1) What are the converse, inverse, and contrapositive of the statement? Which statements are true? If the figure is a rectangle with sides 2 cm and 3 cm, then it has a perimeter of 10cm.
My A: the statement is not true.
2) What are the hypothesis and the conclusion of the following statement? Write it as a conditional. Residents of Key West live in Florida.
My A: That residents in Key West always live in Florida.
3) Test the statement to see if it is reversible. If so, write it as a true biconditional. If not, write not reversible. A perpendicular bisector of a segment is a line, segment, or ray that is perpendicular to a segment at its midpoint.
My A: A perpendicular bisector is a segment of a line and ray.
4) Why is the following statement a poor definition? Elephants are gigantic animals.
My A: Not all elephants are gigantic
5) Name the property of equality or congruence that justifies going from the first statement to the second statement. m A = m S and m S = m K, then m A = m K
My A: I'm very confused
6) ššµāāāāā bisects ā¦RPT so that mā¦RPB = x + 2 and mā¦TPB = 2x ā 6. What is mā¦RPT?
My A: -2
1) The converse of the statement is: If a figure has a perimeter of 10cm, then it is a rectangle with sides 2 cm and 3 cm. The inverse is: If the figure is not a rectangle with sides 2 cm and 3 cm, then it does not have a perimeter of 10cm. The contrapositive is: If the figure does not have a perimeter of 10cm, then it is not a rectangle with sides 2 cm and 3 cm. The true statement is the contrapositive, which means that if the figure does not have a perimeter of 10cm, then it is not a rectangle with sides 2 cm and 3 cm.
2) The hypothesis is: Residents of Key West and the conclusion is: live in Florida. The conditional statement would be: If residents of Key West, then live in Florida.
3) The statement is reversible, so the true biconditional would be: A line, segment, or ray that is perpendicular to a segment at its midpoint is a perpendicular bisector of a segment.
4) The statement is a poor definition because not all elephants are gigantic. There are different sizes and species of elephants, so the term "gigantic" is subjective and not specific enough to define elephants accurately.
5) The property of equality or congruence that justifies going from the first statement to the second statement is the Transitive Property of Equality or Congruence. This property states that if two things are equal to the same thing, then they are equal to each other.
6) Unfortunately, I cannot determine the value of mā¦RPT with the given information. More information is needed to solve for that angle.
1) The converse of the statement is: If the figure has a perimeter of 10cm, then it is a rectangle with sides 2 cm and 3 cm.
The inverse of the statement is: If the figure is not a rectangle with sides 2 cm and 3 cm, then it does not have a perimeter of 10cm.
The contrapositive of the statement is: If the figure does not have a perimeter of 10cm, then it is not a rectangle with sides 2 cm and 3 cm.
The statement is true because a rectangle with sides 2 cm and 3 cm does indeed have a perimeter of 10cm.
2) The hypothesis of the statement is: Residents of Key West.
The conclusion of the statement is: live in Florida.
The conditional statement would be: If residents are from Key West, then they live in Florida.
3) The statement is reversible because a perpendicular bisector of a segment is both a line and a ray. Therefore, it can be written as a true biconditional statement:
A perpendicular bisector of a segment is a line, segment, or ray that is perpendicular to a segment at its midpoint if and only if it is a perpendicular bisector.
4) The statement "Elephants are gigantic animals" is a poor definition because it is not specific enough. It does not provide a clear and distinct characteristic or property that defines what an elephant is. Elephants can vary in size, so not all elephants are gigantic.
5) The property of equality or congruence that justifies going from the first statement to the second statement is the Transitive Property of Equality. This property states that if two things are equal to the same thing, then they are equal to each other. In this case, m A is equal to m S and m S is equal to m K, so the Transitive Property allows us to conclude that m A is equal to m K.
6) From the given information, we have:
mā¦RPB = x + 2
mā¦TPB = 2x - 6
To find mā¦RPT, we know that the sum of the angles in ā¦RPT is 180 degrees because it is a straight angle.
Therefore, we can set up an equation:
mā¦RPB + mā¦TPB + mā¦RPT = 180
Substituting the given values:
(x + 2) + (2x - 6) + mā¦RPT = 180
Combining like terms:
3x - 4 + mā¦RPT = 180
Simplifying:
3x + mā¦RPT = 184
To find mā¦RPT, we need to isolate it by subtracting 3x from both sides:
mā¦RPT = 184 - 3x
Therefore, mā¦RPT is equal to 184 - 3x.
1) The converse of the statement is: If a figure has a perimeter of 10 cm, then it is a rectangle with sides 2 cm and 3 cm.
The inverse of the statement is: If a figure is not a rectangle with sides 2 cm and 3 cm, then it does not have a perimeter of 10 cm.
The contrapositive of the statement is: If a figure does not have a perimeter of 10 cm, then it is not a rectangle with sides 2 cm and 3 cm.
The original statement is false because a rectangle with sides 2 cm and 3 cm has a perimeter of 10 cm.
2) The hypothesis of the statement is "Residents of Key West" and the conclusion is "live in Florida." So, we can write it as a conditional statement: If someone is a resident of Key West, then they live in Florida.
3) The statement is reversible because the definition of a perpendicular bisector states that it can be a line, segment, or ray that is perpendicular to a segment at its midpoint. Therefore, we can write it as a true biconditional statement: A segment is a perpendicular bisector if and only if it is a line, segment, or ray that is perpendicular to a segment at its midpoint.
4) The statement "Elephants are gigantic animals" is a poor definition because it does not cover all possibilities. Not all elephants are gigantic, as there are different species and sizes of elephants. A better definition would be something like "Elephants are large mammals with distinctive long trunks and tusks."
5) The property of equality or congruence that justifies going from the first statement to the second statement is the Transitive Property. This property states that if two angles are congruent to a third angle, then they are congruent to each other. Therefore, if m A = m S and m S = m K, then by the Transitive Property we can conclude that m A = m K.
6) To find mā¦RPT, we need to use the Angle Bisector Theorem. According to the theorem, if a line divides an angle into two congruent angles, then it is an angle bisector.
So, from the given information:
mā¦RPB = x + 2
mā¦TPB = 2x - 6
Since PB is an angle bisector, we know that mā¦RPB = mā¦TPB. Therefore, we can set the two expressions equal to each other and solve for x:
x + 2 = 2x - 6
Solving this equation, we find:
x = 8
Now that we have the value of x, we can substitute it back into either of the given angle measures to find mā¦RPT. Let's use mā¦RPB:
mā¦RPT = mā¦RPB = x + 2 = 8 + 2 = 10
Therefore, mā¦RPT is 10 degrees.