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) 𝑃𝐵⃗⃗⃗⃗⃗ bisects ⦟RPT so that m⦟RPB = x + 2 and m⦟TPB = 2x – 6. What is m⦟RPT?
My A: -2
1) To find the converse of a conditional statement, you switch the hypothesis and conclusion. So the converse of the statement "If the figure is a rectangle with sides 2 cm and 3 cm, then it has a perimeter of 10 cm" would be "If the 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 when you negate both the hypothesis and conclusion. So the inverse of the original statement would be "If the 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 when you switch the hypothesis and conclusion, and negate both. So the contrapositive of the original statement would be "If the figure does not have a perimeter of 10 cm, then it is not a rectangle with sides 2 cm and 3 cm."
In this case, the original statement is false because not all rectangles with sides 2 cm and 3 cm have a perimeter of 10 cm. Therefore, the converse, inverse, and contrapositive are all also false.
2) The hypothesis of the statement is "Residents of Key West" and the conclusion is "live in Florida." So the conditional statement would be "If residents of Key West, then they live in Florida."
3) To test if a statement is reversible, you need to check if the converse of the statement is true. In this case, the statement "A perpendicular bisector of a segment is a line, segment, or ray that is perpendicular to a segment at its midpoint" is true, and its converse "A line, segment, or ray that is perpendicular to a segment at its midpoint is a perpendicular bisector of the segment" is also true. Therefore, the statement is reversible. To write it as a true biconditional, you would say "A line, segment, or ray is a perpendicular bisector of a segment if and only if it is perpendicular to the segment at its midpoint."
4) The statement "Elephants are gigantic animals" is a poor definition because not all elephants are gigantic. There are different sizes and species of elephants, some of which are much smaller than others. A better definition would accurately describe what all elephants have in common or their distinguishing characteristics without making assumptions about their size.
5) Without more information about the properties of equality or congruence being referred to, it is difficult to determine which property justifies going from the first statement to the second statement. In geometry, properties of equality or congruence could include the reflexive property, symmetric property, transitive property, substitution property, or properties specific to angles or sides of shapes. It would depend on the context and specific properties being discussed.
6) To find the measure of angle RPT, you can use the fact that angle RPB and angle TPB are known. Since PB bisects angle RPT, you can set up an equation using the angle bisector theorem, which states that the measure of angle RPB divided by the measure of angle TPB is equal to PR divided by PT.
In this case, you have m angle RPB = x + 2 and m angle TPB = 2x - 6. Setting up the equation:
(x + 2) / (2x - 6) = PR / PT
Solving for PR / PT, you can cross-multiply:
(x + 2) * PT = (2x - 6) * PR
Now you can substitute the values of PT and PR into the equation. Given that PB bisects angle RPT, PT is equal to PB, and PR is equal to PB as well.
So the equation becomes:
(x + 2) * PB = (2x - 6) * PB
Since both sides of the equation are multiplied by PB, you can cancel PB from both sides:
x + 2 = 2x - 6
Solving for x:
x = 8
Now that you know the value of x, you can substitute it into one of the angle measures to find the measure of angle RPT. Let's use m angle RPB = x + 2:
m angle RPT = (8 + 2)
m angle RPT = 10
Therefore, m angle RPT is equal to 10.