Question

state if each statement is true or false, write its converse, and state if its converse is true or false.

1. if the measures of two angles add up to 180, then they are supplementary.

a. converse:

1. if a triangle is equilateral, then it has no congruent sides.

a. converse:
for numbers 7 and 8, determine if each statement and its converse can be written as a biconditional or not. if yes, write the biconditional. if not, explain why they cannot.

1. statement: if a triangle is isosceles, then it has 3 congruent sides.

converse: if a triangle has 3 congruent sides, then it is isosceles.

1. statement: if a number is divisible by 4, then it is divisible by 2.

converse: if a number is divisible by 2, then it is divisible by 4.

1. find the next 2 numbers in the pattern and circle if this uses inductive or deductive reasoning

pattern: 20, 19, 17, 14, 10, 5, _, _
inductive
deductive

Explanation:

5.

Step1: Determine truth - value of original statement

The original statement "If the measures of two angles add up to 180, then they are supplementary" is true by the definition of supplementary angles.

Step2: Find the converse

The converse of "If p, then q" is "If q, then p". So the converse is "If two angles are supplementary, then the measures of the two angles add up to 180", which is also true.

6.

Step1: Determine truth - value of original statement

An equilateral triangle has 3 congruent sides. So the statement "If a triangle is equilateral, then it has no congruent sides" is false.

Step2: Find the converse

The converse is "If a triangle has no congruent sides, then it is equilateral". This is false because an equilateral triangle has congruent sides and a triangle with no congruent sides is a scalene triangle.

7.

Step1: Determine truth - value of original statement and converse

The original statement "If a triangle is isosceles, then it has 3 congruent sides" is false because an isosceles triangle has at least 2 congruent sides. The converse "If a triangle has 3 congruent sides, then it is isosceles" is true. Since the original statement and its converse are not both true, they cannot be written as a biconditional.

8.

Step1: Determine truth - value of original statement and converse

The original statement "If a number is divisible by 4, then it is divisible by 2" is true because 4 = 2×2. The converse "If a number is divisible by 2, then it is divisible by 4" is false (e.g., 2 is divisible by 2 but not by 4). So they cannot be written as a biconditional.

9.

Step1: Analyze the pattern

The differences between consecutive terms are 1, 2, 3, 4, 5. The next difference should be 6 and then 7.

Step2: Find the next terms

5 - 6=-1 and - 1-7=-8. This is inductive reasoning as we are making a generalization based on observed patterns.

Answer:

1. Converse: If two angles are supplementary, then the measures of the two angles add up to 180 (True)
2. Converse: If a triangle has no congruent sides, then it is equilateral (False)
3. Cannot be written as a biconditional because the original statement is false and the converse is true.
4. Cannot be written as a biconditional because the original statement is true and the converse is false.
5. Next two numbers: - 1, - 8; Inductive