at night makes me less cranky.
2) Identify the hypothesis and conclusion of the following statement: If two angles are congruent, then their measures are equal.
3) Write the converse of the following statement: If three points are collinear, then they lie on the same line.
4) Determine if the statement is true or false. If it is false, give a counterexample. If a triangle is acute, then one of the angles is \(60^{\circ}\).
Quick Check Solutions