Question

1. a true conditional statement is given below. write its contrapositive and state whether the contrapositive is true or false.

if an object is a cylinder, then its volume is the area of its base multiplied by its height.
a. if an object is not a cylinder, then its volume is not the area of its base multiplied by its height. true
b. if an object is not a cylinder, then its volume is not the area of its base multiplied by its height. false
c. if its volume is not the area of its base multiplied by its height, then the object is not a cylinder. true
d. if its volume is not the area of its base multiplied by its height, then the object is not a cylinder. false

1. a true conditional statement is given below. write its converse and state whether the converse is true or false.

given: if two angles form a linear pair, then they are supplementary.
a. if they are supplementary, then two angles form a linear pair. true
b. if they are supplementary, then two angles form a linear pair. false
c. if two angles do not form a linear pair, then they are not supplementary. true
d. if two angles do not form a linear pair, then they are not supplementary. false

1. where did the first error occur in the following algebraic proof?

statements	reasons

|2. 7 = 4t + 3|2. subtraction prop. of =
|3. 4 = 4t|3. subtraction prop. of =
|4. 1 = 1|4. division prop. of =
a. statement 1
b. statement 3
c. statement 2
d. statement 4

Explanation:

12.

Step1: Recall contra - positive rule

The contra - positive of a conditional statement "If \(p\), then \(q\)" is "If not \(q\), then not \(p\)". Here, \(p\) is "an object is a cylinder" and \(q\) is "its volume is the area of its base multiplied by its height".

Step2: Form the contra - positive

The contra - positive is "If its volume is not the area of its base multiplied by its height, then the object is not a cylinder". This is a true statement because the volume formula for a cylinder is well - defined and exclusive to cylinders in this context.

Step1: Recall converse rule

The converse of a conditional statement "If \(p\), then \(q\)" is "If \(q\), then \(p\)". Here, \(p\) is "two angles form a linear pair" and \(q\) is "they are supplementary".

Step2: Form the converse

The converse is "If they are supplementary, then two angles form a linear pair". Supplementary angles are two angles whose sum is \(180^{\circ}\), but they don't have to be adjacent (form a linear pair). So this is false.

Step1: Analyze step 1

Starting with \(4t−7 = 8t + 3\).

Step2: Analyze step 2

If we want to isolate the variable terms, from \(4t−7 = 8t + 3\), when we try to get rid of \(4t\) on the left - hand side, we should subtract \(4t\) from both sides. The correct step would be \(-7=4t + 3\), not \(7 = 4t+3\). The error occurred in Statement 2.

Answer:

C. If its volume is not the area of its base multiplied by its height, then the object is not a cylinder. True

13.